Research ArticleHamburger and Stieltjes Moment Problems for Operators

L Lemnete-Ninulescu

Department of Mathematics ldquoPolitehnicardquo University of Bucharest Splaiul Independentei No 313 060042 Bucharest Romania

Correspondence should be addressed to L Lemnete-Ninulescu luminita lemneteyahoocom

Received 2 December 2013 Accepted 8 January 2014 Published 30 April 2014

Academic Editors D D Hai and S Pilipovic

Copyright copy 2014 L Lemnete-Ninulescu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Solutions to some operator-valued unidimensional Hamburger and Stieltjes moment problems in this paper are given Necessaryand sufficient conditions on some sequences of bounded operators beingHamburger respectively Stieltjes operator-valuedmomentsequences are obtained The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied

1 Introduction

A function 119864(120582) 119886 le 120582 le 119887 is called a spectral function if(a) 119864(120582) is a bounded positive operator(b) 119864(120582) le 119864(120583) for any 120582 le 120583(c) 119864(120582 + 0) = 119864(120582)(c1015840) 119864(119886 + 0) = 0 andor 119864(119887 minus 0) = 119864(119887) in case 119886 = minusinfin

andor 119887 = +infin

The spectral function 119864(120582) is called an orthogonal spectralfunction if every 119864(120582) is an orthogonal projection [1 page322]

(1) A sequence 119860119899+infin

119899=0of bounded self-adjoint opera-

tors acting on an arbitrary complex Hilbert spaceH subject on the condition 119860

0= IdH is

called a Hamburger unidimensional operator-valuedmoment sequence if there exists an orthogonal spec-tral function 119864(120582) minusinfin le 120582 le +infin such that 119860

119899=

int+infin

minusinfin120582119899d119864(119905) 119899 = 0 1 2 or

(2) a sequence 119860119899+infin

119899=0 1198600

= IdH of bounded self-adjoint operators is called a unidimensional operator-valued Hamburger moment sequence if there exists apositive operator-valued measure 119865(120582) minusinfin le 120582 le

+infin measure generated by a spectral function suchthat 119860

119899= int+infin

minusinfin120582119899d119865(119905) 119899 = 0 1 2

A sequence 119860119899+infin

119899=0of bounded positive operators is

called a Stieltjes unidimensional operator-valued moment

sequence if there exists a positive operator-valued measure119865(120582) 0 le 120582 le +infin (generated by a spectral function) suchthat 119860

119899= int+infin

0120582119899d119865(120582) 119899 = 0 1 2 The passage from the

integral representation (1) to an integral representation (2)

is done usually by applying Naimarkrsquos dilation theorem ormodified forms of it as in [1]

In both cases (1) and (2) the operator-valued measures119864(120582) or 119865(120582) are called the representing measures for thesequence 119860

119899+infin

119899=0 Necessary and sufficient conditions for

representing scalar sequences or operator-valued sequencesin one or several variables as Hamburger or Stieltjes momentsequences with respect to scalar respectively operator-valued positive measures represent the subject of manyoutstanding papers such as [1ndash4] to quote only few ofthem

In the present paper in Section 3 we give a necessaryand sufficient condition on a sequence of bounded self-adjoint operators to be a Hamburger operator-valued uni-dimensional moment sequence In Section 4 we discuss theuniqueness of the representing measures of the operator-valued Hamburger moment sequence both in (1) and (2)

forms In Section 5 we give some necessary and sufficientconditions on a sequence of positive operators to be a Stielt-jes operator-valued unidimensional moment sequence withrespect to a positive operator-valued measure The positiverepresenting measures in Sections 3 and 5 are obtained byapplying Kolmogorovrsquos theorem of decomposition of thepositive definite kernels

Hindawi Publishing CorporationISRN Mathematical AnalysisVolume 2014 Article ID 836839 7 pageshttpdxdoiorg1011552014836839

2 ISRNMathematical Analysis

2 Preliminaries

Let 119905 isin R denote the real variable in the real Euclidean spacefor H an arbitrary complex Hilbert space 119871(H) represents thealgebra of bounded operators anH we denote with 120575

119894sdot N rarr

0 1 the function

120575119894119895=

1 119894 = 119895

0 119894 = 119895

(1)

for 119870 a Hilbert space 119861(H 119870) represents the set of boundedoperators from H in 119870 We consider the C-vector space ofvectorial functions 119865 = 119891 0 1 119899 rarr H 119891(sdot) =sum119899isinN 120575119899sdot119891(119899)119891with finite support119891(119899) isin HWe define also

the convolution 119891 lowast 1205751sdotisin 119865 as

[119891 lowast 1205751sdot] (119899) = sum

119896isinZ(119891 (119899 minus 119896) sdot 120575

1119896) 119899 isin Nlowast (2)

and make the convention119891 lowast 1205751sdot(0) = 0H We have 119891 lowast 120575

1sdot=

sum119899isinN 120575(119899+1)sdot

119891(119899) 119891 with finite supportIn Section 3 a necessary and sufficient condition on a

sequence of self-adjoint operators to be a Hamburger oper-ator-valued moment sequence is given In Section 5 we givenecessary and sufficient conditions on a sequence of positiveoperators to be a Stieltjes operator-valued moment sequenceIn Section 4 the problemof the uniqueness of the representedmeasures in Sections 3 and 5 is studied The representingmeasures in Sections 3 and 5 are obtained by applying Kol-mogorovrsquos theorem on decomposition of the positive kernelsClassical Kolmogorovrsquos theorem for the decomposition ofpositive kernels is as follows

ldquoLet Γ 119878 times 119878 rarr 119871(119867) be a nonnegative-definite function where 119878 is an arbitrary set and119867a Hilbert space namelysum119899

119894119895=1⟨Γ(119904119894 119904119895)119909119895 119909119894⟩119867ge

0 for any finite number of points 1199041 119904

119899isin 119878

and any vectors 1199091 119909

119899isin 119867 In this case there

exists a Hilbert space 119870 (essentially unique) anda function ℎ 119878 rarr 119861(119867119870) such that Γ(119904 119905) =ℎ(119905)lowastℎ(119904) for any 119904 119905 isin 119878rdquo

We apply this theorem for a particular set 119878 and a particu-lar positive-definite operator-valued function to give an inte-gral representation as Hamburger operator-valued momentsequence and Stieltjes operator-valued moment sequencerespectively to some sequences of self-adjoint and positiveoperators respectively

3 An Operator-Valued HamburgerMoment Sequence Main Result

Let Γ = Γ119899119899isinN be a sequence of bounded self-adjoint opera-

tors acting on an arbitrary complex separable Hilbert spacethat is Γ

119899isin 119871(H) Γ

119899= Γlowast

119899 for all 119899 isin N Γ

0= IdH

subject on the following conditions for any finite vectorsrsquosequence 119909

119899119899isin119868subN sub H there exists another vector sequence

1199101015840

119899119899isin119868subN sub H such that the following two equations are

satisfied(A)

sum

119901119902isin119868

⟨Γ119901+119902+2

(119909119902minus 1199101015840

119902) (119909119901minus 1199101015840

119901)⟩

H

minus 2 Im sum

119901119902isin119868

⟨Γ119901+119902+1

(119909119902+ 1199101015840

119902) (119909119901minus 1199101015840

119901)⟩

H

+ sum

119901119902isin119868

⟨Γ119901+119902

(119909119902+ 1199101015840

119902) (119909119901+ 1199101015840

119901)⟩

H= 0

(3)

and for any finite vectorsrsquo sequence 119909119899119899isin119868subN sub H there

exists another vectorsrsquo sequence 11991010158401015840119899119899isin119868subN sub H such that

(B)

sum

119901119902isin119868

⟨Γ119901+119902+2

(119909119902minus 11991010158401015840

119902) (119909119901minus 11991010158401015840

119901)⟩

H

+ 2 Im sum

119901119902isin119868

⟨Γ119901+119902+1

(119909119902+ 11991010158401015840

119902) (119909119901minus 11991010158401015840

119901)⟩

H

+ sum

119901119902isin119868

⟨Γ119901+119902

(119909119902+ 11991010158401015840

119902) (119909119901+ 11991010158401015840

119901)⟩

H= 0

(4)

Proposition 1 Let Γ = Γ119899119899isinN be a sequence of bounded self-

adjoint operators acting on an arbitrary complex separableHilbert space H subject on the conditions Γ

0= IdH (A) and

(B) satisfied The following statements are equivalent

(i) We have

sum

119899119898isinN⟨Γ119899+119898

119909119898 119909119899⟩H ge 0 (5)

for all sequences 119909119899119899isin H with finite support

(ii) There exists a positive operator-valued measure 119864119860

(spectral function) defined on Bor(R) such that

Γ119899= int

+infin

minusinfin

119905119899d119864119860(119905) 119899 = 0 1 2 119896 (6)

Proof When 119865 = 119891 0 1 119899 rarr H 119891(sdot) =

sum119899isin01119899

120575119899sdot119891(119899) 119891 with finite support is the C-vector

space of functions defined on N with vectorial values weconsider the kernel Γ as a double indexed symmetric one

Γ 0 1 119899 times 0 1 119899 997888rarr 119861 (H)

Γ119899+119898

= Γ (119899119898)

(7)

With the aid of Γ we introduce the Hermitian squarepositive functional Λ

Γ 119865 times 119865 rarr C Λ

Γ(119891 119892) =

sum119898119899isin01119899

⟨Γ119899+119898

119891(119898) 119892(119899)⟩H From property (i) of thekernel Γ as well as from the properties of the scalar productin H ΛΓ satisfies the following conditions

(10) ΛΓ is C-linear in the first argument(20) ΛΓ(119891 119892) = Λ

Γ(119892 119891) for all 119891 119892 isin 119865

ISRNMathematical Analysis 3

(30) ΛΓ(119891 119891) ge 0 for all 119891 isin 119865 and moreover being aHermitian square positive functional on 119865 times 119865 ΛΓ satisfiesthe Cauchy-Buniakovski-Schwarz inequality respectively

(40)10038161003816100381610038161003816ΛΓ(119891 119892)

10038161003816100381610038161003816le ΛΓ(119891 119891)

12

ΛΓ(119892 119892)

12

forall119891 119892 isin 119865 (8)

Also from the construction of the Hermitian functional ΛΓand the symmetry of the kernel Γ (Γ

119899+1+119898= Γ(119899 + 1119898) =

Γ(119899119898 + 1) = Γ119899+119898+1

) the functional ΛΓN satisfies theequalities

(50)

ΛΓ(119891 lowast 120575

1sdot 119892) = Λ

Γ(119891 119892 lowast 120575

1sdot) forall119891 119892 isin 119865 (9)

With these assumptions 119865 ni 119891 rarr ⟨119891 119891⟩12

119865= ΛΓ(119891 119891)

12

is a seminorm on 119865 Let 119878 be the subset in 119865 defined as119878 = 119891 isin 119865 with Λ

Γ(119891 119891) = 0 If follows using the Cauchy-

Buniakovski-Schwarz inequality that if 1198911 1198912isin 119878 we have

also ΛΓ(12057211198911+ 12057221198912 12057211198911+ 12057221198912) = 0 that is 119878 sub 119865 is a

vector subspace in 119865 We consider the separated completionspace of 119865 with respect to 119878 that is in this case the quotientcompletion space119870 = 119865119878

sdotΓ Obviously119870 is a Hilbert spacewith the usual norm in the completion Hilbert space and119863 = 119865119878 is a dense subspace of it (ie ℎ

119870= lim

119899ℎ119899119909 =

lim119899⟨120575119899sdot119909 + 119878 120575

119899sdot119909 + 119878 ⟩

12

119865 where ℎ119909 = ℎ

119899119909 is a Cauchy

sequence of elements in 119865119878 and ℎ119899119909 = 120575119899sdot119909 = 120575119899sdot119909 + 119878) The

Hilbert space 119870 is uniquely defined and it is also describedas 119870 = 119881Ranℎ

119899119909 (the closed linear span of the ranges of

the operators ℎ119899 H rarr 119870 ℎ

119899119909 = 120575

119899sdot119909 = 120575

119899sdot119909 + 119878) From

Kolmogorovrsquos decomposition theorem of positively definedkernels with the above construction the decompositionsΓ119899+119898

= Γ(119899119898) = ℎlowast

119899ℎ119898hold for any 119899119898 isin N Let us consider

the densely defined subspace of 119870 119863 = 119865119878 = sum119899isin119868subN ℎ

119899119909119899

119868 finite 119909119899isin H 119863 sub 119870 and the operator 119860 119863 rarr 119863

defined by119860(sum119899isin119868subN ℎ

119899119909119899) = sum119899isin119868subN ℎ

119899+1119909119899 We prove that119860

is correctly defined Consequently we consider the elementssum119898isin119869subN ℎ

119898119909119898

= sum119898isin119869subN(120575119898sdot119909119898 + 119878) and sum

119899isin119868subN ℎ119899119909119899

=

sum119899isin119868subN(120575119899sdot119909119899 + 119878) such that sum

119899isin119868subN 120575119899sdot119909119899minus sum119898isin119869subN 120575

119898sdot119909119898

isin

119878 and show that 119860(sum119899isin119868subN ℎ

119899119909119899) = 119860(sum

119898isin119869subN ℎ119898119909119898) The

above equality is the same as the equality sum119899isin119868subN ℎ

119899+1119909119899=

sum119898isin119869subN ℎ

119898+1119909119898(modulo 119878) Indeed from (50) we have

100381610038161003816100381610038161003816100381610038161003816

ΛΓ(sum

119899

120575119899+1sdot

119909119899minussum

119898

120575119898+1sdot

119909119898sum

119899

120575119899+1sdot

119909119899minussum

119898

120575119898+1sdot

119909119898)

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

ΛΓ(sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898sum

119899

120575119899+2sdot

119909119899minussum

119898

120575119898+2sdot

119909119898)

100381610038161003816100381610038161003816100381610038161003816

le ΛΓ(sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898)

12

sdot ΛΓ

(119897 119897)12

= 0

(10)

where 119897 = sum119899120575119899+2sdot

119909119899minus sum119898120575119898+2sdot

119909119898 From the above defini-

tion we have

(60)

⟨119860119909 119910⟩119870= ⟨119860( sum

119899isin119868subNℎ119899119909119899) sum

119898isin119869subNℎ119898119910119898⟩

119870

= ⟨ sum

119899isin119868subNℎ119899+1

119909119899 sum

119898isin119869subNℎ119898119910119898⟩

119870

= ⟨ sum

119898119899isin119868cup119869subNℎlowast

119898ℎ119899+1

119909119899 119910119898⟩

119870

= sum

119898119899

⟨Γ119898+119899+1

119909119899 119910119898⟩H

= sum

119898119899

⟨Γ119899+119898+1

119909119899 119910119898⟩H

(11)

and also(70)

⟨119909 119860119910⟩119870= ⟨ sum

119899isin119868subNℎ119899119909119899 119860( sum

119898isin119869subNℎ119898119910119898)⟩

119870

= ⟨ sum

119899isin119868subNℎ119899119909119899 sum

119898isin119869subNℎ119898+1

119910119898⟩

119870

= ⟨ sum

119898119899isin119868cup119869subNℎlowast

119898+1ℎ119899119909119899 119910119898⟩

119870

= sum

119898119899

⟨Γ119898+1+119899

119909119899 119910119898⟩H

= sum

119898119899

⟨Γ119899+119898+1

119909119899 119910119898⟩H

(12)

From (60) and (7

0) ⟨119860119909 119910⟩

119870= ⟨119909 119860119910⟩

119870for 119909 119910 isin 119863

arbitrary we infer that 119860 is a densely defined symmetricoperator We prove that 119860 has equal deficiency indices in 119870consequently 119860rsquos Cayley transform is a partial isometry on119870+= 119877(119860 + i119868

119870) with values in 119870

minus= 119877(119860 minus i119868

119870) Indeed

let 119870plusmn= 119877(119860 plusmn i119868

119870) = 119881

119899isin119868finitesubN(ℎ119899+1

119909119899plusmn iℎ119899119909119899) 119909119899isin H

arbitrary be the ranges in 119870 of the operators (119860 plusmn i119868119870) We

prove that 119870plusmnare vector subspaces in 119870 For this request

we consider the elements 119891plusmn= sum119899isin1finitesubN(ℎ119899+1119909119899 plusmn iℎ

119899119909119899)

119892 = sum119899isin1198682finitesubN(ℎ119899+1119910119899 plusmn iℎ119899119910119899) in119870plusmn and 120572 120573 isin C arbitrary

Let us define the elements119909119899119899isin1198681cup1198682=119868

119909119899=

119909119899 119899 isin 119868

1

0H 119899 isin 1198682minus (1198681cap 1198682)

(13)

119910119899119899isin1198681cup1198682

119910119899=

119910119899 119899 isin 119868

2

0H 119899 isin 1198681minus (1198681cap 1198682)

(14)

also the elements 119891plusmn

= sum119899isin1finitesubN(ℎ119899+1119909119899 plusmn iℎ

119899119909119899)

119892 = sum119899isin1198682finitesubN(ℎ119899+1119910119899 plusmn iℎ

119899119910119899) in119870

plusmnand 120572 120573 isin C It results

4 ISRNMathematical Analysis

that 119891minus119891 = 0119870 119892minus119892 = 0

119870and [120572119891

plusmn+120573119892plusmn] = [120572119891

plusmn+120573119892plusmn] =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899) + iℎ

119899(120572119909119899) + ℎ

119899+1(120573119910119899) plusmn iℎ

119899(120573119910119899)] +

sum119899isin1198681minus(1198681cap1198682)

[ℎ119899+1

(120572119909119899) + iℎ

119899(120573119910119899) + ℎ119899+1

(120573119910119899) plusmn iℎ

119899(120572119909119899+

120573119910119899] + sum

119899isin1198682minus(1198681cap1198682)[ℎ119899+1

(120572119909119899) + 120573119910

119899plusmn iℎ119899(120572119909119899) + 120573119910

119899) =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899

+ 120573119910119899) plusmn iℎ

119899(120572119909119899) + 120573119910

119899] isin 119870

plusmn

Because 119860 is a symmetric operator it results also that119877(119860 plusmn i119868

119870) = 119870

plusmnare closed subspaces in 119870 We prove

that in conditions (A) and (B) for the kernel Γ we have119870+= 119877(119860+i119868

119870) = 119877(119860minusi119868

119870) = 119870minus Indeed given an arbitrary

element119891 = sum119899isin119868finitesubN(ℎ119899+1119909119899+iℎ119899119909119899)we look for an element

119892 isin 119870minus

= 119877(119860 minus i119868119870) 119892 = sum

119899isin119868finitesubN(ℎ119898+11199101015840

119898minus iℎ1198981199101015840

119898)

such that (119891 minus 119892) = 0119870 For a construction of the elements

119909119899119899isin1198681cup1198682=119868

119910119899119899isin1198681cup1198682=119868

sub H like the previous one wehave (119891 minus 119892) = sum[ℎ

119899+1(119909119899minus 1199101015840

119899) + iℎ

119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119902)

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119901)gtH = 0 hArr

sum119901119902isin119868119901119902ge1

Γ119901+119902

[(119909119902minus1

minus 1199101015840

119902minus1) + i(119909

119902+ 1199101015840

119902)] (119909119901minus1

minus 1199101015840

119901minus1+

i(119909119901+ 1199101015840

119901)gtH = 0

119870hArr sum119901119902ge0

⟨Γ119901+119902+2

(119909119902minus 1199101015840

119902) (119909119901minus 1199101015840

119901)⟩

Hminus

2 Imsum119901119902isin119868

⟨Γ119901+119902+1

(119909119902+ 1199101015840

119902) (119909119901

minus 1199101015840

119901)⟩H + sum

119901119902isin119868(119909119902+

1199101015840

119902) (119909119901+ 1199101015840

119901)gtH = 0

119870 According to condition (A) on the

kernel Γ119899119899 such an element exists We have 119870

+sub 119870minus

Conversely let 119891 isin 119870minus 119891 = sum

119899isin119868(ℎ119899+1

119909119899minus iℎ119899119909119899) we search

for an element 119892 isin 119870+ 119892 = sum

119899isin119868(ℎ119899+1

11991010158401015840

119899+ iℎ11989911991010158401015840

119899) with the

property that (119891 minus 119892) = 0119870 Consequently we have to find an

element 119892 isin 119870+such thatsum[120575

(119899+1)(119909119899minus1199101015840

119899) + iℎ119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119902)

sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119901)gtH = 0

119870hArr

sum119901119902isin119868

⟨Γ119901+119902+2

(119909119902minus 11991010158401015840

119902) (119909119901minus 11991010158401015840

119901)⟩

H+ 2 Imsum

119901119902isin119868⟨Γ119901+119902+1

(119909119902+11991010158401015840

119902) (119909119901minus11991010158401015840

119901)⟩H+sum119901119902isin119868 ⟨Γ119901+119902(119909119902 + 119910

10158401015840

119902) (119909119901+ 11991010158401015840

119901)⟩

H=

0 We prove with these computations that119870+= 119877(119860 + i119868

119870) =

119870minus= 119877(119860 minus i119868

119870) sub 119870 That is dim 119870

perp

+= dim119870

perp

minusrArr 119860rsquos

Cayley transform has equal deficiency indices andrArr 119860 admits a self-adjoint extension 119860 Let 119864

119860be the

spectral measure of the self-adjoint operator 119860 Becauseℎ119898119909 = 119860

119898(ℎ0119909) for all 119909 isin H and Γ

119898119909 = ℎ

lowast

0ℎ119898119909 it results

that Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 for all 119909 isin H and the integral

representations Γ119898119909 = ℎ

lowast

0intR 119905119898d119864119860(119905)ℎ0119909 for all 119909 isin H for

all119898 isin NWe consider the positive operator-valued measure119865119860(119905) = ℎ

lowast

0119864119860ℎ0 With respect to this positive operator-

valued measure we have Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 = intR 119905

119898d119865119860(119905)119909

for all 119909 isin H and all 119898 isin N That is Γ119898

= intR 119905119898d119865119860(119905) for

all 119898 isin N the required Hamburger moment integralrepresentations

Conversely If the terms Γ119898119898admit the integral representa-

tions Γ119898= int+infin

minusinfin119905119898d119865119860(119905) for all119898 = 0 1 2 for a positive

operator-valued measure on R we have

sum

119899119898

⟨Γ119899+119898

119909119898 119909119899⟩H

= sum

119899119898

⟨int

+infin

minusinfin

119905119899+119898d119865

119860(119905)119909119898 119909119899⟩

H

= int

+infin

minusinfin

d(⟨sum

119898

11990511989811986512

119860(119905) 119909119898sum

119899

11990511989911986512

119860(119905) 119909119899⟩

H)

= intRd(

100381710038171003817100381710038171003817100381710038171003817

11986512

119860sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

2

) ge 0

(15)

as it is required by (i)

4 About the Uniqueness ofthe Hamburger Operator-Valued MomentSequencesrsquo Representations

Let us consider a sequence of bounded operators 119860119899119899isinN

119860119899isin 119871(H) subject on the condition 119860

119899= 119860lowast

119899 1198600= IdH

119899 = 0 1 2 H an arbitrary complex Hilbert space Forthe sequence 119860

119899119899isinN we get two operator-valued integral

representing measures (or spectral functions) 119864119860 119864119861

Bor(H) rarr 119860(H) that is

119860119899= int

+infin

minusinfin

119905119899d119864119860(119905) = int

+infin

minusinfin

119905119899d119864119861(119905) (16)

for all 119899 = 0 1 2 The operator-valued measures allow usto define the scalar measures 120583119909 ]119909 Bor(R) rarr [0 +infin]120583119909(119863) = ⟨119864

119860(119863)119909 119909⟩H respectively ]

119909(119863) = ⟨119864

119861(119863)119909 119909⟩H

when119909 isin H is arbitraryWith respect to these scalarmeasureswe obtain

119886119909

119899= ⟨119860119899119909 119909⟩H = int

+infin

minusinfin

119905119899d120583119909 (119905)

= int

+infin

minusinfin

119905119899d]119909 (119905) forall119899 isin N

(17)

From [5 page 283] the Hamburger scalar moment prob-lem is indeterminate (the sequence 119886119909

119899119899does not uniquely

determine the scalar representing measure) It followsthat the operator-valued representing measure does notuniquely determine theHamburger operator-valuedmomentsequence

However under some additional conditions about theoperator-valued representing measure the Stieltjes (Ham-burger) operator-valued moment sequence is determined [3pages 509 510 511]

Moreover if the representing measure is that associatedwith a self-adjoint extension of a symmetric operator withdeficiency indices (00) the self-adjoint extension is thecanonical closure of the given operator and is defined on thewhole space Indeed if 119878 119863(119878) rarr H is symmetric with119877(119878 plusmn i) = H and 119860 sup 119878 the canonical closure of 119878 it followsthat H supe 119877(119860 plusmn i) sup 119877(119878 plusmn i) are closed subspaces in H thatis 119877(119860 plusmn i) = H In this case the canonical closure of 119878 isthe smallest self-adjoint extension of 119878 and is defined on thewhole space H (as in Section 3 of this paper Proposition 1)The same arguments are in [4 page 1267 Lemma 21]

ISRNMathematical Analysis 5

Proposition 2 (1) Let 119860119899+infin

119899=0 119860119899isin 119871(H) for all 119899 isin N H

an arbitrary complex Hilbert space subject on the conditions119860119899= 119860lowast

119899 1198600= IdH and 1198641 1198642 Bor(R) rarr 119860(H) two ortho-

gonal spectral functions on R such that

119860119899= int

+infin

minusinfin

1199051198991198641(119905) = int

+infin

minusinfin

1199051198991198642(119905) 119899 = 0 1 2 (18)

Then 1198641= 1198642on Bor(R)

Proof Because1198601isin 119871(H) and119860

1= 119860lowast

1 the existence of the

representation 1198601

= int+infin

minusinfin119905d119864119860(119905) with 119864

119860 Bor(R) rarr

119860(R) 119864119860(R) = 119860

0= IdH is the usual one and is unique The

spectral orthogonal measures coincide that is119864119860= 1198641= 1198642

The representing measure is the spectral orthogonal measureassociated with the self-adjoint operator 119860

1 From 119864

119860(120582)rsquos

multiplicative property it follows that 119860119899= int+infin

minusinfin119905119899119864119860(119905) =

(int+infin

minusinfin119905119864119860(119905))119899 for all 119899 isin N The uniqueness of the integral

representations with respect to spectral functions is assuredtrivially only in case 119860

119899= 119860119899 for all 119899 isin N when the

representation is possible

5 Stieltjes Operator-ValuedMoment Sequences

A sequence of bounded operators Γ = Γ119899119899 acting on an

arbitrary Hilbert space H is called a Stieltjes operator-valuedmoment sequence if there exists a positive operator-valuedmeasure 119864

Γon [0 +infin) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (19)

Proposition 3 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH with conditions (A) and

(B) in Proposition 1 satisfied The following assertions areequivalent

(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(20)

for all sequences 119909119899+infin

119899=0subH with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (21)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same as (i) in Proposition 1 Con-

sequently there exists a positive operator-valuedmeasure119864Γ

Bor(R) rarr 119860(H) such that Γ119899= int+infin

minusinfin119905119899d119864Γ(119905) 119899 = 0 1 2

In the statement (119895) (2) if we consider the sequence with

finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H arbitrary for

all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905)119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905) 119909 119909⟩H ge 0

(22)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same as(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1d⟨119864

Γ(119905) 119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(23)

that is (119895) (2)We give another second characterization on an operator

sequence Γ119899+infin

119899=0to be an operator-valued Stieltjes moment

sequence

Remark 4 In the sequel we argue like in [1 page 329]Between the Hamburger operator-valued moment sequencesand Stieltjes moment sequences we can establish the follow-ing bijection

(A) If Γ = Γ119899+infin

119899=0is a Stieltjes moment sequence with

respect to the spectral measure 119864Γ(119905) on [0 +infin) for the

homeomorphism 1205871

[0 +infin) rarr (minusinfin 0] 1205871(119905) =

minusradic119905 there corresponds a spectral measure 1198651

Γon (minusinfin 0]

defined by 1198651

Γ(119905) = (12)[119864

Γ(infin) minus 119864

Γ∘ 120587minus1

1(119905)] such that

int0

minusinfin1205822119896d1198651Γ(120582) = (12) int

+infin

0120582119896d119864Γ

For the homeomorphism 1205872

[0 +infin) rarr [0 +infin)1205872(119905) = radic119905 there corresponds a spectral measure 119865

2

Γon

[0 +infin) defined by 1198652

Γ(119905) = (12)119864

Γ(infin) + (12)119864

Γ∘ 120587minus1

2(119905)

such that intinfin0

1205822119896d1198652Γ(120582) = (12) int

+infin

0120582119896d119864Γ

6 ISRNMathematical Analysis

We define

119865Γ=

1198651

Γ(119905) 119905 lt 0

1198652

Γ(119905) 119905 ge 0

(24)

For 119865Γ(119905) we have the representations int+infin

minusinfin1205822119896d119865Γ(120582) =

int+infin

0120582119896d119864Γ(120582) 119896 = 0 1 119899 and int

+infin

minusinfin1205822119896+1d119865

Γ(120582) = 0

119896 = 0 1 119899

(B) Conversely If we have the operator-valued Hamburgermoment sequence Γ

0 0 Γ1 0 Γ2 0 with respect to the

spectral representing measure 119865120582 respectively the sequence

119861119899+infin

119899=0 defined by 119861

119899= (Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 for all119899 isin N admits the integral representation 119861

2119899= Γ119899

=

int+infin

minusinfin1199052119899d119865(120582) and 119861

2119899+1= 0 = int

+infin

minusinfin1199052119899+1d119865(119905) We can con-

struct a spectral measure 119864(119905) on [0 +infin) that is for 120587

(minusinfin +infin) rarr [0 +infin) 120587(119905) = 1199052 we have 119864(119905) = 119865 ∘ 120587

minus1(119905)

withint+infinminusinfin

119905119899d119865(119905) = int

+infin

0(1199052)119899d119864(119905) = 119861

2119899= Γ119899 119899 = 0 1 2

For obtaining positive operator-valued measures fromorthogonal spectral functions in both cases we composethe orthogonal spectral measure associated with a self-adjoinoperator with the same projections respectively ℎ

0 ℎlowast0in

our case Summing the above conditions (A) and (B) inRemark 4 for an operator-valued sequence Γ

119899infin

119899=0isin 119871(H)

we construct the operator-valued sequence 119861119899

= (Γ[1198992]

+

(minus1)119899Γ[1198992]

)2 for all 119899 isin 119873 and conversely for a sequence119861119899+infin

119899=0isin 119871(H) we construct the operator-valued sequence

Γ119899+infin

119899=0 with Γ

119899= 1198612119899 119899 isin N

With the above construction we have the following

Proposition 5 The sequence Γ119899+infin

119899=0that satisfies conditions

(A) and (B) in Proposition 1 is a Stieltjes operator-valuedmoment sequence if and only if

119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

Hge 0

(25)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

Proposition 51015840 (reformulated) The sequence Γ119899+infin

119899=0is a

Stieltjes operator-valued moment sequence if and only if

119902

sum

119899119898=0

⟨119861119899+119898

119909119899 119909119898⟩H ge 0 (26)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

119861119899 for all 119899 isin N defined above

Proof Let Γ119899+infin

119899=0be an operatorsrsquo sequence Γ

119899isin 119871(H)

H an arbitrary complex Hilbert space we define 119861119899

=

(Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 119899 isin N that is 1198612119899

= Γ119899and

1198612119899+1

= 0 for all 119899 isin N In the condition of the hypothesis(Propositions 5 and 51015840) we have sum

119899119898⟨119861119899+119898

119909119899 119909119898⟩H ge 0

for all sequences 119909119899119899

sub H with finite support From

Proposition 1 there exists a positive operator-valuedmeasureon R such that

119861119899= int

+infin

minusinfin

119905119899d119865 (119905) 119899 = 0 1 2 (27)

From Remark 4 there exists a positive operator-valued mea-sure 119864(119905) such that

1198612119899

= int

+infin

minusinfin

1199052119899d119865 (119905) = Γ

[21198992]= Γ119899= int

+infin

0

119905119899d119864 (119905)

119899 = 0 1 2

(28)

That is Γ119899119899isinN is a Stieltjes operator-valued moment

sequence

Conversely If Γ119899

= int+infin

0119905119899d119864(119905) 119899 = 0 1 2 we con-

struct a measure 119865(119905) positively defined on (minusinfin +infin) asin Remark 4 with the property that Γ

119899= int+infin

minusinfin1199052119899d119865(119905) =

int+infin

0119905119899d119864(119905) = 119861

2119899 and 119861

2119899+1= int+infin

minusinfin1199052119899+1d119865(119905) = 0 In this

case119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

H

=

119902

sum

119899119898=0

⟨int

+infin

minusinfin

119905119899+119898d119865(119905)119909

119899 119909119898⟩

H

= int

+infin

minusinfin

d100381710038171003817100381710038171003817100381710038171003817

sum

119899

11990511989911986512

(119905) 119909119899

100381710038171003817100381710038171003817100381710038171003817

2

ge 0

(29)

Proposition 6 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH The following assertions are

equivalent(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(30)

for all sequences 119909119899+infin

119899=0sub H with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (31)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same with (i) in Proposition 1

Consequently there exists a positive operator-valued mea-sure 119864

Γ Bor(R) rarr 119860(H) such that Γ

119899= int+infin

minusinfin119905119899d119864Γ(119905)

119899 = 0 1 2 In the statement (119895) (2) if we consider the

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

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ISRNMathematical Analysis 3

(30) ΛΓ(119891 119891) ge 0 for all 119891 isin 119865 and moreover being aHermitian square positive functional on 119865 times 119865 ΛΓ satisfiesthe Cauchy-Buniakovski-Schwarz inequality respectively

(40)10038161003816100381610038161003816ΛΓ(119891 119892)

10038161003816100381610038161003816le ΛΓ(119891 119891)

12

ΛΓ(119892 119892)

12

forall119891 119892 isin 119865 (8)

Also from the construction of the Hermitian functional ΛΓand the symmetry of the kernel Γ (Γ

119899+1+119898= Γ(119899 + 1119898) =

Γ(119899119898 + 1) = Γ119899+119898+1

) the functional ΛΓN satisfies theequalities

(50)

ΛΓ(119891 lowast 120575

1sdot 119892) = Λ

Γ(119891 119892 lowast 120575

1sdot) forall119891 119892 isin 119865 (9)

With these assumptions 119865 ni 119891 rarr ⟨119891 119891⟩12

119865= ΛΓ(119891 119891)

12

is a seminorm on 119865 Let 119878 be the subset in 119865 defined as119878 = 119891 isin 119865 with Λ

Γ(119891 119891) = 0 If follows using the Cauchy-

Buniakovski-Schwarz inequality that if 1198911 1198912isin 119878 we have

also ΛΓ(12057211198911+ 12057221198912 12057211198911+ 12057221198912) = 0 that is 119878 sub 119865 is a

vector subspace in 119865 We consider the separated completionspace of 119865 with respect to 119878 that is in this case the quotientcompletion space119870 = 119865119878

sdotΓ Obviously119870 is a Hilbert spacewith the usual norm in the completion Hilbert space and119863 = 119865119878 is a dense subspace of it (ie ℎ

119870= lim

119899ℎ119899119909 =

lim119899⟨120575119899sdot119909 + 119878 120575

119899sdot119909 + 119878 ⟩

12

119865 where ℎ119909 = ℎ

119899119909 is a Cauchy

sequence of elements in 119865119878 and ℎ119899119909 = 120575119899sdot119909 = 120575119899sdot119909 + 119878) The

Hilbert space 119870 is uniquely defined and it is also describedas 119870 = 119881Ranℎ

119899119909 (the closed linear span of the ranges of

the operators ℎ119899 H rarr 119870 ℎ

119899119909 = 120575

119899sdot119909 = 120575

119899sdot119909 + 119878) From

Kolmogorovrsquos decomposition theorem of positively definedkernels with the above construction the decompositionsΓ119899+119898

= Γ(119899119898) = ℎlowast

119899ℎ119898hold for any 119899119898 isin N Let us consider

the densely defined subspace of 119870 119863 = 119865119878 = sum119899isin119868subN ℎ

119899119909119899

119868 finite 119909119899isin H 119863 sub 119870 and the operator 119860 119863 rarr 119863

defined by119860(sum119899isin119868subN ℎ

119899119909119899) = sum119899isin119868subN ℎ

119899+1119909119899 We prove that119860

is correctly defined Consequently we consider the elementssum119898isin119869subN ℎ

119898119909119898

= sum119898isin119869subN(120575119898sdot119909119898 + 119878) and sum

119899isin119868subN ℎ119899119909119899

=

sum119899isin119868subN(120575119899sdot119909119899 + 119878) such that sum

119899isin119868subN 120575119899sdot119909119899minus sum119898isin119869subN 120575

119898sdot119909119898

isin

119878 and show that 119860(sum119899isin119868subN ℎ

119899119909119899) = 119860(sum

119898isin119869subN ℎ119898119909119898) The

above equality is the same as the equality sum119899isin119868subN ℎ

119899+1119909119899=

sum119898isin119869subN ℎ

119898+1119909119898(modulo 119878) Indeed from (50) we have

100381610038161003816100381610038161003816100381610038161003816

ΛΓ(sum

119899

120575119899+1sdot

119909119899minussum

119898

120575119898+1sdot

119909119898sum

119899

120575119899+1sdot

119909119899minussum

119898

120575119898+1sdot

119909119898)

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

ΛΓ(sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898sum

119899

120575119899+2sdot

119909119899minussum

119898

120575119898+2sdot

119909119898)

100381610038161003816100381610038161003816100381610038161003816

le ΛΓ(sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898sum

119899

120575119899sdot119909119899minussum

119898

120575119898sdot119909119898)

12

sdot ΛΓ

(119897 119897)12

= 0

(10)

where 119897 = sum119899120575119899+2sdot

119909119899minus sum119898120575119898+2sdot

119909119898 From the above defini-

tion we have

(60)

⟨119860119909 119910⟩119870= ⟨119860( sum

119899isin119868subNℎ119899119909119899) sum

119898isin119869subNℎ119898119910119898⟩

119870

= ⟨ sum

119899isin119868subNℎ119899+1

119909119899 sum

119898isin119869subNℎ119898119910119898⟩

119870

= ⟨ sum

119898119899isin119868cup119869subNℎlowast

119898ℎ119899+1

119909119899 119910119898⟩

119870

= sum

119898119899

⟨Γ119898+119899+1

119909119899 119910119898⟩H

= sum

119898119899

⟨Γ119899+119898+1

119909119899 119910119898⟩H

(11)

and also(70)

⟨119909 119860119910⟩119870= ⟨ sum

119899isin119868subNℎ119899119909119899 119860( sum

119898isin119869subNℎ119898119910119898)⟩

119870

= ⟨ sum

119899isin119868subNℎ119899119909119899 sum

119898isin119869subNℎ119898+1

119910119898⟩

119870

= ⟨ sum

119898119899isin119868cup119869subNℎlowast

119898+1ℎ119899119909119899 119910119898⟩

119870

= sum

119898119899

⟨Γ119898+1+119899

119909119899 119910119898⟩H

= sum

119898119899

⟨Γ119899+119898+1

119909119899 119910119898⟩H

(12)

From (60) and (7

0) ⟨119860119909 119910⟩

119870= ⟨119909 119860119910⟩

119870for 119909 119910 isin 119863

arbitrary we infer that 119860 is a densely defined symmetricoperator We prove that 119860 has equal deficiency indices in 119870consequently 119860rsquos Cayley transform is a partial isometry on119870+= 119877(119860 + i119868

119870) with values in 119870

minus= 119877(119860 minus i119868

119870) Indeed

let 119870plusmn= 119877(119860 plusmn i119868

119870) = 119881

119899isin119868finitesubN(ℎ119899+1

119909119899plusmn iℎ119899119909119899) 119909119899isin H

arbitrary be the ranges in 119870 of the operators (119860 plusmn i119868119870) We

prove that 119870plusmnare vector subspaces in 119870 For this request

we consider the elements 119891plusmn= sum119899isin1finitesubN(ℎ119899+1119909119899 plusmn iℎ

119899119909119899)

119892 = sum119899isin1198682finitesubN(ℎ119899+1119910119899 plusmn iℎ119899119910119899) in119870plusmn and 120572 120573 isin C arbitrary

Let us define the elements119909119899119899isin1198681cup1198682=119868

119909119899=

119909119899 119899 isin 119868

1

0H 119899 isin 1198682minus (1198681cap 1198682)

(13)

119910119899119899isin1198681cup1198682

119910119899=

119910119899 119899 isin 119868

2

0H 119899 isin 1198681minus (1198681cap 1198682)

(14)

also the elements 119891plusmn

= sum119899isin1finitesubN(ℎ119899+1119909119899 plusmn iℎ

119899119909119899)

119892 = sum119899isin1198682finitesubN(ℎ119899+1119910119899 plusmn iℎ

119899119910119899) in119870

plusmnand 120572 120573 isin C It results

4 ISRNMathematical Analysis

that 119891minus119891 = 0119870 119892minus119892 = 0

119870and [120572119891

plusmn+120573119892plusmn] = [120572119891

plusmn+120573119892plusmn] =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899) + iℎ

119899(120572119909119899) + ℎ

119899+1(120573119910119899) plusmn iℎ

119899(120573119910119899)] +

sum119899isin1198681minus(1198681cap1198682)

[ℎ119899+1

(120572119909119899) + iℎ

119899(120573119910119899) + ℎ119899+1

(120573119910119899) plusmn iℎ

119899(120572119909119899+

120573119910119899] + sum

119899isin1198682minus(1198681cap1198682)[ℎ119899+1

(120572119909119899) + 120573119910

119899plusmn iℎ119899(120572119909119899) + 120573119910

119899) =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899

+ 120573119910119899) plusmn iℎ

119899(120572119909119899) + 120573119910

119899] isin 119870

plusmn

Because 119860 is a symmetric operator it results also that119877(119860 plusmn i119868

119870) = 119870

plusmnare closed subspaces in 119870 We prove

that in conditions (A) and (B) for the kernel Γ we have119870+= 119877(119860+i119868

119870) = 119877(119860minusi119868

119870) = 119870minus Indeed given an arbitrary

element119891 = sum119899isin119868finitesubN(ℎ119899+1119909119899+iℎ119899119909119899)we look for an element

119892 isin 119870minus

= 119877(119860 minus i119868119870) 119892 = sum

119899isin119868finitesubN(ℎ119898+11199101015840

119898minus iℎ1198981199101015840

119898)

such that (119891 minus 119892) = 0119870 For a construction of the elements

119909119899119899isin1198681cup1198682=119868

119910119899119899isin1198681cup1198682=119868

sub H like the previous one wehave (119891 minus 119892) = sum[ℎ

119899+1(119909119899minus 1199101015840

119899) + iℎ

119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119902)

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119901)gtH = 0 hArr

sum119901119902isin119868119901119902ge1

Γ119901+119902

[(119909119902minus1

minus 1199101015840

119902minus1) + i(119909

119902+ 1199101015840

119902)] (119909119901minus1

minus 1199101015840

119901minus1+

i(119909119901+ 1199101015840

119901)gtH = 0

119870hArr sum119901119902ge0

⟨Γ119901+119902+2

(119909119902minus 1199101015840

119902) (119909119901minus 1199101015840

119901)⟩

Hminus

2 Imsum119901119902isin119868

⟨Γ119901+119902+1

(119909119902+ 1199101015840

119902) (119909119901

minus 1199101015840

119901)⟩H + sum

119901119902isin119868(119909119902+

1199101015840

119902) (119909119901+ 1199101015840

119901)gtH = 0

119870 According to condition (A) on the

kernel Γ119899119899 such an element exists We have 119870

+sub 119870minus

Conversely let 119891 isin 119870minus 119891 = sum

119899isin119868(ℎ119899+1

119909119899minus iℎ119899119909119899) we search

for an element 119892 isin 119870+ 119892 = sum

119899isin119868(ℎ119899+1

11991010158401015840

119899+ iℎ11989911991010158401015840

119899) with the

property that (119891 minus 119892) = 0119870 Consequently we have to find an

element 119892 isin 119870+such thatsum[120575

(119899+1)(119909119899minus1199101015840

119899) + iℎ119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119902)

sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119901)gtH = 0

119870hArr

sum119901119902isin119868

⟨Γ119901+119902+2

(119909119902minus 11991010158401015840

119902) (119909119901minus 11991010158401015840

119901)⟩

H+ 2 Imsum

119901119902isin119868⟨Γ119901+119902+1

(119909119902+11991010158401015840

119902) (119909119901minus11991010158401015840

119901)⟩H+sum119901119902isin119868 ⟨Γ119901+119902(119909119902 + 119910

10158401015840

119902) (119909119901+ 11991010158401015840

119901)⟩

H=

0 We prove with these computations that119870+= 119877(119860 + i119868

119870) =

119870minus= 119877(119860 minus i119868

119870) sub 119870 That is dim 119870

perp

+= dim119870

perp

minusrArr 119860rsquos

Cayley transform has equal deficiency indices andrArr 119860 admits a self-adjoint extension 119860 Let 119864

119860be the

spectral measure of the self-adjoint operator 119860 Becauseℎ119898119909 = 119860

119898(ℎ0119909) for all 119909 isin H and Γ

119898119909 = ℎ

lowast

0ℎ119898119909 it results

that Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 for all 119909 isin H and the integral

representations Γ119898119909 = ℎ

lowast

0intR 119905119898d119864119860(119905)ℎ0119909 for all 119909 isin H for

all119898 isin NWe consider the positive operator-valued measure119865119860(119905) = ℎ

lowast

0119864119860ℎ0 With respect to this positive operator-

valued measure we have Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 = intR 119905

119898d119865119860(119905)119909

for all 119909 isin H and all 119898 isin N That is Γ119898

= intR 119905119898d119865119860(119905) for

all 119898 isin N the required Hamburger moment integralrepresentations

Conversely If the terms Γ119898119898admit the integral representa-

tions Γ119898= int+infin

minusinfin119905119898d119865119860(119905) for all119898 = 0 1 2 for a positive

operator-valued measure on R we have

sum

119899119898

⟨Γ119899+119898

119909119898 119909119899⟩H

= sum

119899119898

⟨int

+infin

minusinfin

119905119899+119898d119865

119860(119905)119909119898 119909119899⟩

H

= int

+infin

minusinfin

d(⟨sum

119898

11990511989811986512

119860(119905) 119909119898sum

119899

11990511989911986512

119860(119905) 119909119899⟩

H)

= intRd(

100381710038171003817100381710038171003817100381710038171003817

11986512

119860sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

2

) ge 0

(15)

as it is required by (i)

4 About the Uniqueness ofthe Hamburger Operator-Valued MomentSequencesrsquo Representations

Let us consider a sequence of bounded operators 119860119899119899isinN

119860119899isin 119871(H) subject on the condition 119860

119899= 119860lowast

119899 1198600= IdH

119899 = 0 1 2 H an arbitrary complex Hilbert space Forthe sequence 119860

119899119899isinN we get two operator-valued integral

representing measures (or spectral functions) 119864119860 119864119861

Bor(H) rarr 119860(H) that is

119860119899= int

+infin

minusinfin

119905119899d119864119860(119905) = int

+infin

minusinfin

119905119899d119864119861(119905) (16)

for all 119899 = 0 1 2 The operator-valued measures allow usto define the scalar measures 120583119909 ]119909 Bor(R) rarr [0 +infin]120583119909(119863) = ⟨119864

119860(119863)119909 119909⟩H respectively ]

119909(119863) = ⟨119864

119861(119863)119909 119909⟩H

when119909 isin H is arbitraryWith respect to these scalarmeasureswe obtain

119886119909

119899= ⟨119860119899119909 119909⟩H = int

+infin

minusinfin

119905119899d120583119909 (119905)

= int

+infin

minusinfin

119905119899d]119909 (119905) forall119899 isin N

(17)

From [5 page 283] the Hamburger scalar moment prob-lem is indeterminate (the sequence 119886119909

119899119899does not uniquely

determine the scalar representing measure) It followsthat the operator-valued representing measure does notuniquely determine theHamburger operator-valuedmomentsequence

However under some additional conditions about theoperator-valued representing measure the Stieltjes (Ham-burger) operator-valued moment sequence is determined [3pages 509 510 511]

Moreover if the representing measure is that associatedwith a self-adjoint extension of a symmetric operator withdeficiency indices (00) the self-adjoint extension is thecanonical closure of the given operator and is defined on thewhole space Indeed if 119878 119863(119878) rarr H is symmetric with119877(119878 plusmn i) = H and 119860 sup 119878 the canonical closure of 119878 it followsthat H supe 119877(119860 plusmn i) sup 119877(119878 plusmn i) are closed subspaces in H thatis 119877(119860 plusmn i) = H In this case the canonical closure of 119878 isthe smallest self-adjoint extension of 119878 and is defined on thewhole space H (as in Section 3 of this paper Proposition 1)The same arguments are in [4 page 1267 Lemma 21]

ISRNMathematical Analysis 5

Proposition 2 (1) Let 119860119899+infin

119899=0 119860119899isin 119871(H) for all 119899 isin N H

an arbitrary complex Hilbert space subject on the conditions119860119899= 119860lowast

119899 1198600= IdH and 1198641 1198642 Bor(R) rarr 119860(H) two ortho-

gonal spectral functions on R such that

119860119899= int

+infin

minusinfin

1199051198991198641(119905) = int

+infin

minusinfin

1199051198991198642(119905) 119899 = 0 1 2 (18)

Then 1198641= 1198642on Bor(R)

Proof Because1198601isin 119871(H) and119860

1= 119860lowast

1 the existence of the

representation 1198601

= int+infin

minusinfin119905d119864119860(119905) with 119864

119860 Bor(R) rarr

119860(R) 119864119860(R) = 119860

0= IdH is the usual one and is unique The

spectral orthogonal measures coincide that is119864119860= 1198641= 1198642

The representing measure is the spectral orthogonal measureassociated with the self-adjoint operator 119860

1 From 119864

119860(120582)rsquos

multiplicative property it follows that 119860119899= int+infin

minusinfin119905119899119864119860(119905) =

(int+infin

minusinfin119905119864119860(119905))119899 for all 119899 isin N The uniqueness of the integral

representations with respect to spectral functions is assuredtrivially only in case 119860

119899= 119860119899 for all 119899 isin N when the

representation is possible

5 Stieltjes Operator-ValuedMoment Sequences

A sequence of bounded operators Γ = Γ119899119899 acting on an

arbitrary Hilbert space H is called a Stieltjes operator-valuedmoment sequence if there exists a positive operator-valuedmeasure 119864

Γon [0 +infin) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (19)

Proposition 3 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH with conditions (A) and

(B) in Proposition 1 satisfied The following assertions areequivalent

(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(20)

for all sequences 119909119899+infin

119899=0subH with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (21)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same as (i) in Proposition 1 Con-

sequently there exists a positive operator-valuedmeasure119864Γ

Bor(R) rarr 119860(H) such that Γ119899= int+infin

minusinfin119905119899d119864Γ(119905) 119899 = 0 1 2

In the statement (119895) (2) if we consider the sequence with

finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H arbitrary for

all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905)119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905) 119909 119909⟩H ge 0

(22)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same as(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1d⟨119864

Γ(119905) 119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(23)

that is (119895) (2)We give another second characterization on an operator

sequence Γ119899+infin

119899=0to be an operator-valued Stieltjes moment

sequence

Remark 4 In the sequel we argue like in [1 page 329]Between the Hamburger operator-valued moment sequencesand Stieltjes moment sequences we can establish the follow-ing bijection

(A) If Γ = Γ119899+infin

119899=0is a Stieltjes moment sequence with

respect to the spectral measure 119864Γ(119905) on [0 +infin) for the

homeomorphism 1205871

[0 +infin) rarr (minusinfin 0] 1205871(119905) =

minusradic119905 there corresponds a spectral measure 1198651

Γon (minusinfin 0]

defined by 1198651

Γ(119905) = (12)[119864

Γ(infin) minus 119864

Γ∘ 120587minus1

1(119905)] such that

int0

minusinfin1205822119896d1198651Γ(120582) = (12) int

+infin

0120582119896d119864Γ

For the homeomorphism 1205872

[0 +infin) rarr [0 +infin)1205872(119905) = radic119905 there corresponds a spectral measure 119865

2

Γon

[0 +infin) defined by 1198652

Γ(119905) = (12)119864

Γ(infin) + (12)119864

Γ∘ 120587minus1

2(119905)

such that intinfin0

1205822119896d1198652Γ(120582) = (12) int

+infin

0120582119896d119864Γ

6 ISRNMathematical Analysis

We define

119865Γ=

1198651

Γ(119905) 119905 lt 0

1198652

Γ(119905) 119905 ge 0

(24)

For 119865Γ(119905) we have the representations int+infin

minusinfin1205822119896d119865Γ(120582) =

int+infin

0120582119896d119864Γ(120582) 119896 = 0 1 119899 and int

+infin

minusinfin1205822119896+1d119865

Γ(120582) = 0

119896 = 0 1 119899

(B) Conversely If we have the operator-valued Hamburgermoment sequence Γ

0 0 Γ1 0 Γ2 0 with respect to the

spectral representing measure 119865120582 respectively the sequence

119861119899+infin

119899=0 defined by 119861

119899= (Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 for all119899 isin N admits the integral representation 119861

2119899= Γ119899

=

int+infin

minusinfin1199052119899d119865(120582) and 119861

2119899+1= 0 = int

+infin

minusinfin1199052119899+1d119865(119905) We can con-

struct a spectral measure 119864(119905) on [0 +infin) that is for 120587

(minusinfin +infin) rarr [0 +infin) 120587(119905) = 1199052 we have 119864(119905) = 119865 ∘ 120587

minus1(119905)

withint+infinminusinfin

119905119899d119865(119905) = int

+infin

0(1199052)119899d119864(119905) = 119861

2119899= Γ119899 119899 = 0 1 2

For obtaining positive operator-valued measures fromorthogonal spectral functions in both cases we composethe orthogonal spectral measure associated with a self-adjoinoperator with the same projections respectively ℎ

0 ℎlowast0in

our case Summing the above conditions (A) and (B) inRemark 4 for an operator-valued sequence Γ

119899infin

119899=0isin 119871(H)

we construct the operator-valued sequence 119861119899

= (Γ[1198992]

+

(minus1)119899Γ[1198992]

)2 for all 119899 isin 119873 and conversely for a sequence119861119899+infin

119899=0isin 119871(H) we construct the operator-valued sequence

Γ119899+infin

119899=0 with Γ

119899= 1198612119899 119899 isin N

With the above construction we have the following

Proposition 5 The sequence Γ119899+infin

119899=0that satisfies conditions

(A) and (B) in Proposition 1 is a Stieltjes operator-valuedmoment sequence if and only if

119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

Hge 0

(25)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

Proposition 51015840 (reformulated) The sequence Γ119899+infin

119899=0is a

Stieltjes operator-valued moment sequence if and only if

119902

sum

119899119898=0

⟨119861119899+119898

119909119899 119909119898⟩H ge 0 (26)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

119861119899 for all 119899 isin N defined above

Proof Let Γ119899+infin

119899=0be an operatorsrsquo sequence Γ

119899isin 119871(H)

H an arbitrary complex Hilbert space we define 119861119899

=

(Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 119899 isin N that is 1198612119899

= Γ119899and

1198612119899+1

= 0 for all 119899 isin N In the condition of the hypothesis(Propositions 5 and 51015840) we have sum

119899119898⟨119861119899+119898

119909119899 119909119898⟩H ge 0

for all sequences 119909119899119899

sub H with finite support From

Proposition 1 there exists a positive operator-valuedmeasureon R such that

119861119899= int

+infin

minusinfin

119905119899d119865 (119905) 119899 = 0 1 2 (27)

From Remark 4 there exists a positive operator-valued mea-sure 119864(119905) such that

1198612119899

= int

+infin

minusinfin

1199052119899d119865 (119905) = Γ

[21198992]= Γ119899= int

+infin

0

119905119899d119864 (119905)

119899 = 0 1 2

(28)

That is Γ119899119899isinN is a Stieltjes operator-valued moment

sequence

Conversely If Γ119899

= int+infin

0119905119899d119864(119905) 119899 = 0 1 2 we con-

struct a measure 119865(119905) positively defined on (minusinfin +infin) asin Remark 4 with the property that Γ

119899= int+infin

minusinfin1199052119899d119865(119905) =

int+infin

0119905119899d119864(119905) = 119861

2119899 and 119861

2119899+1= int+infin

minusinfin1199052119899+1d119865(119905) = 0 In this

case119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

H

=

119902

sum

119899119898=0

⟨int

+infin

minusinfin

119905119899+119898d119865(119905)119909

119899 119909119898⟩

H

= int

+infin

minusinfin

d100381710038171003817100381710038171003817100381710038171003817

sum

119899

11990511989911986512

(119905) 119909119899

100381710038171003817100381710038171003817100381710038171003817

2

ge 0

(29)

Proposition 6 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH The following assertions are

equivalent(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(30)

for all sequences 119909119899+infin

119899=0sub H with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (31)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same with (i) in Proposition 1

Consequently there exists a positive operator-valued mea-sure 119864

Γ Bor(R) rarr 119860(H) such that Γ

119899= int+infin

minusinfin119905119899d119864Γ(119905)

119899 = 0 1 2 In the statement (119895) (2) if we consider the

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

4 ISRNMathematical Analysis

that 119891minus119891 = 0119870 119892minus119892 = 0

119870and [120572119891

plusmn+120573119892plusmn] = [120572119891

plusmn+120573119892plusmn] =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899) + iℎ

119899(120572119909119899) + ℎ

119899+1(120573119910119899) plusmn iℎ

119899(120573119910119899)] +

sum119899isin1198681minus(1198681cap1198682)

[ℎ119899+1

(120572119909119899) + iℎ

119899(120573119910119899) + ℎ119899+1

(120573119910119899) plusmn iℎ

119899(120572119909119899+

120573119910119899] + sum

119899isin1198682minus(1198681cap1198682)[ℎ119899+1

(120572119909119899) + 120573119910

119899plusmn iℎ119899(120572119909119899) + 120573119910

119899) =

sum119899isin1198681cap1198682

[ℎ119899+1

(120572119909119899

+ 120573119910119899) plusmn iℎ

119899(120572119909119899) + 120573119910

119899] isin 119870

plusmn

Because 119860 is a symmetric operator it results also that119877(119860 plusmn i119868

119870) = 119870

plusmnare closed subspaces in 119870 We prove

that in conditions (A) and (B) for the kernel Γ we have119870+= 119877(119860+i119868

119870) = 119877(119860minusi119868

119870) = 119870minus Indeed given an arbitrary

element119891 = sum119899isin119868finitesubN(ℎ119899+1119909119899+iℎ119899119909119899)we look for an element

119892 isin 119870minus

= 119877(119860 minus i119868119870) 119892 = sum

119899isin119868finitesubN(ℎ119898+11199101015840

119898minus iℎ1198981199101015840

119898)

such that (119891 minus 119892) = 0119870 For a construction of the elements

119909119899119899isin1198681cup1198682=119868

119910119899119899isin1198681cup1198682=119868

sub H like the previous one wehave (119891 minus 119892) = sum[ℎ

119899+1(119909119899minus 1199101015840

119899) + iℎ

119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119902)

[sum119899120575(119899+1)

(119909119899minus 1199101015840

119899) + i120575

119899(119909119899minus 1199101015840

119899)](119901)gtH = 0 hArr

sum119901119902isin119868119901119902ge1

Γ119901+119902

[(119909119902minus1

minus 1199101015840

119902minus1) + i(119909

119902+ 1199101015840

119902)] (119909119901minus1

minus 1199101015840

119901minus1+

i(119909119901+ 1199101015840

119901)gtH = 0

119870hArr sum119901119902ge0

⟨Γ119901+119902+2

(119909119902minus 1199101015840

119902) (119909119901minus 1199101015840

119901)⟩

Hminus

2 Imsum119901119902isin119868

⟨Γ119901+119902+1

(119909119902+ 1199101015840

119902) (119909119901

minus 1199101015840

119901)⟩H + sum

119901119902isin119868(119909119902+

1199101015840

119902) (119909119901+ 1199101015840

119901)gtH = 0

119870 According to condition (A) on the

kernel Γ119899119899 such an element exists We have 119870

+sub 119870minus

Conversely let 119891 isin 119870minus 119891 = sum

119899isin119868(ℎ119899+1

119909119899minus iℎ119899119909119899) we search

for an element 119892 isin 119870+ 119892 = sum

119899isin119868(ℎ119899+1

11991010158401015840

119899+ iℎ11989911991010158401015840

119899) with the

property that (119891 minus 119892) = 0119870 Consequently we have to find an

element 119892 isin 119870+such thatsum[120575

(119899+1)(119909119899minus1199101015840

119899) + iℎ119899(119909119899+ 1199101015840

119899)] =

0119870

hArr sum119901119902isin119868

Γ119901+119902

[sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119902)

sum119899120575(119899+1)

(119909119899minus 11991010158401015840

119899) minus i120575

119899(119909119899+ 11991010158401015840

119899)](119901)gtH = 0

119870hArr

sum119901119902isin119868

⟨Γ119901+119902+2

(119909119902minus 11991010158401015840

119902) (119909119901minus 11991010158401015840

119901)⟩

H+ 2 Imsum

119901119902isin119868⟨Γ119901+119902+1

(119909119902+11991010158401015840

119902) (119909119901minus11991010158401015840

119901)⟩H+sum119901119902isin119868 ⟨Γ119901+119902(119909119902 + 119910

10158401015840

119902) (119909119901+ 11991010158401015840

119901)⟩

H=

0 We prove with these computations that119870+= 119877(119860 + i119868

119870) =

119870minus= 119877(119860 minus i119868

119870) sub 119870 That is dim 119870

perp

+= dim119870

perp

minusrArr 119860rsquos

Cayley transform has equal deficiency indices andrArr 119860 admits a self-adjoint extension 119860 Let 119864

119860be the

spectral measure of the self-adjoint operator 119860 Becauseℎ119898119909 = 119860

119898(ℎ0119909) for all 119909 isin H and Γ

119898119909 = ℎ

lowast

0ℎ119898119909 it results

that Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 for all 119909 isin H and the integral

representations Γ119898119909 = ℎ

lowast

0intR 119905119898d119864119860(119905)ℎ0119909 for all 119909 isin H for

all119898 isin NWe consider the positive operator-valued measure119865119860(119905) = ℎ

lowast

0119864119860ℎ0 With respect to this positive operator-

valued measure we have Γ119898119909 = ℎ

lowast

0119860119898ℎ0119909 = intR 119905

119898d119865119860(119905)119909

for all 119909 isin H and all 119898 isin N That is Γ119898

= intR 119905119898d119865119860(119905) for

all 119898 isin N the required Hamburger moment integralrepresentations

Conversely If the terms Γ119898119898admit the integral representa-

tions Γ119898= int+infin

minusinfin119905119898d119865119860(119905) for all119898 = 0 1 2 for a positive

operator-valued measure on R we have

sum

119899119898

⟨Γ119899+119898

119909119898 119909119899⟩H

= sum

119899119898

⟨int

+infin

minusinfin

119905119899+119898d119865

119860(119905)119909119898 119909119899⟩

H

= int

+infin

minusinfin

d(⟨sum

119898

11990511989811986512

119860(119905) 119909119898sum

119899

11990511989911986512

119860(119905) 119909119899⟩

H)

= intRd(

100381710038171003817100381710038171003817100381710038171003817

11986512

119860sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

2

) ge 0

(15)

as it is required by (i)

4 About the Uniqueness ofthe Hamburger Operator-Valued MomentSequencesrsquo Representations

Let us consider a sequence of bounded operators 119860119899119899isinN

119860119899isin 119871(H) subject on the condition 119860

119899= 119860lowast

119899 1198600= IdH

119899 = 0 1 2 H an arbitrary complex Hilbert space Forthe sequence 119860

119899119899isinN we get two operator-valued integral

representing measures (or spectral functions) 119864119860 119864119861

Bor(H) rarr 119860(H) that is

119860119899= int

+infin

minusinfin

119905119899d119864119860(119905) = int

+infin

minusinfin

119905119899d119864119861(119905) (16)

for all 119899 = 0 1 2 The operator-valued measures allow usto define the scalar measures 120583119909 ]119909 Bor(R) rarr [0 +infin]120583119909(119863) = ⟨119864

119860(119863)119909 119909⟩H respectively ]

119909(119863) = ⟨119864

119861(119863)119909 119909⟩H

when119909 isin H is arbitraryWith respect to these scalarmeasureswe obtain

119886119909

119899= ⟨119860119899119909 119909⟩H = int

+infin

minusinfin

119905119899d120583119909 (119905)

= int

+infin

minusinfin

119905119899d]119909 (119905) forall119899 isin N

(17)

From [5 page 283] the Hamburger scalar moment prob-lem is indeterminate (the sequence 119886119909

119899119899does not uniquely

determine the scalar representing measure) It followsthat the operator-valued representing measure does notuniquely determine theHamburger operator-valuedmomentsequence

However under some additional conditions about theoperator-valued representing measure the Stieltjes (Ham-burger) operator-valued moment sequence is determined [3pages 509 510 511]

Moreover if the representing measure is that associatedwith a self-adjoint extension of a symmetric operator withdeficiency indices (00) the self-adjoint extension is thecanonical closure of the given operator and is defined on thewhole space Indeed if 119878 119863(119878) rarr H is symmetric with119877(119878 plusmn i) = H and 119860 sup 119878 the canonical closure of 119878 it followsthat H supe 119877(119860 plusmn i) sup 119877(119878 plusmn i) are closed subspaces in H thatis 119877(119860 plusmn i) = H In this case the canonical closure of 119878 isthe smallest self-adjoint extension of 119878 and is defined on thewhole space H (as in Section 3 of this paper Proposition 1)The same arguments are in [4 page 1267 Lemma 21]

ISRNMathematical Analysis 5

Proposition 2 (1) Let 119860119899+infin

119899=0 119860119899isin 119871(H) for all 119899 isin N H

an arbitrary complex Hilbert space subject on the conditions119860119899= 119860lowast

119899 1198600= IdH and 1198641 1198642 Bor(R) rarr 119860(H) two ortho-

gonal spectral functions on R such that

119860119899= int

+infin

minusinfin

1199051198991198641(119905) = int

+infin

minusinfin

1199051198991198642(119905) 119899 = 0 1 2 (18)

Then 1198641= 1198642on Bor(R)

Proof Because1198601isin 119871(H) and119860

1= 119860lowast

1 the existence of the

representation 1198601

= int+infin

minusinfin119905d119864119860(119905) with 119864

119860 Bor(R) rarr

119860(R) 119864119860(R) = 119860

0= IdH is the usual one and is unique The

spectral orthogonal measures coincide that is119864119860= 1198641= 1198642

The representing measure is the spectral orthogonal measureassociated with the self-adjoint operator 119860

1 From 119864

119860(120582)rsquos

multiplicative property it follows that 119860119899= int+infin

minusinfin119905119899119864119860(119905) =

(int+infin

minusinfin119905119864119860(119905))119899 for all 119899 isin N The uniqueness of the integral

representations with respect to spectral functions is assuredtrivially only in case 119860

119899= 119860119899 for all 119899 isin N when the

representation is possible

5 Stieltjes Operator-ValuedMoment Sequences

A sequence of bounded operators Γ = Γ119899119899 acting on an

arbitrary Hilbert space H is called a Stieltjes operator-valuedmoment sequence if there exists a positive operator-valuedmeasure 119864

Γon [0 +infin) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (19)

Proposition 3 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH with conditions (A) and

(B) in Proposition 1 satisfied The following assertions areequivalent

(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(20)

for all sequences 119909119899+infin

119899=0subH with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (21)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same as (i) in Proposition 1 Con-

sequently there exists a positive operator-valuedmeasure119864Γ

Bor(R) rarr 119860(H) such that Γ119899= int+infin

minusinfin119905119899d119864Γ(119905) 119899 = 0 1 2

In the statement (119895) (2) if we consider the sequence with

finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H arbitrary for

all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905)119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905) 119909 119909⟩H ge 0

(22)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same as(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1d⟨119864

Γ(119905) 119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(23)

that is (119895) (2)We give another second characterization on an operator

sequence Γ119899+infin

119899=0to be an operator-valued Stieltjes moment

sequence

Remark 4 In the sequel we argue like in [1 page 329]Between the Hamburger operator-valued moment sequencesand Stieltjes moment sequences we can establish the follow-ing bijection

(A) If Γ = Γ119899+infin

119899=0is a Stieltjes moment sequence with

respect to the spectral measure 119864Γ(119905) on [0 +infin) for the

homeomorphism 1205871

[0 +infin) rarr (minusinfin 0] 1205871(119905) =

minusradic119905 there corresponds a spectral measure 1198651

Γon (minusinfin 0]

defined by 1198651

Γ(119905) = (12)[119864

Γ(infin) minus 119864

Γ∘ 120587minus1

1(119905)] such that

int0

minusinfin1205822119896d1198651Γ(120582) = (12) int

+infin

0120582119896d119864Γ

For the homeomorphism 1205872

[0 +infin) rarr [0 +infin)1205872(119905) = radic119905 there corresponds a spectral measure 119865

2

Γon

[0 +infin) defined by 1198652

Γ(119905) = (12)119864

Γ(infin) + (12)119864

Γ∘ 120587minus1

2(119905)

such that intinfin0

1205822119896d1198652Γ(120582) = (12) int

+infin

0120582119896d119864Γ

6 ISRNMathematical Analysis

We define

119865Γ=

1198651

Γ(119905) 119905 lt 0

1198652

Γ(119905) 119905 ge 0

(24)

For 119865Γ(119905) we have the representations int+infin

minusinfin1205822119896d119865Γ(120582) =

int+infin

0120582119896d119864Γ(120582) 119896 = 0 1 119899 and int

+infin

minusinfin1205822119896+1d119865

Γ(120582) = 0

119896 = 0 1 119899

(B) Conversely If we have the operator-valued Hamburgermoment sequence Γ

0 0 Γ1 0 Γ2 0 with respect to the

spectral representing measure 119865120582 respectively the sequence

119861119899+infin

119899=0 defined by 119861

119899= (Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 for all119899 isin N admits the integral representation 119861

2119899= Γ119899

=

int+infin

minusinfin1199052119899d119865(120582) and 119861

2119899+1= 0 = int

+infin

minusinfin1199052119899+1d119865(119905) We can con-

struct a spectral measure 119864(119905) on [0 +infin) that is for 120587

(minusinfin +infin) rarr [0 +infin) 120587(119905) = 1199052 we have 119864(119905) = 119865 ∘ 120587

minus1(119905)

withint+infinminusinfin

119905119899d119865(119905) = int

+infin

0(1199052)119899d119864(119905) = 119861

2119899= Γ119899 119899 = 0 1 2

For obtaining positive operator-valued measures fromorthogonal spectral functions in both cases we composethe orthogonal spectral measure associated with a self-adjoinoperator with the same projections respectively ℎ

0 ℎlowast0in

our case Summing the above conditions (A) and (B) inRemark 4 for an operator-valued sequence Γ

119899infin

119899=0isin 119871(H)

we construct the operator-valued sequence 119861119899

= (Γ[1198992]

+

(minus1)119899Γ[1198992]

)2 for all 119899 isin 119873 and conversely for a sequence119861119899+infin

119899=0isin 119871(H) we construct the operator-valued sequence

Γ119899+infin

119899=0 with Γ

119899= 1198612119899 119899 isin N

With the above construction we have the following

Proposition 5 The sequence Γ119899+infin

119899=0that satisfies conditions

(A) and (B) in Proposition 1 is a Stieltjes operator-valuedmoment sequence if and only if

119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

Hge 0

(25)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

Proposition 51015840 (reformulated) The sequence Γ119899+infin

119899=0is a

Stieltjes operator-valued moment sequence if and only if

119902

sum

119899119898=0

⟨119861119899+119898

119909119899 119909119898⟩H ge 0 (26)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

119861119899 for all 119899 isin N defined above

Proof Let Γ119899+infin

119899=0be an operatorsrsquo sequence Γ

119899isin 119871(H)

H an arbitrary complex Hilbert space we define 119861119899

=

(Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 119899 isin N that is 1198612119899

= Γ119899and

1198612119899+1

= 0 for all 119899 isin N In the condition of the hypothesis(Propositions 5 and 51015840) we have sum

119899119898⟨119861119899+119898

119909119899 119909119898⟩H ge 0

for all sequences 119909119899119899

sub H with finite support From

Proposition 1 there exists a positive operator-valuedmeasureon R such that

119861119899= int

+infin

minusinfin

119905119899d119865 (119905) 119899 = 0 1 2 (27)

From Remark 4 there exists a positive operator-valued mea-sure 119864(119905) such that

1198612119899

= int

+infin

minusinfin

1199052119899d119865 (119905) = Γ

[21198992]= Γ119899= int

+infin

0

119905119899d119864 (119905)

119899 = 0 1 2

(28)

That is Γ119899119899isinN is a Stieltjes operator-valued moment

sequence

Conversely If Γ119899

= int+infin

0119905119899d119864(119905) 119899 = 0 1 2 we con-

struct a measure 119865(119905) positively defined on (minusinfin +infin) asin Remark 4 with the property that Γ

119899= int+infin

minusinfin1199052119899d119865(119905) =

int+infin

0119905119899d119864(119905) = 119861

2119899 and 119861

2119899+1= int+infin

minusinfin1199052119899+1d119865(119905) = 0 In this

case119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

H

=

119902

sum

119899119898=0

⟨int

+infin

minusinfin

119905119899+119898d119865(119905)119909

119899 119909119898⟩

H

= int

+infin

minusinfin

d100381710038171003817100381710038171003817100381710038171003817

sum

119899

11990511989911986512

(119905) 119909119899

100381710038171003817100381710038171003817100381710038171003817

2

ge 0

(29)

Proposition 6 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH The following assertions are

equivalent(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(30)

for all sequences 119909119899+infin

119899=0sub H with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (31)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same with (i) in Proposition 1

Consequently there exists a positive operator-valued mea-sure 119864

Γ Bor(R) rarr 119860(H) such that Γ

119899= int+infin

minusinfin119905119899d119864Γ(119905)

119899 = 0 1 2 In the statement (119895) (2) if we consider the

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

ISRNMathematical Analysis 5

Proposition 2 (1) Let 119860119899+infin

119899=0 119860119899isin 119871(H) for all 119899 isin N H

an arbitrary complex Hilbert space subject on the conditions119860119899= 119860lowast

119899 1198600= IdH and 1198641 1198642 Bor(R) rarr 119860(H) two ortho-

gonal spectral functions on R such that

119860119899= int

+infin

minusinfin

1199051198991198641(119905) = int

+infin

minusinfin

1199051198991198642(119905) 119899 = 0 1 2 (18)

Then 1198641= 1198642on Bor(R)

Proof Because1198601isin 119871(H) and119860

1= 119860lowast

1 the existence of the

representation 1198601

= int+infin

minusinfin119905d119864119860(119905) with 119864

119860 Bor(R) rarr

119860(R) 119864119860(R) = 119860

0= IdH is the usual one and is unique The

spectral orthogonal measures coincide that is119864119860= 1198641= 1198642

The representing measure is the spectral orthogonal measureassociated with the self-adjoint operator 119860

1 From 119864

119860(120582)rsquos

multiplicative property it follows that 119860119899= int+infin

minusinfin119905119899119864119860(119905) =

(int+infin

minusinfin119905119864119860(119905))119899 for all 119899 isin N The uniqueness of the integral

representations with respect to spectral functions is assuredtrivially only in case 119860

119899= 119860119899 for all 119899 isin N when the

representation is possible

5 Stieltjes Operator-ValuedMoment Sequences

A sequence of bounded operators Γ = Γ119899119899 acting on an

arbitrary Hilbert space H is called a Stieltjes operator-valuedmoment sequence if there exists a positive operator-valuedmeasure 119864

Γon [0 +infin) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (19)

Proposition 3 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH with conditions (A) and

(B) in Proposition 1 satisfied The following assertions areequivalent

(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(20)

for all sequences 119909119899+infin

119899=0subH with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (21)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same as (i) in Proposition 1 Con-

sequently there exists a positive operator-valuedmeasure119864Γ

Bor(R) rarr 119860(H) such that Γ119899= int+infin

minusinfin119905119899d119864Γ(119905) 119899 = 0 1 2

In the statement (119895) (2) if we consider the sequence with

finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H arbitrary for

all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905)119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905) 119909 119909⟩H ge 0

(22)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same as(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1d⟨119864

Γ(119905) 119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(23)

that is (119895) (2)We give another second characterization on an operator

sequence Γ119899+infin

119899=0to be an operator-valued Stieltjes moment

sequence

Remark 4 In the sequel we argue like in [1 page 329]Between the Hamburger operator-valued moment sequencesand Stieltjes moment sequences we can establish the follow-ing bijection

(A) If Γ = Γ119899+infin

119899=0is a Stieltjes moment sequence with

respect to the spectral measure 119864Γ(119905) on [0 +infin) for the

homeomorphism 1205871

[0 +infin) rarr (minusinfin 0] 1205871(119905) =

minusradic119905 there corresponds a spectral measure 1198651

Γon (minusinfin 0]

defined by 1198651

Γ(119905) = (12)[119864

Γ(infin) minus 119864

Γ∘ 120587minus1

1(119905)] such that

int0

minusinfin1205822119896d1198651Γ(120582) = (12) int

+infin

0120582119896d119864Γ

For the homeomorphism 1205872

[0 +infin) rarr [0 +infin)1205872(119905) = radic119905 there corresponds a spectral measure 119865

2

Γon

[0 +infin) defined by 1198652

Γ(119905) = (12)119864

Γ(infin) + (12)119864

Γ∘ 120587minus1

2(119905)

such that intinfin0

1205822119896d1198652Γ(120582) = (12) int

+infin

0120582119896d119864Γ

6 ISRNMathematical Analysis

We define

119865Γ=

1198651

Γ(119905) 119905 lt 0

1198652

Γ(119905) 119905 ge 0

(24)

For 119865Γ(119905) we have the representations int+infin

minusinfin1205822119896d119865Γ(120582) =

int+infin

0120582119896d119864Γ(120582) 119896 = 0 1 119899 and int

+infin

minusinfin1205822119896+1d119865

Γ(120582) = 0

119896 = 0 1 119899

(B) Conversely If we have the operator-valued Hamburgermoment sequence Γ

0 0 Γ1 0 Γ2 0 with respect to the

spectral representing measure 119865120582 respectively the sequence

119861119899+infin

119899=0 defined by 119861

119899= (Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 for all119899 isin N admits the integral representation 119861

2119899= Γ119899

=

int+infin

minusinfin1199052119899d119865(120582) and 119861

2119899+1= 0 = int

+infin

minusinfin1199052119899+1d119865(119905) We can con-

struct a spectral measure 119864(119905) on [0 +infin) that is for 120587

(minusinfin +infin) rarr [0 +infin) 120587(119905) = 1199052 we have 119864(119905) = 119865 ∘ 120587

minus1(119905)

withint+infinminusinfin

119905119899d119865(119905) = int

+infin

0(1199052)119899d119864(119905) = 119861

2119899= Γ119899 119899 = 0 1 2

For obtaining positive operator-valued measures fromorthogonal spectral functions in both cases we composethe orthogonal spectral measure associated with a self-adjoinoperator with the same projections respectively ℎ

0 ℎlowast0in

our case Summing the above conditions (A) and (B) inRemark 4 for an operator-valued sequence Γ

119899infin

119899=0isin 119871(H)

we construct the operator-valued sequence 119861119899

= (Γ[1198992]

+

(minus1)119899Γ[1198992]

)2 for all 119899 isin 119873 and conversely for a sequence119861119899+infin

119899=0isin 119871(H) we construct the operator-valued sequence

Γ119899+infin

119899=0 with Γ

119899= 1198612119899 119899 isin N

With the above construction we have the following

Proposition 5 The sequence Γ119899+infin

119899=0that satisfies conditions

(A) and (B) in Proposition 1 is a Stieltjes operator-valuedmoment sequence if and only if

119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

Hge 0

(25)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

Proposition 51015840 (reformulated) The sequence Γ119899+infin

119899=0is a

Stieltjes operator-valued moment sequence if and only if

119902

sum

119899119898=0

⟨119861119899+119898

119909119899 119909119898⟩H ge 0 (26)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

119861119899 for all 119899 isin N defined above

Proof Let Γ119899+infin

119899=0be an operatorsrsquo sequence Γ

119899isin 119871(H)

H an arbitrary complex Hilbert space we define 119861119899

=

(Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 119899 isin N that is 1198612119899

= Γ119899and

1198612119899+1

= 0 for all 119899 isin N In the condition of the hypothesis(Propositions 5 and 51015840) we have sum

119899119898⟨119861119899+119898

119909119899 119909119898⟩H ge 0

for all sequences 119909119899119899

sub H with finite support From

Proposition 1 there exists a positive operator-valuedmeasureon R such that

119861119899= int

+infin

minusinfin

119905119899d119865 (119905) 119899 = 0 1 2 (27)

From Remark 4 there exists a positive operator-valued mea-sure 119864(119905) such that

1198612119899

= int

+infin

minusinfin

1199052119899d119865 (119905) = Γ

[21198992]= Γ119899= int

+infin

0

119905119899d119864 (119905)

119899 = 0 1 2

(28)

That is Γ119899119899isinN is a Stieltjes operator-valued moment

sequence

Conversely If Γ119899

= int+infin

0119905119899d119864(119905) 119899 = 0 1 2 we con-

struct a measure 119865(119905) positively defined on (minusinfin +infin) asin Remark 4 with the property that Γ

119899= int+infin

minusinfin1199052119899d119865(119905) =

int+infin

0119905119899d119864(119905) = 119861

2119899 and 119861

2119899+1= int+infin

minusinfin1199052119899+1d119865(119905) = 0 In this

case119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

H

=

119902

sum

119899119898=0

⟨int

+infin

minusinfin

119905119899+119898d119865(119905)119909

119899 119909119898⟩

H

= int

+infin

minusinfin

d100381710038171003817100381710038171003817100381710038171003817

sum

119899

11990511989911986512

(119905) 119909119899

100381710038171003817100381710038171003817100381710038171003817

2

ge 0

(29)

Proposition 6 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH The following assertions are

equivalent(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(30)

for all sequences 119909119899+infin

119899=0sub H with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (31)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same with (i) in Proposition 1

Consequently there exists a positive operator-valued mea-sure 119864

Γ Bor(R) rarr 119860(H) such that Γ

119899= int+infin

minusinfin119905119899d119864Γ(119905)

119899 = 0 1 2 In the statement (119895) (2) if we consider the

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRNMathematical Analysis

We define

119865Γ=

1198651

Γ(119905) 119905 lt 0

1198652

Γ(119905) 119905 ge 0

(24)

For 119865Γ(119905) we have the representations int+infin

minusinfin1205822119896d119865Γ(120582) =

int+infin

0120582119896d119864Γ(120582) 119896 = 0 1 119899 and int

+infin

minusinfin1205822119896+1d119865

Γ(120582) = 0

119896 = 0 1 119899

(B) Conversely If we have the operator-valued Hamburgermoment sequence Γ

0 0 Γ1 0 Γ2 0 with respect to the

spectral representing measure 119865120582 respectively the sequence

119861119899+infin

119899=0 defined by 119861

119899= (Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 for all119899 isin N admits the integral representation 119861

2119899= Γ119899

=

int+infin

minusinfin1199052119899d119865(120582) and 119861

2119899+1= 0 = int

+infin

minusinfin1199052119899+1d119865(119905) We can con-

struct a spectral measure 119864(119905) on [0 +infin) that is for 120587

(minusinfin +infin) rarr [0 +infin) 120587(119905) = 1199052 we have 119864(119905) = 119865 ∘ 120587

minus1(119905)

withint+infinminusinfin

119905119899d119865(119905) = int

+infin

0(1199052)119899d119864(119905) = 119861

2119899= Γ119899 119899 = 0 1 2

For obtaining positive operator-valued measures fromorthogonal spectral functions in both cases we composethe orthogonal spectral measure associated with a self-adjoinoperator with the same projections respectively ℎ

0 ℎlowast0in

our case Summing the above conditions (A) and (B) inRemark 4 for an operator-valued sequence Γ

119899infin

119899=0isin 119871(H)

we construct the operator-valued sequence 119861119899

= (Γ[1198992]

+

(minus1)119899Γ[1198992]

)2 for all 119899 isin 119873 and conversely for a sequence119861119899+infin

119899=0isin 119871(H) we construct the operator-valued sequence

Γ119899+infin

119899=0 with Γ

119899= 1198612119899 119899 isin N

With the above construction we have the following

Proposition 5 The sequence Γ119899+infin

119899=0that satisfies conditions

(A) and (B) in Proposition 1 is a Stieltjes operator-valuedmoment sequence if and only if

119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

Hge 0

(25)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

Proposition 51015840 (reformulated) The sequence Γ119899+infin

119899=0is a

Stieltjes operator-valued moment sequence if and only if

119902

sum

119899119898=0

⟨119861119899+119898

119909119899 119909119898⟩H ge 0 (26)

for all sequences 119909119899+infin

119899=0sub Hwith finite support and all 119902 isin N

119861119899 for all 119899 isin N defined above

Proof Let Γ119899+infin

119899=0be an operatorsrsquo sequence Γ

119899isin 119871(H)

H an arbitrary complex Hilbert space we define 119861119899

=

(Γ[1198992]

+ (minus1)119899Γ[1198992]

)2 119899 isin N that is 1198612119899

= Γ119899and

1198612119899+1

= 0 for all 119899 isin N In the condition of the hypothesis(Propositions 5 and 51015840) we have sum

119899119898⟨119861119899+119898

119909119899 119909119898⟩H ge 0

for all sequences 119909119899119899

sub H with finite support From

Proposition 1 there exists a positive operator-valuedmeasureon R such that

119861119899= int

+infin

minusinfin

119905119899d119865 (119905) 119899 = 0 1 2 (27)

From Remark 4 there exists a positive operator-valued mea-sure 119864(119905) such that

1198612119899

= int

+infin

minusinfin

1199052119899d119865 (119905) = Γ

[21198992]= Γ119899= int

+infin

0

119905119899d119864 (119905)

119899 = 0 1 2

(28)

That is Γ119899119899isinN is a Stieltjes operator-valued moment

sequence

Conversely If Γ119899

= int+infin

0119905119899d119864(119905) 119899 = 0 1 2 we con-

struct a measure 119865(119905) positively defined on (minusinfin +infin) asin Remark 4 with the property that Γ

119899= int+infin

minusinfin1199052119899d119865(119905) =

int+infin

0119905119899d119864(119905) = 119861

2119899 and 119861

2119899+1= int+infin

minusinfin1199052119899+1d119865(119905) = 0 In this

case119902

sum

119899119898=0

⟨(Γ[(119899+119898)2]

+ (minus1)119899+119898

Γ[(119899+119898)2]

2)119909119899 119909119898⟩

H

=

119902

sum

119899119898=0

⟨int

+infin

minusinfin

119905119899+119898d119865(119905)119909

119899 119909119898⟩

H

= int

+infin

minusinfin

d100381710038171003817100381710038171003817100381710038171003817

sum

119899

11990511989911986512

(119905) 119909119899

100381710038171003817100381710038171003817100381710038171003817

2

ge 0

(29)

Proposition 6 Let Γ119899+infin

119899=0be an operator sequence with Γ

119899isin

119860(119867) for all 119899 isin N and Γ0= IdH The following assertions are

equivalent(119895)

(1) sum

119899119898

⟨Γ119899+119898

119909119899 119909119898⟩H ge 0

(2) sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H ge 0

(30)

for all sequences 119909119899+infin

119899=0sub H with finite support

(119895119895) There exists a positive operator-valued measure 119864Γ

Bor(R) rarr 119860(H) such that

Γ119899= int

+infin

0

119905119899d119864Γ(119905) 119899 = 0 1 2 (31)

Proof We prove that condition (119895) [(1) and (2)] is sufficientCondition (119895) (1) is the same with (i) in Proposition 1

Consequently there exists a positive operator-valued mea-sure 119864

Γ Bor(R) rarr 119860(H) such that Γ

119899= int+infin

minusinfin119905119899d119864Γ(119905)

119899 = 0 1 2 In the statement (119895) (2) if we consider the

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Analysis 7

sequence with finite support 119909119899119899as 119909119899= 120585119899119909 120585119899isin C 119909 isin H

arbitrary for all 119899 isin N we obtain

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

120585119899120585119898d⟨119864Γ(119905) 119909 119909⟩H

= int

+infin

minusinfin

119905

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119899isin119868

119905119899120585119899

1003816100381610038161003816100381610038161003816100381610038161003816

2

d⟨119864Γ(119905)119909 119909⟩H ge 0

(32)

for all polynomials sum119899119905119899120585119899with complex coefficients and

all 119909 isin H It follows that the representing measure 119864Γis

concentrated on [0 +infin)

Conversely The implication (119895119895) rArr (119895) (1) is the same with(ii) rArr (i) from Proposition 1 Moreover from (119895119895) it resultsthat for any sequence 119909

119899119899sub H with finite support we have

sum

119899119898

⟨Γ119899+119898+1

119909119899 119909119898⟩H

= sum

119899119898

int

+infin

minusinfin

119905119899+119898+1

119889⟨119864Γ(119905)119909119899 119909119898⟩H

= int

+infin

0

119905d⟨119864Γ(119905)sum

119899

119905119899119909119899sum

119898

119905119898119909119898⟩

H

= int

+infin

0

119905d100381710038171003817100381710038171003817100381710038171003817

11986412

Γ(119905)sum

119899

119905119899119909119899

100381710038171003817100381710038171003817100381710038171003817

ge 0

(33)

that is (119895) (2)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Ando ldquoTruncated moment problems for operatorsrdquo ActaScientiarum Mathematicarum vol 31 pp 319ndash334 1970

[2] JW Helton andM Putinar ldquoPositive polynomials in scalar andmatrix variables the spectral theorem and optimizationrdquo inOperator Theory Structured Matrices and Dilations vol 7 pp229ndash307 Theta Bucharest Romania 2007

[3] F J Narcowich ldquo119877-operators II On the approximation ofcertain operator-valued analytic functions and the Hermitianmoment problemrdquo Indiana UniversityMathematics Journal vol26 no 3 pp 483ndash513 1977

[4] F-H Vasilescu ldquoHamburger and Stieltjes moment problems inseveral variablesrdquo Transactions of the American MathematicalSociety vol 354 no 3 pp 1265ndash1278 2002

[5] G Choquet Lectures on Analysis vol 2 Benjamin New YorkNY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of