The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on l2

Systems & Control Letters 16 (1991) 219-227 219 North-Holland

The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups o n l 2

S c o t t H a n s e n * a n d G e o r g e W e i s s * * unto) and e lements of the dua l l 2 . a s inf ini te row matr ices . Department of Mathematics, Virginia Polytechnic Institute,

Blacksburg~ VA 24061, USA W e a r e conce rned wi th scr ibed by

Received 1 July 1990 Revised 27 October 1990 .~( t ) = A x ( t ) + Bu( t ) ,

Abstract: Suppose a control system is described by a diagonal semigroup on the state space l 2 and by an unbounded control operator B defined on the input space l 2. We formulate a condition called the operator Carleson measure criterion, and show that this condition is necessary for the admissibility of B. If the semigroup is analytic or invertible, then the condition is sufficient as well. Our results extend the ones of Ho, Russell and Weiss concerning the case where the input space is one dimensional.

Keywords: Admissible control operators, operator Carleson measures.

1. Introduction and statement of the main results

In the papers b y H o and Russel l [1] and Weiss [9], a necessary and sufficient cond i t ion for admis- s ibi l i ty of cont ro l opera to rs for d iagona l semigroups on 12 was proved. This condi t ion , ca l led the Car leson measure cri ter ion, concerns only con- t rol opera to rs def ined on the one d imens iona l space C (such cont ro l opera to r s are also referred to as input elements). In this pape r we prove extensions of the above men t ioned resul t for con- t rol opera to rs def ined o n l 2.

All vector spaces cons idered here are complex. W e represent e lements of l 2 a s inf in i te co lumn

matr ices (i.e., matr ices consis t ing of a single col-

* Supported in part by the Air Force Office of Scientific Research under contract AFOSR-89-0268.

* * Supported mainly by the Weizraarm Fellowship, and also by the Air Force Office of Scientific Research under contract AFOSR-89-0031.

con t ro l systems de-

(1.1)

where x( t ) ~ 12 is the s ta te and u ~ L2([0, ~ ) , l 2) is the inpu t funct ion. The opera to r s A and B are represen ted by inf ini te matr ices . A is d iagonal and its d iagona l e lements ~,k (its eigenvalues) sat isfy

sup Re )~k = 00 < 0. (1.2) k E N

Therefore A genera tes a s t rongly cont inuous diagonal semigroup T = (Tt)t > 0 on 12:

(TtX)k = eXktxk, Vk ~ N, (1.3)

where x k denotes the k - th c o m p o n e n t of x. The inf ini te ma t r ix B represents an admissible control operator for T if for any input funct ion u, (1.1) has an 12-valued s t rong so lu t ion (for more detai ls on this see Sect ion 2).

W e wan t to f ind necessary a n d / o r sufficient cond i t ions for B to be admiss ible . The as sumpt ion that T is d iagona l is qui te restr ict ive in that not all semigroups are i somorph ic to d iagona l ones. On the o ther hand , we lose no general i ty by assuming in (1.2) tha t P0 < 0 (which means that T is ex- poonen t i a l l y stable); see (2.2) and the text following it. Likewise, the a s sumpt ions that u is 12-val - ued and that B is r epresen ted by an inf ini te ma t r ix are no t restr ict ive; see R e m a r k 2.4.

W e need a no t a t i on for cer ta in rectangles in C:

R ( h , ~ ) = ( z ~ C l O < R e z <h,

I Im z - o ~ l _<h}, (1.4)

for any h > 0 and any ~ ~ R. W e denote by b k the k - th row of B. Then b~bk is an inf ini te ma t r ix of r ank one.

0167-6911/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

220 S. Hansen, G. Weiss / Operator Carleson measure criterion for admissibili O,

Definition 1.1. The infinite matrix B, with rows bk, satisfies the operator Carleson measure criterion for the semigroup T defined by (1.3), if b k ~ l 2. for any k E N, and there is some M_> 0 such that for any h > 0 and any ~0 ~ R,

Y'. b~b k < M . h . (1.5) - Xk ~ R( h ,~ ) _L~'( I 2)

where , 8 ~ p ( A ) ( X t does not depend upon /3). For such t , ( i l l - A ) 1 extends to an isomor- phism from X ~ to X. T extends to a strongly continuous semigroup on X 1, whose generator is an extension of A, with domain X. The extended semigroup is isomorphic to the initial one. We denote the extensions of T and A by the same symbols.

We denote by OCM(T) the space of infinite matrices which satisfy the operator Carleson measure criterion for T. Equivalent, perhaps more intuitive formulations of this criterion will be given in Section 3 (see Proposition 3.3 and Remarks 3.4 and 3.5). The name of the criterion will be justi- fied in Section 4. Our main results are the following.

Definition 2.1. With the above notations, let U be a Hilbert space and B ~.LP(U, X 1). Then B is said to be an admissible control operator for T, if for some ~ > 0 and any u~L2( [0 , ~ ) , U) we have ~ u ~ X, where qbu is defined by

@u = f0~T,_oBu(o) do. (2.1)

Theorem 1.2. I f the infinite matrix B represents an admissible control operator for the semigroup T defined in (1.3), then B ~ OCM(T).

The second theorem is a partial converse of the first.

Theorem 1.3. I f T defined in (1.3) is analytic or invertible, and B ~ OCM(T), then B represents an admissible control operator for T.

The proof of these theorems will be given in Section 3, along with some general results on admissibility of control operators for invertible semigroups. In Section 4 we introduce operator Carleson measures and we state a conjecture on them which, if true, would imply that Theorem 1.3 holds for any diagonal semigroup. In Section 5 we discuss some extensions of our main results.

If B is admissible then for any ~" > 0, q~, defined by (2.1) is a bounded linear operator from L2([0, oo), U) to X. Moreover, for any x 0 ~ X and any u ~ L2([0, oo), U), the function x defined on [0, oo) by x ( t ) = Ttx 0 + Ctu is continuous, and is a strong solution of the differential equation (1.1). The importance of admissible control operators is due to the fact that a very general class of control systems (called abstract linear control systems in [8]) is described by (1.1), with admissible B. In this context, X is referred to as the state space and U as the input space.

The space ~ ( U , X, T) of all admissible control operators for T which are defined on U is a subspace of L~'(U, X 1 ) . This space becomes a Banach space with the norm

IIInlll, = sup It¢,ullx, IlullL:--<l

2. Some background

We briefly recall the definition and some facts about admissible control operators in general, following Ho and Russell [1], S a l m o n [6,7] and Weiss [8] (our notation follows [8]).

Suppose A is the generator of a strongly continuous semigroup T = (T t ) ,> 0 on the Hilbert space X. Let the space X_ z be the completion of X with respect to the norm

II x I1-1 = I I ( B I - a ) - l x l l ,

where the choice of T > 0 is unimportant for the topology of ~ ( U , X, T).

Let p ~ R. We define a new semigroup on X by Tt = e - P t ~ • It then follows easily from Defini- tion 2.1 that

x, v) = x, %. (2.2)

In particular, p can be picked so that T is ex- ponentiaUy stable. This shows that the restriction P0 < 0 appearing in (1.2), and thus imposed in Theorems 1.2 and 1.3 is unessential. In the sequel we assume that T is exponentially stable. For any

S. Hansen, G. Weiss / Operator Carleson measure criterion for admissibility 221

B ~.~(a(U, X_I) we defined ~: L2([0, oc), U)-- , X_ 1 by

~u = fo°°T,,Bu(o) do. (2.3)

Then B is admissible if and only if ~u ~ X for any u ~ L2([0,oo), U) as above. If B is admissible then (~ .~a(L2([0 , oo), U), X). We can define another equivalent norm on ~ (U , X, T) by

III n IIioo = sup II ~ u IIx, (2.4) IlullL2-<l

and it is easy to see that

Ill B IIl~ -- t im III B III,. "r --'~ OO

If U---C then we use the notation b(X, T) instead of ~ ( C , X, T), and we identify b(X, T) with a subspace of X_ 1- We have

X c b ( X , T ) c X _ l ,

with continuous embeddings. The elements of b(X, T) are called admissible input elements. It is easy to see that for any B ~ ( U , X, T) and v ~ U we have Bv ~ b (X, T) and

III By III, -< III n II1," II o II, (2.5)

in particular, B ~Za(U, b(X, T)). We recall from Ho and Russell [1] and Weiss

[9] the scalar Carleson measure criterion. If X = l 2 and T is given by (1.3), then in the definition of the norm II ' I1-1 we may take fl -- 0, so that for any x ~ l 2, II x II-a is given by the 12-norm of the sequence (Xk/hk) . It follows that the space X_ 1 = 12_1 can be identified with the Hilbert space of those C-valued sequences b = (b k) for which

bk 2 Ilbll 2 1 = ,., ~ < oo. (2.6)

k = l

Definition 2.2. We say that a C-valued sequence b satisfies the scalar Carleson measure criterion for the semigroup T defined in (1.3), if there is some M > 0 such that for any h > 0 and any ~a ~ R,

E I bk I 2--- M . h (2.7) -hk ~R(h,a 0

(R (h , ~) was defined in (1.4)).

The analogy between (1.5) and (2.7) is clear. If the infinite matrix in Definition 1.1 has a single nonzero column, then (1.5) means that this column satisfies (2.7). The following result was proved in [1] and [9].

Theorem 2.3. Assume the semigroup T is given by (1.3). A C-valued sequence b = ( bk ) is an admissible input element for T if and only if it satisfies the scalar Carleson measure criterion (2.7).

In particular, (2.7) implies b ~ 121 (i.e., the inequality (2.6)). Similarly, we shall see that OCM(T) c ~ ( l 2, l 2_ 1) (see Remark 3.4).

Remark 2.4. The scalar Carleson measure criterion can be used to check admissibility of control operators B defined on finite dimensional spaces; we only have to check if each of the (finitely many) columns of B satisfies (2.7). In what concerns admissibility of control operators defined on an arbitrary infinite dimensional Hilbert space U, when the state space X = 12, we may restrict our attention, without loss of generality, to the case U = 12 (the approach taken in this paper). The reason for this is as follows: Since 121 is separable, the orthogonal complement of Ker B in U is separable as well (it is the range of B*) . Re- stricting B to this subspace then is equivalent to having B defined on 12. Since any operator from l 2 to 121 is represented by an infinite matrix, there is no loss of generality in considering only operators B which are represented by infinite matrices.

3. Proof of the main results

In this section we prove Theorems 1.2 and 1.3 after some preliminaries.

Definition 3.1. Let the semigroup T be defined by (1.3) and assume b - - ( b k ) is a sequence of complex numbers. The Carleson measure norm of b with respect to T, denoted III b IIIcM, is defined by

1 12" III b 1112M = sup ~ ~ I b, h > 0 , t o ~ R - h k ~ R ( h , t ~ )

2 2 2 S. Hansen, G. Weiss / Operator Carleson measure criterion for admissibill D'

It is easy to check that II1" IIIcM is a norm on the space of sequences which satisfy the Carleson measure criterion (2.7), i.e., on b(X, "g).

Proposition 3.2. The above defined Carleson measure norm II1" IIIcM on b(X, 1-) is equivalent to the norm II1" IIioo defined in (2.4). More precisely, for any b ~ b(X, ~),

m~ III b IIIcM -< III b Ill~ -< m2 III b IItcM, (3.1)

where m 1 > 0.6 and m 2 < 20.

Proof. The first inequality in (3.1) was obtained in [9, proof of Theorem 3.2] with the following bound for m~ :

m 1 > min - z ~ R ( 1 , 0 )

(for z = 0 extend by continuity). Elementary calculus gives m 1 > 0.6.

The second inequality in (3.1) was obtained in [1, proof of Corollary 2.5], with the following bound:

2000 m 2< _ _

A factor of 5 in the right-hand side above comes from [1, Lemma 4.1] (a covering lemma). A careful look at the proof of this lemma shows that the number 5 in its statement can be replaced by 3 + c, for any c > 0. This leads to the bound on m 2 given above. []

Proposition 3.3. For T defined by (1.3),

OCM(T) = 2 ' ( l 2, ~,(x, r ) ) .

Proof. It is easy to show that both spaces men- tioned in the proposition consist of infinite matrices with rows in l 2.. Let B be such a matrix, with rows b k. By Proposi t ion 3.2, B Z~'(l 2, b(X, T)) if and only if

sup Ill Bv IIIcM < oo. II vii _<1

Since (BV)k = bkv, the above inequality holds if and only if there is an M > 0 such that for any h > 0 and any to ~ R,

1 12 sup ~ ~ I bkV < M . (3.2) Ilvll < 1 -XkER(h,w )

We next observe that

sup ~ l bkt, 12 li ~' II -< 1 - ~ ER(h,w)

= sup Z (bZbkv, v) tl v II -< 1 -~ , , , c : : R ( h , ~ )

Ilvll-<l - X k ~ R ( h , w )

-Xk ~R(h,~o)

This shows that (3.2) is equivalent to (1.5). []

Remark 3.4. With the natural norm of ,£P(I 2, b (X, T)), OCM(T) becomes a Banach space. This norm is equivalently given by the square-root of the smallest M which can be used in (1.5). We can state an equivalent definition of OCM(T):

OCM(T) = { B ~,£a( l 2, l 2_ 1)I Ran B c b( X, r ) }

(Ran stands for range). This follows from Proposi- tion 3.3, the continuity of the embedding b(X, -~) c 12-1 and the closed graph theorem.

Remark 3.5. The proof of Proposition 3.3 also shows that the left-hand side of (1.5) can be written as II Bh,~, II 2, where Bh,,o is the submatrix of B consisting of those rows b k for which - X k R(h , to). This gives an alternative formulation of the operator Carleson measure criterion which may be easier to verify in some cases.

Proof of Theorem 1.2. If B is admissible for T, then B ~,£a(12, b(X, T)) (see (2.5)). It then follows from Proposition 3.3 that B ~ OCM(T). []

The following proposition is a general admissibility criterion for invertible semigroups. Its proof uses a 'regularity lifting' result from Weiss [8]. For simplicity, we assume that the semigroup is exponentially stable, but this restriction is unessential (see (2.2)).

Proposition 3.6. Let A generate a strongly continuous and exponentially stable semigroup T on the Hilbert space X and assume that T is invertible. Let U be another Hilbert space and suppose B .~(U, X 1). Then B ~ ~(U , X, T) if and only if

sup II(sI-A)-lBIl~e~v,x)< oo. (3.3) R e s > 0


Proof. Since T is exponentially stable, there exist numbers a > 0 and m > 1 such that

II1,11 < m e -at , Vt>__O.

First assume that B is admissible. Let the semigroup "[" be defined by Tt = eat/2"[t. Then by (2.2), B is admissible for T. Since "[" is exponentially stable, by (2.4) there is a number K > 0 such that

fo~TtBu( t ) dt x < K II ullL~ (3.4)

for any u ~ L2([0, ~ ) , U). Let v ~ U and s ~ C with Re s > 0. Define

u( t ) = e-(~+~/~)'~.

Then with this u, (3.4) becomes

K II(sI-h)- lBvl lx <_ K II ulIL :-< ~-a II vllv,

which implies (3.3). Conversely, assume (3.3) holds. Let ~t be de-

fined by (2.1). Let u ~ LZ([0, o0), U) and denote x ( t ) = ~tu. The exponential stability of T implies that x ~ L2([0, ~ ) , X_l). In particular, its Lapace transform ~ is well defined. By simple calculus,

~ ( s ) = ( s I - A ) - l B ~ ( s ) .

Let H2(X) denote the Hardy space of X-valued functions on the right half-plane, as defined, e.g., in Rosenblum and Rovnyak [4, p. 79]. By the Hilbert space valued version of the Paley-Wiener theorem (see, e.g., [4, p. 91]) we have ~ ~ H2(U). Now (3.3) implies that 02 ~ H2(X), hence again by the Paley-Wiener theorem, x ~ L2([0, oo), X). In particular, x ( t ) ~ X for almost every t > 0. By [8, Proposition 4.3], this implies that B is admissible. []

Corollary 3.7. With X, T and B as in Proposition 3.6, B is admissible if and only if

Ran B c b (X, T) . (3.5)

Proof. If B is admissible then (3.5) follows from (2.5). Conversely, if (3.5) holds then for any v ~ U, by the 'only if' part of Proposition 3.6 applied to By, we have

sup II (sI - A ) - I B v II < oo. Res>0

By the uniform boundedness theorem it follows that (3.3) holds, so by the 'if ' part of Proposition 3.6, B is admissible. []

We will be ready to prove Theorem 1.3 after the following lemma.

Lenuna 3.8. There exists C > 0 such that for any 0 E5 L2[0, ~ ) ,

2

f0 ° Y'~ e -2 '2" /2v ( t ) dt _<CIIvll 2. (3.6) hEN

Proof. Let S be the diagonal semigroup on l 2 defined by

( S , x ) . = e-2" 'x . , Vn e N.

The sequence (2 "/2) satisfies the Carleson measure criterion (2.7) for S, and thus by Theorem 2.3 it is an admissible input element (for S). The inequality (3.6) now follows from (2.3) and (2.4) (with S in place of T). rn

Proof of Theorem 1.3. First assume that T is invertible. Since B ~ OCM(T), Proposition 3.3 implies that B ~Ae(l 2, b(X, T)). Corollary 3.7 then shows that B is admissible.

Now assume that T is analytic. Since T is also exponentially stable (see 1.2)), there is a ~ [0, ½tr) such that

Jarg( -X~) l -<Y, V k ~ N (3.7)

(see Pazy [3, p. 61]). Let /% = -00 (see (1.2)), and for n ~ N let/~, = 2". Denote A = { - X k l k ~ N), and for n ~ {0, 1, 2 . . . . } define

An= ( s ~ A l lz ,< Re s < tz,+a}.

Let (ek)k~ N be the standard basis for l 2 and let X" be the subspace of 12 defined by

X" = closed span( e k J - ?~k ~ A" }"

These spaces are T-invariant and their direct sum is 12. Let P, denote the orthogonal projection of 12 onto X ". These projections have bounded extensions to lZ__ 1, and for any x ~ 121 we have

x ~ l 2 ,,* £ II P.x II 2 < ~ . (3 .8 ) n=0

224 S. Hansen, G. Weiss / Operator Carleson measure criterion for admissibifitv

(Actually, if x ~ l 2 then [I x 1[~ is given by the sum on the r ight-hand side of (3.8).) Define

T/~=P,,~P,, * , B"=PnB.

It is easy to see that for any n ~ (0, 1, 2 . . . . },

1[ ]'," I[ < e-U°t, Vt > 0. (3.9)

For n ~ { 0 , 1 , 2 . . . . }, let h , be the smallest number such that A" c R(h , , 0). By simple geom- etry (use (3.7)), there exists x > 0 such that

Vn ~ {0, 1, 2 , . . . ) . (3.10) h , _< t¢2 n,

We have

II B " II = = sup E I (P.Bv)k l 2 Ilvll <1 k~N

= sup ~., I (Bv)kl 2 Ilvll <1 -~k~A"

_< sup E I (Bo)~ [ 2" II vii <1 -hk~R(h.,O)

By the same calculation as used in the p roof of Proposi t ion 3.3, for n ~ (0, 1, 2 . . . . } we get

II B" II z < Y~ b~bk ze(F)" --~k ~ R ( h . , 0 )

Thus f rom (1.5) and (3.10),

II B" II 2 < Mh, < Mr2 ~. (3.11)

Let u ~ L2([0, oo), l 2) and x = ~ u (see (2.3)). By (3.9) and (3.11),

II e.x II z = r t = 0

2

< £ (fo°°e-~'t(MIc2n)l/2llu(t)ll d t ) . n = 0

F r o m (3.6) and (3.8) it now follows that x ~ / 2 . Thus B is admissible. []

4. Operator Carleson measures

In the sequel, C O will denote the open fight half-plane (Re s > 0), and off will denote an arbi- t rary Hi lber t space. As in Section 3, H 2 ( O f ) will denote the H a r d y space of 3~evalued functions on

C O .

Recall f rom Ho and Russell [1] or Koosis [2] that a posit ive measure F on the Borel subsets of C o is a Carleson measure if there is an M >_ 0 such that

t~( R( h, ¢o )) <_ Mh (4.1)

for any h > 0 and any ~0 ~ R. The Carleson measure theorem (see [1] or [2]) says that if /~ is a Carleson measure then there is a C >__ 0 such that for any f ~ H2(C) ,

fc l f I e dlx < C II f II 2- (4.2) o

We now recall some concepts on operator-valued measures following Zeman ian [11, Ch. 2]. For any Borel set S c C, 2 ( S ) will denote the set of Borel subsets of S. PO will s tand for 'pos i t ive opera tor -va lued ' . Fo r S O ~ X(C), a funct ion I~: 2 ( S o ) ~ A a ( o f ) is called a PO measure on S O if for any x ~ o f , S ~ Qz(S)x, x ) is a positive measure on S 0. Any PO measure is countably addit ive with respect to the strong topology of Ae(Jg ' ) , see [11, p. 27].

Let S O be a Borel set in C and let ~ (S0) denote the Banach space of bounded measurable C-valued funct ions def ined on So with the sup-norm, which we denote I1" I1~- Let fg(S0)Ogff be the space of all the funct ions f : S O --* Jff of the fo rm

N

f ( z ) = ~_~ f k ( z ) a k , (4.3) k = l

where fk ~ re(So) and a k ~ o f . (This space is iso- morph ic to the algebraic tensor p roduc t of ~ (S0) and ~,~ff.) We equip ~ ( S o ) ( 3 ) f f with the n o r m

( 1 II f II1 --- inf ~ II fk I1~" II ak II , k = l

where the in f imum is taken over all possible repre- sentat ions of the fo rm (4.3) for f . We denote by ~(S0)~3) f f the comple t ion of ~ ( S o ) Q J g ' with respect to II" II1. For f , g ~ ~(S0)(2))ff and ~ a finite PO measure on S o, the integral

J = fso(d#f , g) (4.4)

is defined by first considering f , g ~ ~(S0)(S).gff (in which case there is a na tura l definit ion for J ) and then extending by cont inui ty (see [11, p. 44]).


Proposition 4.1. Let f ~ H2(~,~), and for n ~ N define

S . = { s ~ C l < R e s < n , I l m s l < n } .

Then the restriction o f f to S, is in f¢( S,)C)~.

Conjecture 4.3. Let i x be an operator Carleson measure with values in .~(o~'). Then there is a C > 0 such that for a n y f ~ H2(o~),

( d # f , f ) < C II f II 2- (4.6) 0

Proof. By the Paley-Wiener theorem (see [4, p. 91]), f - - ~3 for some v ~ Z2([0, oo), o~), so we can write

It is easy to see that (4.6) is a generalization of (4.2). If true, Conjecture 4.3 would imply the following.

f ( s ) = fo°°e -(,-1/~)t e-(1/n)tv(t) dt.

The function t --* e-O/n)tv(t) is in LI([0, oo), o,~) and the integral kernel e x p ( - ( s - 1 / n ) t ) is bounded for (s, t) ~ Sn x [0, oo). Now the conclu- sion follows from [11, Theorem 2.5-1]. []

Definition 4.2. Let Z b denote the set of bounded Borel subsets of C 0. A function ~: Z b ~.oqa(.g ') is called an operator Carleson measure if it satisfies the following:

(i) For any S o ~ Zb, the restriction of /~ to Z(S0) is a PO measure on S 0.

(ii) There exists M > 0 such that for any h > 0 and any to ~ R,

11 #(R(h , to)) II ~ Mh. (4.5)

It is easy to see that an operator Carleson measure is a o-finite PO measure, as defined in [11, p. 30]. If , ~= 12, (4.5) reduces to (4.1), and can be extended to all of Z(C0) by putting #(S) = ~ for certain unbounded sets S. In this sense, Definition 4.2 reduces to the definition of a (scalar-valued) Carleson measure. In the operator- valued case one can not, in general, extend # to Z(C0) in this way without losing strong countable additivity.

Proposition 4.1 allows us to define for any f ~ H2(,,W) and any operator Carleson measure #,

fc(d#f'o f ) = "~lim f s S d # f ' f ) '

Conjecture 4.4. Theorem 1.3 remains true without the assumption that T is analytic or invertible, i.e., for any semigroup T given by (1.3),

~(12, l 2, T) = OCM(T) . (4.7)

Indeed, with the notation of Definitions 1.1 and 4.2 we can define # : Z b -'-~ . ~ ( l 2) by

g t ( S ) = Y'~ b~b k. (4.8) --)~k ~ S

Then (1.5) implies that /~ is an (atomic) operator Carleson measure and (4.6) would imply

E (b~bkf(--Xk), f(--)~k)) <-- CII f l l 2 k ~ N

for any f~H2(12). Taking f = ~, where u ~ L2([0, oo), 12), we get by the Paley-Wiener theorem

E Ibka(--Xk)12 ~ (2~rC)II ull 2" (4.9) k ~ N

Let t~u be defined by (2.3). Then the k-th compo- nent of ~u is b k ~ ( - - h k ) , SO by (4.9), ~u ~ l 2. This shows that B is admissible (and Ill B 11100-< (2,1rC)1/2).

Note that if the input space is t2 (instead of 12), then b~b k in (4.8) becomes I bk 12, so that /x extends to a Carleson measure on ~(C0). In this case Conjecture 4.4 reduces to the 'if ' part of Theorem 2.3, and the above argument reduces to its proof, as given in [1].

where the integrals on the right-hand side are defined as in (4.4).

Note. The authors offer $100 for the first proof or disproof of (4.7).

226 S. Hansen, G. Weiss / Operator Carleson measure criterion for admissibility

5. Extensions and related results

The following simple proposition tends to support Conjecture 4.4, but it imposes a restriction upon the control operator B.

Proposition 5.1. Assume T is given by (1.3) and B is an infinite matrix. Suppose that there is an m ~ N such that each row of B has at most m nonzero elements. Then B is admissible if and only /f B = OCM(T).

Sketch of proof. The 'only if' part follows from Theorem 1.2. Conversely, suppose B ~ OCM(T). It is enough to consider the case m = 1. By re- arranging the order of the eigenvalues of the generator, and hence also the order of the rows of B, we may assume that B is block diagonal, with each block consisting of one (possibly infinite) column. By Proposition 3.3, the II1" IIIoo norms of these columns are uniformly bounded. The corre- sponding operator ~ (see (2.3)) is block diagonal as well and its diagonal blocks are uniformly bounded as operators from L2[0, oo) to l 2. There- fore ~ is a bounded operator from L2([0, oo), l z) to l 2. []

There are some other, more technical recent results of the authors which also tend to support Conjecture 4.4. These results will be published elsewhere, but we give a hint of what they are about.

Let T be an exponentially stable diagonal semigroup on l 2. In other words, T is given by (1.3) a n d the numbers X k satisfy (1.2). Theorems 1.2 and 1.3 tell us that (4.7) holds if the numbers --%k lie in a vertical strip or in an angular sector (see (3.7)) in C 0. It is easy to conclude that (4.7) holds if the number - X k lie in the union of a vertical strip and an angular sector. We can show that there are also other subsets S of C O with the property that - h k ~ S implies (4.7). For example,

S = { z ~ C 0 l a l l m z l ~ < R e z < b l l m zl ~}

where a > 0 and 0 < a < b, has this property (only the case 0 < a < 1 is interesting, for a = 0 or a > 1 it follows from Theorems 1.2 and 1.3).

Remark 5.2. All the results in this paper can be generalized for normal semigroups on arbitrary

Hilbert spaces. We give a brief outline of how to proceed. Let -0- be a normal semigroup on the Hilbert space X, with generator A. We again assume for simplicity that 3- is exponentially stable. By the spectral theorem for normal semigroups (see Rudin, [5, p. 360]), we have

T~=£e ,XdE(2t), 0

where E is the spectral decomposition of - A , extended from o ( - A ) to C o by E ( S ) = 0 if S n o ( - A ) = g , Diagonal semigroups correspond to the particular case when E is atomic. Using that f o r f l ~ C o a n d x ~ X ,

1 II ( # / - A ) - ' x II 2 = fc0 I P + X 12 (dE(X)x, x)

(see [5, p. 343]), it is easy to show that for any bounded Borel set S c Co, E ( S ) has an extension to an operator in &,°(X_~, X).

Let U be another Hilbert space and B £P(U, X_1). Then B satisfies the operator Carle- son measure criterion for T if there is some M > 0 such that

H E ( R ( h , (o))B II .~<u,x> ~ Mh, (5.1)

for any h > 0 and any ~0 ~ R. The generalized version of Theorem 2.3 says that if U = C then (5.1) is necessary and sufficient for the admissibility of B. For arbitrary U, the generalizations of Theorems 1.2 and 1.3 say that (5.1) is necessary for the admissibility of B, and it is also sufficient if T is analytic or invertible. The proofs go along the same lines as for diagonal semigroups, but are notationally more involved. If B satisfies (5.1) then # ( S ) - - -B*E(S)B defines an operator Carle- son measure, and Conjecture 4.3 would imply that (5.1) is always sufficient for the admissibility of B. We leave it to the reader to formulate the generalizations of the other results.

Remark 5.3. The theory of admissible observation operators is, to a great extent, dual to the theory of admissible control operators, see, e.g., Weiss [10, Sect. 6]. For example, the dual version of Theorem 2.3 is stated in [10, Proposition 7.1]. All the results in this paper have dual versions, which we leave to the reader to formulate.


References

[1] L.F. Ho, D.L. Russell, Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. Control Optim. 21 (1983) 614-640.

[2] P. Koosis, Introduction to lip Spaces, London Mathemati- cal Society Lecture Note Series Vol. 40 (Cambridge Uni- versity Press, Cambridge, 1980).

[3] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences Vol. 44 (Springer-Verlag, New York, 1983).

[4] M. Rosenblum, J. Rovnyak, Hardy Classes and Operator Theory (Oxford University Press, New York, 1985).

[5] W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

[6] D. Salamon, Control and Observation of Neutral Systems, Research Notes in Mathematics Voi. 91 (Pitman, Boston, 1984).

[7] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc. 300 (1987) 383-431.

[8] G. Weiss, Admissibility of unbounded control operators, S l A M J. Control Optim. 27 (1989) 527-545.

[9] G. Weiss, Admissibility of input elements for diagonal semigroups on 12, Systems Control Lett. 10 (1988) 79-82.

[10] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. 65 (1989) 17-43

[11] A.H. Zemanian, Realizability Theory for Continuous Lin- ear Systems (Academic Press, New York, 1972).

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The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on l2