Generalized frames for operators in Hilbert spaces

2nd Reading

May 23, 2014 15:11 WSPC/S0219-0257 102-IDAQPRT 1450013

Infinite Dimensional Analysis, Quantum Probabilityand Related TopicsVol. 17, No. 2 (2014) 1450013 (20 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219025714500131

Generalized frames for operators in Hilbert spaces

Mohammad Sadegh Asgari∗ and Hamidreza Rahimi†

Department of Mathematics, Faculty of Science,Islamic Azad University, Central Tehran Branch,

Tehran, Iran∗[email protected]∗[email protected]†[email protected]

†[email protected]

Received 27 May 2013Accepted 4 March 2014Published 27 May 2014

Communicated by M. Bozejko

In this paper we present a family of analysis and synthesis systems of operators withframe-like properties for the range of a bounded operator on a separable Hilbert space.This family of operators is called a Θ–g-frame, where Θ is a bounded operator on aHilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstructelements from the range of Θ. In general, range of Θ is not a closed subspace. We alsoconstruct new Θ–g-frames by considering Θ–g-frames for its components. We furtherstudy Riesz decompositions for Hilbert spaces, which are a generalization of the notionof Riesz bases. We define the coefficient operators of a Riesz decomposition and we will

show that these coefficient operators are continuous projections. We obtain some resultsabout stability of Riesz decompositions under small perturbations.

Keywords: G-orthonormal bases; G-frames; fusion frames; Riesz decompositions.

AMS Subject Classification: Primary 41A58, Secondary 42C15, 42C40, 46C05

1. Introduction

Traditionally, frames have been used in signal processing, image processing, datacompression, and sampling theory. Frames were first introduced by Duffin and Scha-effer5 in the context of nonharmonic Fourier series and reintroduced in 1986 byDaubechies, Grossmann and Meyer.3 Since then the theory of frames began to bemore widely studied. On the other hand, the generalized frame theory is emergingmathematical theory that provides a natural framework for performing hierarchicaldata processing. Fusion frames were first formally defined by Casazza and Kutyniok

1450013-1

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

http://dx.doi.org/10.1142/S0219025714500131

2nd Reading


M. S. Asgari & H. Rahimi

in Ref. 1 and g-frames were introduced by Sun in Refs. 10 and 11 as generalizationsof frames in Hilbert spaces.

Throughout the paper H,K will be two separable Hilbert spaces and J is acountable index set that has been well-ordered. We shall denote by Jn, n ∈ N, thesubset of the first n indices in J such that

J1 ⊆ J2 ⊆ J3 ⊆ · · · ⊆ Jn ↗ J.

If |J | < ∞ then Jn = J for all n ≥ |J |. {Wj}j∈J is a sequence of closed subspacesof K. Also B(H, Wj) denote the collection of all bounded linear operators from Hinto Wj and Λj ∈ B(H, Wj) for all j ∈ J .

Recall that a family Λ = {Λj}j∈J is called a generalized frame or simply ag-frame for H with respect to {Wj}j∈J if there exist constants 0 < A ≤ B < ∞such that

A‖f‖2 ≤∑j∈J

‖Λjf‖2 ≤ B‖f‖2 ∀ f ∈ H. (1)

The constants A and B are called g-frame bounds. If the right-hand side of (1)holds, then Λ is said to be a g-Bessel sequence for H with respect to {Wj}j∈J , alsoif Wj is a closed subspace of H and Λj = πWj is the orthogonal projection of Honto Wj for all j ∈ J . Then {Wj}j∈J is called a fusion frame for H. We shall denotethe representation space associated with a g-Bessel sequence Λ as follows:

∑j∈J

⊕Wj

�2

=

{gj}j∈J | gj ∈ Wj and

∑j∈J

‖gj‖2 < ∞ .

The mapping TΛ : (∑

j∈J ⊕Wj)�2 → H defined by TΛ({gj}j∈J) =∑

j∈J Λ∗jgj is

called the synthesis operator and the adjoint of TΛ which is given by

T ∗Λ : H →

∑

j∈J

⊕Wj

�2

T ∗Λf = {Λjf}j∈J .

is said to be the analysis operator of Λ. By composing TΛ, T ∗Λ, we obtain the fusion

frame operator

SΛ : H → H SΛf = TΛT ∗Λf =

∑j∈J

Λ∗jΛjf,

which is a bounded, self-adjoint, positive and invertible operator and CIH ≤ SΛ ≤DIH. For more details about the theory and applications of frames and Riesz baseswe refer the readers to Ref. 2 and for fusion frames to Ref. 1, about g-frame theoryand its applications to Refs. 8, 10 and 11.

Atomic systems for subspaces and Pseudo-frames for subspaces were first intro-duced by Feichtinger and Werther in Ref. 6 and Li and Ogawa in Ref. 9 based onexamples arising in sampling theory. Recently Gavruta in Ref. 7 introduced atomicsystems for operators in Hilbert spaces and discussed some properties of them.

1450013-2

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading


Generalized frames for operators

In this paper we introduce g-frames and fusion frames for operators on Hilbertspaces. We develop the atomic systems theory to the situation of g-frames andfusion frames for operators.

The structure of this paper is as follows: In the rest of this section we will reviewthe necessary parts from g-orthonormal bases. In Sec. 2, we present g-frames andfusion frames for operators as generalizations of atomic systems for operators. Inthis section we generalize some of the known results in frame theory to generalizedframes and atomic systems for operators. In Sec. 3, we construct new g-framesassociated with a bounded operator from a given g-frame. Finally, Sec. 4 deals withRiesz decompositions in Hilbert spaces. In this section, we generalized some resultsof bases to Riesz decompositions.

We will start by stating the definitions and basic properties of g-orthonormalbases for a Hilbert space.

Definition 1.1. Let {Ξj ∈ B(H, Wj) | j ∈ J} be a sequence of operators. Then

(i) {Ξj}j∈J is called a g-complete set for H, if H = span{Ξ∗j (Wj)}j∈J .

(ii) {Ξj}j∈J is called a g-orthonormal system for H with respect to {Wj}j∈J , ifΞiΞ∗

jgj = δijgj for all i, j ∈ J and gj ∈ Wj .(iii) A g-complete and g-orthonormal system {Ξj}j∈J is called a g-orthonormal

basis for H with respect to {Wj}j∈J .(iv) If {Wj}j∈J is a family of closed subspaces of H and Ξj is orthogonal projection

on Wj . Then {Wj}j∈J is called orthonormal fusion basis for H.

Example 1.2. Let {ej}j∈N be an orthonormal basis for H, then

(i) The family of operators {Ξj}j∈N given by

Ξj : H → C, Ξjf = 〈f, ej〉is a g-orthonormal basis for H with respect to C.

(ii) Suppose that {Wj}j∈N is a family of subspaces of H which is defined byWj = span{e2j−1 + e2j}. Since the orthogonal projection on Wj is as πWj f =12 〈f, e2j−1 + e2j〉(e2j−1 + e2j). Thus πWiπ

∗Wj

gj = δijgj for all i, j ∈ N,gj ∈ Wj . Also from 〈e1 − e2, e2j−1 + e2j〉 = 0 for all j ∈ N implies thatH = span{π∗

Wj(Wj)}j∈J . Therefore {Wj}j∈N is an orthonormal fusion system

for H, but it is not an orthonormal fusion basis for H.(iii) Let {Wj}j∈N be a family of subspaces of H defined by Wj = span{e2j−1, e2j}.

Since the orthogonal projection on Wj is as πWj f = 〈f, e2j−1〉e2j−1+〈f, e2j〉e2j .Hence πWiπ

∗Wj

gj = δijgj for all i, j ∈ N, gj ∈ Wj . Also for every f ∈ H wehave f =

∑j∈N

πWj f , which implies that {Wj}j∈N is an orthonormal fusionbasis for H.

Example 1.3. Let N ∈ N,H = CN+1 and let {ek}N+1k=1 be the standard orthonor-

mal basis of H. Define Wj = span{∑N+1k=1k �=j

ek} and Ξj({ci}N+1i=1 ) = cj√

N

∑N+1k=1k �=j

ek.

1450013-3

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



Then Ξ∗j ({ci}N+1

i=1 ) = ( 1√N

∑N+1k=1k �=j

ck)ej forall {ci}N+1i=1 ∈ H and 1 ≤ j ≤ N +1. This

shows that span{Ξ∗j (Wj)}j∈J = span{ej}j∈N = H and ΞiΞ∗

jgj = δijgj for all j ∈ N

and gj ∈ Wj . Therefore {Ξj}j∈N is a g-orthonormal basis for H with respect to{Wj}N+1

j=1 .

Example 1.4. For every sequence of closed subspaces {Wj}j∈J of K. The familyof partial isometries {Ξj}j∈J defined by

Ξj :

∑

j∈J

⊕Wj

�2

→ Wj , Ξj({gk}k∈J ) = gj (2)

is called the standard g-orthonormal basis for (∑

j∈J ⊕Wj)�2 with respect to{Wj}j∈J . We also have

ΞjΞ∗i =

{I0 i = j

0 i = jand

∑j∈J

Ξ∗jΞj = I�2 , (3)

where I�2 denotes the identity operator on (∑

j∈J ⊕Wj)�2 and I0 denotes the iden-tity operator on Wj .

Corollary 1.5. Let {Ξj}j∈J be a g-orthonormal system for H with respect to{Wj}j∈J , then Ξj is onto and ‖Ξj‖ = 1 for all j ∈ J .

Proof. This follows from the definition and the fact that for any j ∈ J we haveΞjΞ∗

jΞj = Ξj .

Corollary 1.6. Let {Ξj}j∈J be a g-orthonormal system for H with respect to{Wj}j∈J , then the series

∑j∈J Ξ∗

jgj converges if and only if {gj}j∈J ∈ (∑

j∈J ⊕Wj)�2 and in this case we have∥∥∥∥

∑j∈J

Ξ∗jgj

∥∥∥∥2

=∑j∈J

‖gj‖2.

Proof. The equivalence follows immediately from the definition.

The following theorem is a characterization of orthonormal g-bases for H.

Theorem 1.7. Let Ξ = {Ξj}j∈J be a g-orthonormal system for H with respect to{Wj}j∈J . Then the following conditions are equivalent :

(i) Ξ is a g-orthonormal basis for H with respect to {Wj}j∈J .

(ii) f =∑

j∈J Ξ∗jΞjf ∀ f ∈ H.

(iii) ‖f‖2 =∑

j∈J ‖Ξ∗jΞjf‖2 ∀ f ∈ H.

(iv) ‖f‖2 =∑

j∈J ‖Ξjf‖2 ∀ f ∈ H.

1450013-4

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



(v) 〈f, g〉 =∑

j∈J 〈Ξjf, Ξjg〉 ∀ f, g ∈ H.(vi) H =

⊕j∈J Ξ∗

j (Wj).

Proof. If (i) is satisfied, then Corollary 1.6 guarantees that the operator

T :

∑

j∈J

⊕Wj

�2

→ H with T ({gj}j∈J ) =∑j∈J

Ξ∗jgj

is bounded and invertible. Therefore for every f ∈H there exists a sequence{gj}j∈J ∈ (

∑j∈J ⊕Wj)�2 such that f =

∑j∈J Ξ∗

jgj which implies that gj = Ξjf

and so (ii) holds. The implication (ii) ⇒ (iii) follows from the fact that for each j ∈ J

and f ∈ H we have (Ξ∗jΞj)2f = Ξ∗

jΞjf . The implications (iii) ⇒ (iv) ⇒ (v) followfrom the definition. To prove (v) ⇒ (vi) assume that f ∈ Ξ∗

i (Wi) ∩ Ξ∗j (Wj) for all

i = j ∈ J . Then there exist gi ∈ Wi and gj ∈ Wj such that f = Ξ∗i gi = Ξ∗

jgj hencegj = ΞjΞ∗

i gi = 0 which implies that f = 0. Moreover, we have f =∑

j∈J Ξ∗jΞjf for

every f ∈ H which implies (vi). Also the implication (vi) ⇒ (i) is obvious.

Corollary 1.8. Let T : H → U be a bounded linear operator and let {Ξj}j∈J bea g-orthonormal basis for H with respect to {Wj}j∈J . Then the family {Ξ′

j}j∈J

defined by Ξ′j = ΞjT

∗ for all j ∈ J, is an g-orthonormal basis for U with respect to{Wj}j∈J if and only if T is unitary.

Proof. The equivalence follows from Theorem 1.7.

2. G-Frames for Operators

In this section we introduce g-frames for operators, which allows one to reconstructelements from the range of a bounded linear operator. The frame situation of ourresults was considered by H. G. Feichtinger and T. Werther6 and L. Gavruta.7 Firstwe briefly recall some definitions and basic properties of frames for operators.

Let F = {fi}i∈I in H be a Bessel sequence with Bessel bound B and let Θ ∈B(H). Then F is called a Θ-frame in H if there exists a sequence of linear functionals{ci}i∈I such that

(i) ∃A > 0 with∑

i∈I |ci(f)|2 ≤ A‖f‖2,(ii) Θf =

∑i∈I ci(f)fi,

for all f ∈ H. The constants A and B are called Θ-frame bounds. The representa-tion space associated with a Θ-frame is �2(I) and the synthesis operator for F isdefined by

TF : �2(I) → H, TF ({ci}i∈I) =∑i∈I

cifi,

which is a bounded operator such that 1√A‖Θ‖ ≤ ‖TF‖ ≤ √

B and RΘ ⊆ RTF . Theadjoint T ∗

F is called the analysis operator given by T ∗Ff = {〈f, fi〉}i∈I . By composing

1450013-5

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



TF and T ∗F we obtain the Θ-frame operator

SF : H → H, SFf = TFT ∗Ff =

∑i∈I

〈f, fi〉fi,

which is a bounded self-adjoint operator and 1AΘΘ∗ ≤ SF ≤ BIH. For more details

about the Θ-frames we refer the readers to Ref. 7.

Definition 2.1. Let Θ ∈ B(H) and Λj ∈ B(H, Wj) for all j ∈ J , then {Λj}j∈J iscalled a Θ–g-frame in H with respect to {Wj}j∈J , if the following holds:

(i) The series∑

j∈J Λ∗jgj converges for all {gj}j∈J ∈ (

∑j∈J ⊕Wj)�2 .

(ii) There exists B > 0 such that for every f ∈ H there exists {gj}j∈J ∈(∑

j∈J ⊕Wj)�2 such that

Θf =∑j∈J

Λ∗jgj and

∑j∈J

‖gj‖2 ≤ B‖f‖2. (4)

Moreover, if {Wj}j∈J is a family of closed subspaces of H and Λj = πWj then{Wj}j∈J is called a Θ-fusion frame in H.

Corollary 2.2. Every Θ–g-frame is a g-Bessel sequence for H.

Proof. Suppose that Λ = {Λj}j∈J is a Θ–g-frame in H with respect to {Wj}j∈J

and {Jn}∞n=1 is a family of finite subsets of J such that J1 ⊆ J2 ⊆ J3 ⊆ · · · ⊆Jn ↗ J . Consider the sequence of bounded linear operators Tn : (

∑j∈J ⊕Wj)�2 →

H, Tn({gj}j∈J) =∑

j∈JnΛ∗

jgj . Clearly Tn → TΛ pointwise as n → ∞, where TΛ isthe synthesis operator of Λ. Thus TΛ is bounded by Uniform Boundedness Principleand T ∗

Λ(f) = {Λjf}j∈J . Therefore∑

j∈J ‖Λjf‖2 ≤ ‖TΛ‖2‖f‖2.

The following result shows that every bounded operator Θ ∈ B(H) has a Θ–g-frame.

Theorem 2.3. Every Θ ∈ B(H) has a Θ–g-frame.

Proof. Let {Ξj}j∈J be a g-orthonormal basis for H with respect to {Wj}j∈J . Iffor all j ∈ J and f ∈ H we define Λj = ΞjΘ∗ and gj = Ξjf , then by Theorem 1.7we have

Θf = Θ

∑

j∈J

Ξ∗jΞjf

=

∑j∈J

Λ∗jgj.

Further for every {gj}j∈J ∈ (∑

j∈J ⊕Wj)�2 we have∥∥∥∥∑j∈J

Λ∗jgj

∥∥∥∥2

≤ ‖Θ‖2

∥∥∥∥∑j∈J

Ξ∗jgj

∥∥∥∥2

= ‖Θ‖2∑j∈J

‖gj‖2.

Thus {Λj}j∈J is a Θ–g-frame with respect to {Wj}j∈J .

1450013-6

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



The next proposition is a fundamental result in the study of the Θ–g-frames,which is taken from Ref. 4.

Proposition 2.4. Let T ∈ B(H1,H) and S ∈ B(H2,H), then the following condi-tions are equivalent :

(i) RT ⊂ RS.(ii) TT ∗ ≤ λ2SS∗ for some λ ≥ 0.(iii) There exists a bounded operator U ∈ B(H1,H2) such that T = SU .

Moreover, if (i), (ii) and (iii) are valid, then there exists a unique operator U sothat

(1) ‖U‖2 = inf{µ |TT ∗ ≤ µSS∗}.(2) NT = NU .(3) RU ⊂ RS∗.

The following theorem is a characterization of the Θ–g-frames.

Theorem 2.5. Let Θ ∈ B(H). Then the following statements are equivalent :

(i) {Λj}j∈J is a Θ–g-frame with respect to {Wj}j∈J .(ii) There exist constants 0 < A ≤ B < ∞ such that

A‖Θ∗f‖2 ≤∑j∈J

‖Λjf‖2 ≤ B‖f‖2 ∀ f ∈ H.

(iii) {Λj}j∈J is a g-Bessel sequence for H with respect to {Wj}j∈J and there existsa g-Bessel sequence {Γj}j∈J for H with respect to {Wj}j∈J such that

Θf =∑j∈J

Λ∗jΓjf ∀ f ∈ H.

Proof. (i) ⇒ (ii) By Definition 2.1 there exists B > 0 such that for everyg ∈ H there exists {gj}j∈J ∈ (

∑j∈J ⊕Wj)�2 such that Θg =

∑j∈J Λ∗

jgj and∑j∈J ‖gj‖2 ≤ B‖g‖2. By using Cauchy–Schwarz inequality for all f ∈ H we

compute

‖Θ∗f‖2 = sup‖g‖=1

|〈Θ∗f, g〉|2 = sup‖g‖=1

∣∣∣∣∣∣∑j∈J

〈Λjf, gj〉∣∣∣∣∣∣2

≤ sup‖g‖=1

∑

j∈J

‖Λjf‖2

∑

j∈J

‖gj‖2

≤ B

∑j∈J

‖Λjf‖2.

By setting A = B−1 and Corollary 2.2, (ii) follows.

1450013-7

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



(ii) ⇒ (iii) Since Λ = {Λj}j∈J is a g-Bessel sequence for H with respect to {Wj}j∈J

hence the synthesis operator TΛ is bounded and we have

A‖Θ∗f‖2 ≤∑j∈J

‖Λjf‖2 = ‖T ∗Λf‖2,

for all f ∈ H. This implies that ΘΘ∗ ≤ 1ATΛT ∗

Λ. Now Proposition 2.4 implies thatthere exists a bounded operator Γ : H → (

∑j∈J ⊕Wj)�2 such that Θ = TΛΓ.

If we define Γj = ΞjΓ, where {Ξj}j∈J is the standard g-orthonormal basis for(∑

j∈J ⊕Wj)�2 with respect to {Wj}j∈J . Then {Γj}j∈J is a g-Bessel sequence forH with respect to {Wj}j∈J and for every f ∈ H we have

Θf = TΛΓf = TΛ({ΞjΓf}j∈J) =∑j∈J

Λ∗jΓjf.

(iii) ⇒ (i) Let B be g-Bessel bound of {Γj}j∈J . Taking gj = Γjf for all j ∈ J andf ∈ H, then we have

Θf =∑j∈J

Λ∗jgj and

∑j∈J

‖gj‖2 ≤ B‖f‖2.

This shows that {Λj}j∈J is a Θ–g-frame with respect to {Wj}j∈J .

Let {Λj}j∈J be a Θ–g-frame in H with respect to {Wj}j∈J , then the constantsA and B in Theorem 2.5(ii) is called the Θ–g-frame bounds for it and the g-Besselsequence {Γj}j∈J defined as Theorem 2.5(iii) is called a dual Θ–g-frame for {Λj}j∈J .

Corollary 2.6. Let Λ = {Λj}j∈J be a Θ–g-frame in H with respect to {Wj}j∈J

with bounds A, B and dual Θ–g-frame Γ = {Γj}j∈J . Then Γ is a Θ∗–g-frame in Hwith respect to {Wj}j∈J with lower bound 1

B .

Proof. Since Γ is a dual Θ–g-frame for Λ hence Θ = TΛT ∗Γ and for every f ∈ H

we have

‖Θf‖2 = sup‖g‖=1

|〈Θf, g〉|2 = sup‖g‖=1

∣∣∣∣∣∣⟨∑

j∈J

Λ∗jΓjf, g

⟩∣∣∣∣∣∣2

= sup‖g‖=1

∣∣∣∣∣∣∑j∈J

〈Γjf, Λjg〉∣∣∣∣∣∣2

≤ sup‖g‖=1

∑

j∈J

‖Γjf‖2

∑

j∈J

‖Λjg‖2

≤ B

∑j∈J

‖Γjf‖2.

Now the conclusion follows from Theorem 2.5.

The following theorem is a characterization of Θ–g-frames using bounded linearoperators.

Theorem 2.7. Λ = {Λj}j∈J is a Θ–g-frame with respect to {Wj}j∈J , if and onlyif there exists T ∈ B((

∑j∈J ⊕Wj)�2 ,H) such that RΘ ⊂ RT and Λj = ΞjT

∗ for

1450013-8

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



all j ∈ J, where {Ξj}j∈J is the standard g-orthonormal basis for (∑

j∈J ⊕Wj)�2

with respect to {Wj}j∈J .

Proof. Suppose that Λ is a Θ–g-frame with respect to {Wj}j∈J . Then TΛ isbounded and RΘ ⊂ RTΛ and Λj = ΞjT

∗Λ for all j ∈ J . For the opposite impli-

cation, assume that Λj = ΞjT∗ for all j ∈ J , where T ∈ B((

∑j∈J ⊕Wj)�2 ,H) and

RΘ ⊂ RT . Then by Proposition 2.4 there exists A > 0 such that AΘΘ∗ ≤ TT ∗

hence for all f ∈ H we have

A‖Θ∗f‖2 ≤ ‖T ∗f‖2 =∑j∈J

‖ΞjT∗f‖2 =

∑j∈J

‖Λjf‖2.

Now the conclusion follows from Theorem 2.5.

Corollary 2.8. A g-Bessel sequence Λ = {Λj}j∈J is a Θ–g-frame in H with respectto {Wj}j∈J , if and only if RΘ ⊆ RTΛ . Moreover if Λ is a g-frame sequence, then it isa Θ–g-frame in H with respect to {Wj}j∈J , if and only if RΘ ⊆ span{Λ∗

j(Wj)}j∈J .

Proof. This follows immediately from Proposition 2.4 and Theorem 2.7.

Corollary 2.9. Let {Λj}j∈J be a Θ–g-frame in H with respect to {Wj}j∈J . Thenfor every Ψ ∈ B(H) it is also a ΘΨ–g-frame in H with respect to {Wj}j∈J .

Proof. Since RΘΨ ⊆ RΘ thus the claim follows immediately from Corollary 2.8.

Corollary 2.10. Suppose that Λ = {Λj}j∈J is a Θ–g-frame in H with respect to{Wj}j∈J and Θ is also right-invertible. Then it is a g-frame for H with respect to{Wj}j∈J .

Proof. Since Θ is right-invertible on H hence RΘ = H which implies that RTΛ =H. Therefore Λ is a g-frame for H with respect to {Wj}j∈J .

Theorem 2.11. Let Λ = {Λj}j∈J be a Θ–g-frame with respect to {Wj}j∈J withbounds A and B. Then there exists a g-Bessel sequence Γ = {Γj}j∈J for H withrespect to {Wj}j∈J such that

AΘΘ∗f =∑j∈J

(Λ∗jΛj − Γ∗

jΓj)f ∀ f ∈ H.

Proof. By the hypothesis the Θ–g-frame operator

SΛ : H → H, SΛf =∑j∈J

Λ∗jΛjf

is well-defined, bounded, self-adjoint and AΘΘ∗ ≤ SΛ. Therefore, SΛ − AΘΘ∗ is apositive operator on H. Define Γj = Ξj(SΛ−AΘΘ∗)

12 for each j ∈ J where {Ξj}j∈J

1450013-9

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



is a g-orthonormal basis for H with respect to {Wj}j∈J . For every f ∈ H we alsohave ∑

j∈J

‖Γjf‖2 = ‖(SΛ − AΘΘ∗)12 f‖2 = 〈(SΛ − AΘΘ∗)f, f〉

≤ (‖SΛ‖ + A‖Θ‖2)‖f‖2 ≤ (B + A‖Θ‖2)‖f‖2.

Moreover, we compute∑j∈J

(Λ∗jΛj − Γ∗

jΓj)f =∑j∈J

Λ∗jΛjf −

∑j∈J

(SΛ − AΘΘ∗)12 Ξ∗

jΞj(SΛ − AΘΘ∗)12 f

= SΛf − (SΛ − AΘΘ∗)f = AΘΘ∗f.

Theorem 2.12. A Bessel sequence F = {fj}j∈J is a Θ-frame for H if and only ifRΘ ⊆ RTF . Moreover, if F is a frame sequence, then F is a Θ-frame for H if andonly if RΘ ⊆ span{fj}j∈J .

Proof. The first part follows immediately from Theorem 3 in Ref. 7 and Proposi-tion 2.4. For the other implication, assume that RΘ ⊆ RTF , then by Proposition 2.4there exists an operator Γ ∈ B(H, �2(J)) such that Θ = TFΓ. Now for each f ∈ Hwe have

‖Θ∗f‖2 = ‖Γ∗T ∗Ff‖2 = sup

‖g‖=1

|〈Γ∗T ∗Ff, g〉|2

= sup‖g‖=1

∣∣∣∣∣∣⟨

Γ∗

∑

j∈J

〈f, fj〉δj

, g

⟩∣∣∣∣∣∣2

= sup‖g‖=1

∣∣∣∣∣∣∑j∈J

〈f, fj〉〈Γ∗δj , g〉∣∣∣∣∣∣2

≤ sup‖g‖=1

∑

j∈J

|〈f, fj〉|2∑

j∈J

|〈Γ∗δj , g〉|2

= ‖Γ‖2∑j∈J

|〈f, fj〉|2,

where {δj}j∈J is the standard orthonormal basis for �2(J). This shows that F is aΘ-frame for H.

In the following we define a family of local g-atoms for a closed subspace, whichcan be viewed as a generalization of local atoms for subspaces in Ref. 6.

Definition 2.13. Let Λ = {Λj}j∈J be a g-Bessel sequence for H with respectto {Wj}j∈J and let H0 be a closed subspace of H. Then Λ is called a family oflocal g-atoms for H0 with respect to {Wj}j∈J , if there exists a g-Bessel sequence

1450013-10

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



Γ = {Γj}j∈J for H0 with respect to {Wj}j∈J such that f =∑

j∈J Λ∗jΓjf for all

f ∈ H0. In this case if B is the g-Bessel bound of Γ. Then the family {(Λj, Γj)}j∈J

is called a g-atomic decomposition for H0 with respect to {Wj}j∈J with g-atomicbound B.

In the following we generalize two results of Feichtinger and Werther in Ref. 6to the situation of the local g-atoms.

Theorem 2.14. Let Λ = {Λj}j∈J be a g-Bessel sequence for H with respect to{Wj}j∈J and let H0 be a closed subspace of H. Then the following conditions areequivalent :

(i) Λ is a family of local g-atoms for H0 with respect to {Wj}j∈J .(ii) Λ is a PH0-g-frame with respect to {Wj}j∈J , where PH0 is the orthogonal

projection from H onto H0.(iii) There exists A > 0 such that A‖PH0f‖2 ≤∑j∈J ‖Λjf‖2 for all f ∈ H.(iv) There exists g-Bessel sequence Γ = {Γj}j∈J for H with respect to {Wj}j∈J

such that

PH0f =∑j∈J

Λ∗i Γjf ∀ f ∈ H.

(v) There exists a bounded linear operator T : (∑

j∈J ⊕Wj)�2 →H such that H0 ⊂RT and Λj = ΞjT

∗ for all j ∈ J, where {Ξj}j∈J is the standard g-orthonormalbasis for (

∑j∈J ⊕Wj)�2 with respect to {Wj}j∈J .

Proof. Suppose that (i) holds then there exists a g-Bessel sequence {Ψj}j∈J forH0 with respect to {Wj}j∈J such that f =

∑j∈J Λ∗

i Ψjf for all f ∈ H0. If we definegj = ΨjPH0f then we have PH0f =

∑j∈J Λ∗

i gj for all f ∈ H, hence (i) ⇒ (ii).Also (ii) ⇒ (iii) ⇒ (iv) follows from Theorem 2.5 and the implication (iv) ⇒ (i)obtains from the definition. Finally the equivalence (ii) ⇔ (v) follows immediatelyfrom Theorem 2.7.

3. Constructing New Frames and g-Frames for Operators

In this section we present several ways for constructing a new Θ-frame and Θ–g-frame from a given Θ-frame and Θ–g-frame.

Theorem 3.1. Let Λ = {Λj}j∈J be a Θ–g-frame in H with respect to {Wj}j∈J ,

and let Γj ∈ B(H, Wj) for all j ∈ J . If U : H → H defined by

Uf =∑j∈J

(Γ∗jΓjf − Λ∗

jΛjf) ∀ f ∈ H

is a positive bounded operator. Then Γ = {Γj}j∈J is a Θ–g-frame with respect to{Wj}j∈J .

1450013-11

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



Proof. Let A and B be the Θ–g-frame bounds for Γ. Since U is a bounded self-adjoint operator on H, hence T : H → H defined by T = TΛT ∗

Λ + U is a linear,bounded and self-adjoint operator. Thus for all f ∈ H we have∑

j∈J

‖Γjf‖2 ≤ ‖T ‖‖f‖2 ≤ (‖TΛ‖2 + ‖U‖)‖f‖2 ≤ (B + ‖U‖)‖f‖2.

On the other hand, since U is positive thus we obtain

A‖Θ∗f‖2 ≤∑j∈J

‖Λjf‖2 = 〈TΛT ∗Λf, f〉 ≤ 〈Tf, f〉 =

∑j∈J

‖Γjf‖2.

Theorem 3.2. Suppose that Θ ∈ B(H), Λj ∈ B(H, Wj) for all j ∈ J . If T ∈ B(K)is an operator with closed range such that span{Wj}j∈J ⊆ N⊥

T . Then {Λj}j∈J is aΘ–g-frame for H with respect to {Wj}j∈J if and only if {TΛj}j∈J is a Θ–g-framefor H with respect to {TWj}j∈J .

Proof. Since T †T is orthogonal projection on N⊥T where T † is the pseudo-inverse

of T . Thus for any f ∈ H we have

‖T †‖−2∑j∈J

‖Λjf‖2 ≤∑j∈J

‖TΛjf‖2 ≤ ‖T ‖2∑j∈J

‖Λjf‖2.

From this the result follows at once.

Corollary 3.3. Suppose that Θ ∈ B(H), Λj ∈ B(H, Wj) for all j ∈ J . If T ∈B(K) is an invertible operator. Then {Λj}j∈J is a Θ–g-frame for H with respect to{Wj}j∈J if and only if {TΛj}j∈J is a Θ–g-frame for H with respect to {TWj}j∈J .

The following results are generalizations of Theorem 3.2 of Ref. 1 and Theo-rem 2.2 of Ref. 8.

Theorem 3.4. Let Θ ∈ B(H), Λj ∈ B(H, Wj), j ∈ J and let {fjk}k∈Ij be aframe for Wj with the frame bounds Aj and Bj such that 0 < A = infj∈J Aj ≤supj∈J Bj = B < ∞. Suppose that {ejk}k∈Ij is an orthonormal basis for each Wj.Then the following conditions are equivalent.

(i) {Λj}j∈J is a Θ–g-frame for H with respect to {Wj}j∈J .(ii) {Λ∗

jfjk}j∈J,k∈Ij is a Θ-frame for H.(iii) {Λ∗

jejk}j∈J,k∈Ij is a Θ-frame for H.

Proof. First we prove (i) ⇔ (ii). This claim follows immediately from the fact thatfor each f ∈ H we have

A∑j∈J


Aj‖Λjf‖2 ≤∑j∈J

∑k∈Ij

|〈Λjf, fjk〉|2

=∑j∈J

∑k∈Ij

|〈f, Λ∗jfjk〉|2 ≤

∑j∈J

Bj‖Λjf‖2 ≤ B∑j∈J

‖Λjf‖2.

This shows that if {Λj}j∈J is a Θ–g-frame for H with respect to {Wj}j∈J withbounds C and D. Then {Λ∗

jfjk}j∈J,k∈Ij is a Θ-frame for H with bounds AC

1450013-12

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



and BD . Moreover, if {Λ∗jfjk}j∈J,k∈Ij is a Θ-frame with bounds C and D, the

calculation above implies that {Λj}j∈J is a Θ–g-frame with respect to {Wj}j∈J

with bounds CB and D

A . To prove (i) ⇔ (iii) we also have∑j∈J

∑k∈Ij

|〈f, Λ∗jejk〉|2 =

∑j∈J

‖Λjf‖2, ∀ f ∈ H,

from which it follows that (i) and (iii) are equivalent.

Theorem 3.5. Let Θ ∈ B(H), Λj ∈ B(H, Wj), j ∈ J and let {Γjk}k∈Ij be ag-frame for Wj with respect to {Vjk}k∈Ij with the g-frame bounds Aj and Bj suchthat 0 < A = infj∈J Aj ≤ supj∈J Bj = B < ∞. Suppose that {Ξjk}k∈Ij is ag-orthonormal basis for each Wj with respect to {Vjk}k∈Ij . Then the following con-ditions are equivalent.

(i) {Λj}j∈J is a Θ–g-frame for H with respect to {Wj}j∈J .(ii) {ΓjkΛj}j∈J,k∈Ij is a Θ–g-frame for H with respect to {Vjk}j∈J,k∈Ij .(iii) {ΞjkΛj}j∈J,k∈Ij is a Θ–g-frame for H with respect to {Vjk}j∈J,k∈Ij .

Proof. (i) ⇔ (ii) Since for every j ∈ J , {Γjk}k∈Ij is a g-frame for Wj with respectto {Vjk}k∈Ij with the g-frame bounds Aj and Bj hence we have

A∑j∈J


Aj‖Λjf‖2 ≤∑j∈J

∑k∈Ij

‖ΓjkΛjf‖2

≤∑j∈J

Bj‖Λjf‖2 ≤ B∑j∈J

‖Λjf‖2.

(i) ⇔ (iii) Since for every f ∈ H we have∑j∈J

∑k∈Ij

‖ΞjkΛjf‖2 =∑j∈J

‖Λjf‖2.

Thus (i) and (iii) are equivalent.

Corollary 3.6. Let Θ ∈ B(H), Λj ∈ B(H, Wj), j ∈ J and let {Vjk}k∈Ij be afusion frame for Wj with the fusion frame bounds Aj and Bj such that 0 < A =infj∈J Aj ≤ supj∈J Bj = B < ∞. Then the following conditions are equivalent.

(i) {Λj}j∈J is a Θ–g-frame for H with respect to {Wj}j∈J .(ii) {πVjk

Λj}j∈J,k∈Ij is a Θ–g-frame for H with respect to {Vjk}j∈J,k∈Ij .

Proof. The proof is similar to the proof of Theorem 3.5.

In the sequel, we study two results of stability of Θ–g-frames under perturbation.

Theorem 3.7. Let Λ = {Λj}j∈J be a Θ–g-frame for H with respect to {Wj}j∈J

with bounds A and B. Suppose that Γj ∈ B(H, Wj) for all j ∈ J and there exist

1450013-13

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



constants λ1, λ2, µ ≥ 0 such that max{λ1 + µ√A

, λ2} < 1 and

∑

j∈J

‖(Λj − Γj)f‖2

12

≤ λ1

∑

j∈J

‖Λjf‖2

12

+ λ2

∑

j∈J

‖Γjf‖2

12

+ µ‖Θ∗f‖.

Then Γ = {Γj}j∈J is a Θ–g-frame for H with respect to {Wj}j∈J with bounds

A

(1 −

λ1 + λ2 + µ√A

1 + λ2

)2

and B

(1 +

λ1 + λ2 + µ√A

1 − λ2

)2

.

Proof. For all f ∈ H we have∑

j∈J

‖Γjf‖2

12

≥∑

j∈J

‖Λjf‖2

12

−∑

j∈J

‖(Λj − Γj)f‖2

12

≥(

1 − λ1 − µ√A

)∑j∈J

‖Λjf‖2

12

− λ2

∑

j∈J

‖Γjf‖2

12

.

Hence ∑

j∈J

‖Γjf‖2

12

≥(

1 −λ1 + λ2 + µ√

A

1 + λ2

)∑

j∈J

‖Λjf‖2

12

.

Similarly we obtain∑

j∈J

‖Γjf‖2

12

≤(

1 +λ1 + λ2 + µ√

A

1 − λ2

)∑j∈J

‖Λjf‖2

12

.

From this the result follows at once.

Corollary 3.8. Let F = {fj}j∈J be a Θ-frame for H with bounds A, B and letG = {gj}j∈J be a sequence in H. If there exist constants λ1, λ2, µ ≥ 0 such thatmax{λ1 + µ√

A, λ2} < 1 and

∑

j∈J

|〈f, fj − gj〉|2

12

≤ λ1

∑

j∈J

|〈f, fj〉|2

12

+ λ2

∑

j∈J

|〈f, gj〉|2

12

+ µ‖Θ∗f‖.

Then G = {gj}j∈J is a Θ-frame for H with bounds

A

(1 −

λ1 + λ2 + µ√A

1 + λ2

)2

and B

(1 +

λ1 + λ2 + µ√A

1 − λ2

)2

.

Proof. The proof is similar to the proof of Theorem 3.7.

1450013-14

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



4. Characterizations of Riesz Decompositions

The concept of Riesz decomposition was introduced by Casazza and Kutyniok inRef. 1 as a natural generalization of the notion of Riesz bases in Hilbert spaces. Inthis section, we generalized some results of bases to Riesz decompositions.

Definition 4.1. Let {Wj}j∈J be a family of closed subspaces of H, then {Wj}j∈J

is called a Riesz decomposition or simply a R-decomposition for H if for any f ∈ Hthere exists a unique sequence {gj : gj ∈ Wj}j∈J such that

f =∑j∈J

gj. (5)

It is clear that every gj ∈ Wj in (5) is a linear operator of f . If we denote this linearoperator by

PWj : H → Wj , PWj f = gj . (6)

Then we have f =∑

j∈J PWj f . In this case we say that {(Wj , PWj )}j∈J is aR-decomposition system for H.

Example 4.2. Let H = R3 and define the subspaces W1, W2, W3 by

W1 = {(2x, x, 0) |x ∈ R}, W2 = {(3y, y, 0) | y ∈ R},W3 = {(z, z, z) | z ∈ R}.

It is easy to check that if (a, b, c) = (2x, x, 0) + (3y, y, 0) + (z, z, z) then

x = −a + 3b − 2c, y = a − 2b + c, z = c.

Thus {W1, W2, W3} is a R-decomposition for R3. Now if we define

PW1(x, y, z) = (−2x + 6y − 4z,−x + 3y − 2z, 0),

PW2(x, y, z) = (3x − 6y + 3z, x − 2y + z, 0),

PW3(x, y, z) = (z, z, z).

Then {(Wi, PWi)}3i=1 is a R-decomposition system for H.

Example 4.3. Let N ∈ N, H = CN and let {ek}N

k=1 be the standard orthonormalbasis of C

N . Then

(1) Define the subspaces {Wi}Ni=1 by Wi = span{∑i

k=1 ek}. Since from {ci}Ni=1 =∑N

i=1 xi(∑i

k=1 ek) we obtain

xi = ci − ci+1, (1 ≤ i ≤ N − 1) and xN = cN ,

1450013-15

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



hence {Wi}Ni=1 is a R-decomposition for CN . Moreover, if we define

PWi({cj}Nj=1) =

i∑k=1

(ci − ci+1)ek, (1 ≤ i ≤ N − 1)

PWN ({cj}Nj=1) =

N∑k=1

cNek.

Then {(Wi, PWi)}Ni=1 is a R-decomposition system for CN .

(2) Let N > 1, Wj = span{∑Nk=1,k �=j ek} and

PWj : H → Wj ,

PWj f =1

N − 1

⟨f,

N∑k=1k �=j

ek − (N − 2)ej

⟩ N∑

k=1k �=j

ek

,

for all 1 ≤ j ≤ N . Then {(Wj , PWj )}Nj=1 is a R-decomposition system for CN .

In the next theorem we show that the operators PWi of a R-decompositionsystem are continuous projections.

Theorem 4.4. Let {(Wj , PWj )}j∈J be a R-decomposition system for H. Then

PWj ∈ B(H, Wj) and PWiPWj = δijPWj ∀ i, j ∈ J,

where δij is the Kronecker delta.

Proof. Define the space

A =

{gj}j∈J | gj ∈ Wj ,

∑j∈J

gj is convergent

,

with the norm defined by �{gj}j∈J� = supn∈N ‖∑i∈Jngi‖. Since for all {gj}j∈J ∈ A

we have∑

j∈J gj = limn→∞∑

j∈Jngj and convergent sequences are bounded, hence

�{gj}j∈J� < ∞, which this shows that � · � is well defined. It is easy to see thatA with the pointwise operations and the above norm is a normed space. First weshow that A is complete with respect to this norm. To see this, let {un}n∈N be aCauchy sequence in A and let un = {gnj}j∈J , then given any ε > 0, there exists anumber N1 such that �un − um� < ε for all m, n ≥ N1. This yields that for everyj ∈ Jk, k ≥ 2 and n, m ≥ N1

‖gnj − gmj‖ ≤∥∥∥∥∑i∈Jk

(gni − gmi)∥∥∥∥+

∥∥∥∥∑

i∈Jk−1

(gni − gmi)∥∥∥∥ ≤ 2�un − um� < 2ε.

Also for j ∈ J1 we have ‖gnj − gmj‖ ≤ �un − um�. It follows that {gnj}n∈N

is a Cauchy sequence in Wj and thus convergent. Let gj ∈ Wj such that gj =

1450013-16

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



limn→∞ gnj and u = {gj}j∈J . Our goal is to show that un → u in the norm of A.Fix m, n ≥ N1, k ∈ N and define

vmn =∑j∈Jk

(gmj − gnj) and vn =∑j∈Jk

(gj − gnj).

Since ‖vmn‖ < ε hence we have

‖vmn − vn‖ =∥∥∥∥∑j∈Jk

(gmj − gj)∥∥∥∥ ≤

∑j∈Jk

‖gmj − gj‖ → 0 as m → ∞.

Thus ‖vn‖ = limm→∞ ‖vmn‖ ≤ ε, which implies that

�un − u� = supk∈N

∥∥∥∥∑j∈Jk

(gj − gnj)∥∥∥∥ ≤ ε. (7)

Since uN1 ∈ A so the series∑

j∈J gN1j is convergent. Thus, there is an N2 suchthat ∥∥∥∥

∑j∈Jn−Jm

gN1j

∥∥∥∥ ≤ ε, ∀n ≥ m ≥ N2.

Put N = max{N1, N2}. Then, it is easy to see that∥∥∥∥∑

j∈Jn−Jm

gj

∥∥∥∥ ≤∥∥∥∥∑j∈Jn

(gj − gN1j)∥∥∥∥+

∥∥∥∥∑

j∈Jm

(gj − gN1j)∥∥∥∥

+∥∥∥∥

∑j∈Jn−Jm

gN1j

∥∥∥∥ < ε + ε + ε = 3ε

for all n ≥ m ≥ N . Therefore∑

j∈J gj is convergent, which this shows u ∈ A.Finally, by (7), it follows that un → u in the norm of A, so A is complete. Nowdefine the mapping

T : A → H with T ({gj}j∈J) =∑j∈J

gj.

Since {Wj}j∈J is a R-decomposition for H hence T is clearly well-defined, linear,and it is bijective. Further, if {gj}j∈J ∈ A, then

‖T ({gj}j∈J )‖ =∥∥∥∥∑j∈J

gj

∥∥∥∥ = limn→∞

∥∥∥∥∑j∈Jn

gj

∥∥∥∥ ≤ supk∈N

∥∥∥∥∑j∈Jk

gj

∥∥∥∥ = �{gj}j∈J�,

so T is bounded. The open mapping theorem implies that T is a topological isomor-phism of A onto H. Now suppose that f =

∑j∈J PWj f is a fixed, arbitrary element

of H, where elements PWj f are unique, so T−1 is given by T−1f = {PWj f}j∈J .If Sn =

∑j∈Jn

PWj is the partial sum operators for all n ∈ N, then we have

‖Snf‖ ≤ supk∈N

‖Skf‖ = supk∈N

∥∥∥∥∑j∈Jk

PWj f∥∥∥∥

= �{PWj f}j∈J� = �T−1f� ≤ ‖T−1‖‖f‖.

1450013-17

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



Therefore Sn : H → H is bounded, and ‖Sn‖ ≤ ‖T−1‖. Now for every j ∈ Jn, n ≥ 2we have PWj = Sn − Sn−1 which this implies

‖PWj f‖ ≤ ‖Snf‖ + ‖Sn−1f‖ ≤ 2‖T−1‖‖f‖.Thus ‖PWj‖ ≤ 2‖T−1‖ < ∞. Since for j ∈ J1 we have PWj = S1. Consequently,each PWj is bounded and ‖PWj‖ ≤ 2‖T−1‖ for all j ∈ J . Moreover, by uniquenesswe have PWigj = δijgj for all gj ∈ Wj hence PWiPWj = δijPWj for all i, j ∈ J .

If {(Wj , PWj )}j∈J is a R-decomposition system for H. Then {P ∗Wj

(Wj)}j∈J

is a family of closed subspaces of H. The following theorem shows that{(P ∗

Wj(Wj), P ∗

Wj)}j∈J is also a R-decomposition system for H.

Theorem 4.5. If {(Wj , PWj )}j∈J is a R-decomposition system for H, then {(P ∗Wj

×(Wj), P ∗

Wj)}j∈J is also a R-decomposition system for H.

Proof. By Theorem 4.4 for every i, j ∈ J , we obtain P ∗Wi

P ∗Wj

= δijP∗Wj

. Iff ⊥ span{P ∗

Wj(Wj)}j∈J , then ‖PWj f‖2 = 〈f, P ∗

WjPWj f〉 = 0, and so PWj f = 0

for all j ∈ J , hence f =∑

j∈J PWj f = 0. It follows that H = span{P ∗Wj

(Wj)}j∈J .Now if f ∈ H then f = limn→∞

∑j∈J P ∗

Wjgn

j for some {gnj }∞n=1 ⊆ Wj , where

P ∗Wj

gnj = 0 for at most finitely many indices j ∈ J . Since P ∗

Wjis continu-

ous on H therefore P ∗Wj

f = limn→∞ P ∗Wj

gnj . From this we obtain

∑j∈J P ∗

Wjf =

limn→∞∑

j∈J PWj gnj = f . Since we have P ∗

WiP ∗

Wj= δijP

∗Wj

, it follows that{(P ∗

Wj(Wj), P ∗

Wj)}j∈J is a R-decomposition system for H.

Theorem 4.6. Let {(Wj , PWj )}j∈J be a R-decomposition system for H and let T :H → K be a bounded invertible operator such that Vj = TWj and QVj = TPWj T

−1

for all j ∈ J . Then {(Vj , QVj )}j∈J is a R-decomposition system for K.

Proof. Suppose that f ∈ K, then we can write f = Tg for some g ∈ H. Byhypothesis g has a unique expansion to form g =

∑j∈ gj for some sequence {gj :

gj ∈ Wj}j∈J which implies that f has a unique expansion of the form f =∑

j∈J fj

with fj = Tgj. We also have

QVifj = TPWiT−1fj = TPWigj = T (δijgj) = δijfj .

From this the result follows.

Theorem 4.7. Let {(Wj , PWj )}j∈J be a R-decomposition system for H. Then thereis a fusion orthonormal basis {Vj}j∈J for H and a bounded invertible operatorT ∈ B(H,H) such that

Wj = TVj and PWj = TπVjT−1 ∀ j ∈ J.

Proof. For any j ∈ J , let {ejk}k∈Ij be an orthonormal basis for Wj . ByTheorem 4.5 in Ref. 1 the sequence {ejk}j∈J,k∈Ij is a Riesz basis for H and hence

1450013-18

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



we can write it in the form {Tujk}j∈J,k∈Ij for some orthonormal basis {ujk}j∈J,k∈Ij

of H, where T is an invertible operator on H. If we define Vj = span{ujk}k∈Ij then{Vj}j∈J is a fusion orthonormal basis for H and Wj = TVj. Moreover for everyf ∈ H we have

f = TT−1f = T

∑

j∈J

πVj T−1f

=

∑j∈J

TπVj T−1f,

which implies that PWj = TπVj T−1.

The stability of bases is important in practice and is therefore studied widelyby many authors, e.g., see Ref. 12. In the rest of this section we study the stabilityof R-decompositions under perturbations. First we generalized a result of Paley–Wiener12 to the situation of R-decomposition.

Theorem 4.8. Let {Wj}j∈J be a R-decomposition for H and suppose that {Vj}j∈J

is a family of closed subspaces of H such that∥∥∥∥∑j∈Jn

(gj − fj)∥∥∥∥ ≤ λ

∥∥∥∥∑j∈Jn

gj

∥∥∥∥ ∀ gj ∈ Wj , fj ∈ Vj , n ∈ N,

for some constant 0 ≤ λ < 1. Then {Vj}j∈J is a R-decomposition for H.

Proof. By the assumption the series∑

j∈J (gj − fj) is convergent whenever theseries

∑j∈J gj is convergent. Thus the mapping T : H → H defined by

Tf =∑j∈j

(gj − πVj gj) ∀ f =∑j∈J

gj,

is well-defined and bounded on H and ‖Tf‖ = ‖∑j∈J (gj − πVj gj)‖ ≤ λ‖f‖. Itfollows that ‖T ‖ ≤ λ < 1 and so the operator IH − T is invertible. Since (IH − T )Wj = Vj , the result follows from Theorem 4.6.

Acknowledgments

The authors thank the referee for useful remarks and interesting comments. Thefirst author was supported in part by a grant from Scientific Research of IslamicAzad University, Central Tehran Branch.

References

1. P. G. Casazza and G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Oper-ator Theory, College Park, MD, 2003, Contemp. Math., Vol. 345 (Amer. Math. Soc.,2004), pp. 87–113.

2. O. Christensen, An Introduction to Frames and Riesz Bases (Birkhauser, 2003).3. I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions,

J. Math. Phys. 27 (1986) 1271–1283.

1450013-19

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

2nd Reading



4. R. G. Douglas, On majorization, factorization and range inclusion of operators onHilbert space, Proc. Amer. Math. Soc. 17 (1966) 413–415.

5. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer.Math. Soc. 72 (1952) 341–366.

6. H. G. Feichtinger and T. Werther, Atomic systems for subspaces, in ProceedingsSampTA 2001, Orlando, FL, ed. L. Zayed (2001), pp. 163–165.

7. L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139–144.8. A. Khosravi and K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl.

324 (2008) 1068–1083.9. S. Li and H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal.

Appl. 10 (2004) 409–431.10. W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl. 322 (2006) 437–452.11. W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326 (2007) 858–868.12. R. Young, An Introduction to Nonharmonic Fourier Series (Academic Press, 2001).

1450013-20

Infi

n. D

imen

s. A

nal.

Qua

ntum

. Pro

bab.

Rel

at. T

op. 2

014.

17. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by U

NIV

ER

SIT

Y O

F Q

UE

EN

SLA

ND

on

05/2

9/14

. For

per

sona

l use

onl

y.

Documents

Generalized frames for operators in Hilbert spaces