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The lossless embedding problem for time-varying contractive systems
S) STIMS lb ¢ONTiK)I. IATTIRS
ELSEVIER
Systems & Control Letters 28 (1996) 181-187
The lossless embedding problem for time-varying contractive systems A v r a h a m Feintuch*, Alexander M a r k u s Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Received 9 September 1995; revised 1 May 1996
Abstract
We consider the problem of extending a given causal time-varying contraction T to become the partial input-output operator of a lossless system. We give a necessary and sufficient condition for the existence of such an embedding. In particular, this is the case for T strictly contractive. We show how to construct a large class of extensions for this case and discuss various examples of arbitrary contractions. Keywords." Lossless embedding; Linear time-varying systems; Contractions
1. Introduction
While the idea of embedding a contraction into a 2 x 2 isometric operator matrix is almost a half century old, this problem was solved in the seventies by a number of authors with the added constraints that the entries are no longer simply bounded operators but represent linear time-invariant stable systems [5, 4, 1, 6]. For the time-varying systems the problem was first (as far as we know) formulated in [8, 9] and was related to the causal-anticausal factorization of the non-negative operator I - T* T. This problem has been considered recently for linear time-varying discrete time systems by van der Veen in his doctoral dissertation and subsequently in a joint paper with Dewilde [11]. They show that if the input-output operator T has a state-space representation with certain properties, then T has an embedding in an isometric 2 × 2 operator matrix. We begin as they do with a linear time-varying stable system (a bounded causal operator) in discrete time and give a necessary and sufficient condition for * Corresponding author.
the existence of a 2 × 2 isometric matrix whose entries are from the same class (bounded causal operators) in which T is embedded. We show how to construct a large class of solutions in the case T is a strict contraction using the spectral factorizations of (I - T ' T ) 1/2. We also present examples of contractions for which embeddings do not exist. In the final section we show that if we require the operators to be from the same class as T, then unitary extensions do not in general exist. This fact is crucial in understanding fundamental differences between our results and those of [11]. Van der Veen and Dewilde begin with a causal contraction which has a minimal state-space representation (in a standard infinite-dimensional sense) and seek a unitary causal extension with a minimal state-space representation. In order to accomplish this they interpret causality of the extension in a manner which is conceptually similar but technically quite different from that of the given contraction. As we show, an identical notion of causality does not allow unitary extensions. We, on the other hand, are not concerned with state-space considerations at all.
0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S0167-6911 (96)00026-6
182
A. Feintuch, A. Markusl Systems & Control Letters 28 (1996) 181-187
The existence of isometric extensions is shown in two stages. In the first stage (Lemma 5) we give a necessary and sufficient condition for the existence of a 2 × 1 operator matrix, an isometric enlargement (see [6]) for the given contraction T. In the next stage (Lemma 4 - we stated it earlier for stylistic reasons) we show that the same condition is necessary and sufficient for the 2 × 2 isometric extension. An extension with the same number of inputs and outputs is the classical framework considered in the time-invariant case. But there is an added point here. It is precisely in this lemma where a fundamental difference between the time-invariant theory and the time-varying one is clarified. In the time-invariant case the existence of isometric (unitary) extensions depends on the existence of spectral factorizations. In the time-varying case even where the analogous Cholesky factorizations do not exist, isometric extensions can. The critical issue is the existence of causal isometrics whose ranges are orthogonal to each other. This will be discussed in detail in a future work.
2. Preliminaries We consider the Hilbert space ~vf o f infinite square sequences whose entries are in C k, for k < oc. P, will denote the orthogonal truncation projection
We will make use of the following notion of spectral factorization. If A is an uniformly positive operator on J r , a spectral factorization for A is a factorization A -- B*B with B, B -1 E ~. We will however need to consider the situation where A is only non-negative. For this purpose we must consider the notion of an outer operator introduced by Arveson [2]. Suppose B E ~ and let P8 denote the projection onto the closure [B,~f] o f the range ofB. B is outer if Pe commutes with each Pn and B ( I - Pn)~,uf is dense in (I - P~)~Cf~ A B ~ for each P~. Outer means that the "almost invertible" operator is "almost causally invertible". In particular, if B E cg is invertible, B is outer. We will need the following special case of [3, p. 178, Theorem 14.19]. We state the result for cg. Exactly the same result holds for Mz(Cg). Theorem 1. Every non-negative operator A on gg~ Jactors as B ' B f o r some outer B E c6. Furthermore, B can be chosen so that f o r every operator U, UA E cg i f and only i f UB* E ~g. Note that if UB* E cg so is UA since B E cg and is an algebra. The theorem tells us that there is a particular factorization for which UB*B E c6 only when UB* E c~., We will make use of precisely this factorization.
P,(xO,Xl,. . .,xn,xn+l . . . . ) = (xo . . . . . xn, O. . . . ).
A bounded linear operator T on J f is causal if for all n>~O, PnT = PnTPn. Let 0 Pn
'
acting on ~ ® . ~ in the standard manner,
1
PnY j "
An operator S on ~ ~ ~ is causal if for each n >~O, QnS = Q~SQ~. Now every operator on ~ ® . ~ can be represented as a 2 x 2 operator matrix
S=
[LS21 Sll 322 s121
where Sij is an operator on J r . It is easily checked that S is causal on ~ • off if and only if each Sij is causal on off. ~ will denote the (weakly closed) algebra of causal operators on ~ and it is natural to denote the algebra of causal operators on ~ ® ~ by M2(¢g), the 2 × 2 matrices with entries from ~.
3. The main result We consider T E cg with [IT 11~< 1. Thus ( I - T* T)1/2 is non-negative. Our first result is a special case of [2, Theorem 3.3]. Lemma 2. L e t T be an operator on 9~ such that Nn>~o[T(I-Pn )9 ¢t°] = {0}. Then there exists a partial isometry V which is isometric on the range o f T such that A = VT E c6. The required condition given here is a weak version of invertibility on the nest {(I - P n ) J f } , The next lemma is a converse of the above. Lemma 3. L e t T be a operator on . ~ and V an isometry defined on [TJf]. I f VT E ~ , then (-]n>~o[T(I - P,)gff] = {0}. Proof. Since VT E ~, we have, for n/>0, that [VT(I - P n ) J t ° ] c ( l - P n ) - ~ . Also, since V is an
183
A. Feintueh, A. Markusl Systems & Control Letters 28 (1996) 181-187
isometry on [Tgg~],
= IITTII 2 + Ilnf[I 2 + Ilgll 2
[VT(I - P.)9~] = V[T(I - P.)Jeg]
= Ilfll 2 + Iloll =.
for each n~>0. Since ['].~>0(I - P.)o~cf = {0}, it follows that ~.>.oV[T(I - P n ) ~ ] = {0}. But V is oneto-one and therefore this can be rewritten as VNn~o[V(I - P . ) ~ ] = {0}. Again, since V is one-to-one, ('].~>0[T(I - P n ) ~ ] : {o}.
[]
[]
Remarks. (1) The fact that there exist causal isometrics W,/'3 which have orthogonal ranges is a unique characteristic of time-varying systems. This is not possible for time-invariant systems. (2) Taking T1 = 0 is particularly useful. It gives the property that T n can be embedded in S n, for all n > 0, by the same embedding.
The next lemma simplifies the problem. Lemma 4. Suppose T E cd is a contraction. Then
there exist T1,7"2, T3 E ~ such that T
T~
is an isometry if and only if there exists B E ~ such that IV8 ] is an isometry.
Lemma 5. Given T E cg with IITII ~ 1. There exists B E cg such that [~] is an isometry ifandonly if there exists an isometry V defined on [(I - T* T)I/2~Ct~] = [(I - T * T ) ~ ] such that V(I - T ' T ) 1/2 Ccg. Proof. Clearly [Br ] is an isometry if and only if B = V(I - T ' T ) 1/2 where V is an isometry defined on [ ( I - T* T)1/2 ~ ] . We are concerned with exactly these V such that B E c£. []
Proof. If We can now state the main result.
is an isometry then so is [ ~ ] . So we need only show that if there exists B E ~ such that [r] is an isometry then we can find Tb T2, T3 E ~f such that
S = [ T7/'2 7"3T'] is an isometry. Suppose we have B E ~ such that [~] is an isometry. Define W to be an isometry which, for each k, maps (Pk - Pk-1)ocg onto (P2k - P 2 k - 1 ) o~¢t°.(That is, the kth coordinate space onto the 2kth one. Note that these are o f the same dimension. We take P - I = 0.) Also define T3 to be an isometry which maps (Pk -Pk-1)~,ug onto (P2k+l --P2k)9 ¢t°. Clearly W, T3 E o.K. Take T1 = 0 and/'2 = WB. Then
Theorem 6. Given T E cg with IIT[I ~< 1. There exist operators 7'1,/'2,/'3 c c~ such that
is an isometry if and only if N [ ( I - T*T)I/2(I - Pn)get °] = {0}.
n~>0 Proof. Suppose Nn~>0[(I - T*T)I/2(I - Pn)9~] = {0}. By Lemma 2, there exists an isometry V defined on [ ( I - T*T)I/2~] such that V(I - T ' T ) 1/2 E cg. By Lemma 5 there exists B Ccg such that [Br ] is an isometry. Applying L e m m a 4 gives the required result. Conversely, if there exist/'1, T2,/'3 such that
S [ f ] 2 = llTfl[2 + llT2f + T39112. Since T2f ± T39 for all f ,9 C J~ff (by construction), we can rewrite this as
S[
~'1 2=
IlTf[12 + IIT2fII2+I[T39[, 2
= I I T f l l 2 + II WBfll 2 + IIo[I 2
is an isometry, then so is [ ~ ]. By L e m m a 5, there exists an isometry V defined on [(I - T*T)X/29~] such that 7"2 = V(I - T ' T ) 1/2 E cg. By L e m m a 3, [-]n>~0[(I- T * T ) I / 2 ( I - P n ) g ] = {0}. []
184
A. Feintuch, A. Markus/Systems & Control Letters 28 (1996) 181 187
Remark. We have formulated the results in terms o f one-sided infinite sequences, and fixed finite dimension for the coordinate spaces. Neither of these assumptions are essential. The changes in the proofs are minor and left to the reader.
where [Uj, U2] is an isometry from ~ ® dot~ into ~4'~. Apply Theorem 1 to (I - A ' A ) 1/2 to obtain a spectral factorization (I - A ' A ) 1/2 = B* B with B,B -1 EMz(CK). As above, we want those [U1, U2] with [Ub U2]B* E ~. This is equivalent to
[UI*] n >P n ~ C~ B - I [OP n ~ c' t ~ OUP n g ~] ] ' 4. Strict contractions We begin by noting that if T E cg satisfies IITI[ < 1, then I - T*T and also (I - T ' T ) 1/2 are invertible. Thus (']n~>o[(l- T * T ) I / 2 ( I - P n ) ~ ] =
(I - T*T)I/2An>~O(I - P,)~'(( = {0}. We state this result formally.
Denote the projection onto B-I[P,,Jg ® P,,.z,~] by S.. Then one way to construct
fine FU [ v ; j l to be an isometry from (P2n+l - P 2 ~ - l ) ~ onto (S~ - S~-I )[~(( ® J r ] for each n and let
v;/:Z® u2 J
Corollary 7. I f T E cg and I - T* T is invertible then
there ex&t T1, T2, T3 E ~ such that T
T~
More generally, any sequence o f indices {mn} such that m, - mn-1/> 2 works equally well. We choose to be a co-isometry from (Pm. - P m . _ , ) ~
cZ
is" isometric. We can describe how to construct a large class of such embeddings. (1) We first construct all T1 E c£ such that II[T, T1]ll < 1. It is easier to work with adjoints. We want TI* E ~* such that
[;i] <' The norm condition implies that TI* = L(I - TT*)I/2 with [[Lll < 1. We are concerned with those L such that T~* E ~*. Use Theorem 1 applied to c£,. This gives a spectral factorization (I - TT* )1/2 = BB* with B*, B *-1 E cg, and such that LBB* E ¢£* if and only if LB E ~*. Equivalently, we want those L such that NLII < 1 and L*(I - P n ) W C B * - a ( I - P , ) ~ , for all n>~0. Now B * - I ( I - P n ) W is a closed subspace, so denote by R, the orthogonal projection onto it. Decompose J¢~ as ~ ®(P, - P n - 1 ) ~ and ~ ®(R,_I Rn)W, and let L* be any strict contraction that maps (P, - P , - 1 ) W into (R,-1 - R n ) ~ for n ~>0. We take P - I = 0, R-1 = L Then L* satisfies the required condition and we take Tl = (I - TT*)I/ZL*. (2) Let A = [T, T1]. We extend A to an isometry S E M2(C£) as follows. The second row [/'2, 7"3] of S must be of the form [T2, T3] = [U1,/-72](1 - A ' A ) 1/2
u~* is as follows: De-
onto (S~ - S~-1 )[~¢~ ® ~ ] . In both cases it is easy to check that [ ~i: ] PnJVt~C B-l[pno~f ® PnJf] for all L
~
J
n~>0.
5. Special cases and examples Here we present some results which lead to examples of causal contractions which cannot be embedded into a causal isometry. All of them follow from our main result. Corollary8. Given T E ~q with [ITIl~ o A ° n = {0} if and only if there exists no E N such that ~n0 = {0}, that is,
(I
-
Pno ) o~¢gC Ker(l - T* T) 1/2.
Since Ker(I - T ' T ) = Ker(I - T ' T ) 1/2 (this holds for all non-negative operators), we can rewrite this as (I - P n 0 ) ~ c Ker(I - T ' T ) . Taking orthogonal compliments gives the required result. []
A. Feintuch, A. M a r k u s / S y s t e m s & Control Letters 28 (1996) 181-187
Example 9. It is easy to construct examples which either satisfy or do not satisfy the above property. Let dim(P~ - P ~ - i ) = 1 and {ek}~ be the standard basis on the space ~ = 12(0,c~). Suppose T * T = I - (.,f)f (I - T * T is one-dimensional). Then (I - T * T ) ~ C P n o ~ for some no if and only if f has only finitely many non-zero Fourier coefficients
185
Proof. The implication (a)=~ (b) is obvious. If condition (b) holds then the part of the spectrum of T which lies on the unit circle 121 = 1 is at most finite and consists of isolated eigenvalues of finite multiplicities. If we denote by 9¢t~2the spectral subspace of T corresponding to this part of its spectrum, then we obtain (a). []
( f , ek).
We now consider the case where (I - T*T)~Uf is infinite dimensional. We recall that for a self-adjoint operator A, A is Fredholm if and only i f A ~ is closed and dim Ker A < oo. In particular for a contraction T, I - T * T is Fredholm if and only if the point 2 -- 1 is either in the resolvent set of T * T or is an isolated eigenvalue of finite multiplicity. Also, if I - T * T is Fredholm so is (I - T* T) 1/2.
Corollary 10. I f T Ccg & a contraction and I - T * T is Fredholm, then T has an embeddin9 in an isometry S ~ M2(~).
Proof. By [7, Corollary 4.2.1], ( I - T* T ) U 2 ( I - P n ) ~ is a closed subspace for each n/> 0. Also, since I - T* T is Fredholm, K e r ( l - T * T) 1/2 is finite dimensional and thus there exists a number no such that Ker(l - T ' T ) '/2 A (I - P n o ) ~ = {0}. Thus N [(I - T*T)I/2(I - P n ) ~ ] n >~no
= N (I - T*T)I/2(I - P n ) ~ n >~no
= (I - T ' T ) t/2
(I -Pn),)~ = {0}. n>~no
Since [(1 - T*T)I/2(I - Pn)Jf] = {0}, n>~0
the result follows from Theorem 6.
[]
L e m m a 11. For a contraction T the follow&g are equivalent: (a) ~ = 9ffl ~ 9¢g2 where o~ffi, o~2 a r e invariant subspaces o f T , dimJ~2 < c~, IITl ,ll < 1 and T [ ~ 2 is a n i s o m e t r y . (b) T = T, + T2 where pact operator.
IIZ, II < 1 andT2 i s a c o m -
This lemma gives the following special case of Corollary 10.
Corollary 12. L e t T E ~ be a contraction which can be represented as T = T1 + T2 with IIT~ tl < 1 and T2 compact (T1, T2 are not necessarily causal). Then T has an isometric extension S E M2(cK). We conclude with the following example which is "almost" a strict contraction and yet has no isometric extension.
Example 13. Define A on ~ = 12(0, c¢) by its matrix representation of the form A =
ILek
3j, k=O
oc2
where ek > 0, )-~0 ek < c~, 1 > 20 > 21 > ' " > 0. We can assume HAll < 1. The hypothesis on {2k} gives the following property [12]: For any sequence {wk}k~0 with ~ o Iwkl < ~ , ~ = o 2~wk = 0 (n = O, 1 . . . ) implies wk = Of o r allk. Now fix n and suppose (Aek, y ) = 0 for all k>>.n. This can be expressed as ~k ~-~j~--02~fij = 0 for k ~>n. Thus the function f ( 2 ) = ~-]~j=035j2J, which is analytic in the unit circle, vanishes at 2k, k ~>n. Thus f - 0 and yj -- 0 for all j. Thus {Aek}k>~n is complete in ~¢f. (This construction of an "overcomplete" system is due to Yu. Lyubich.) Note that KerA = {0}: If x E 0~' such that Ax=0, then ~ - - 0 e k 2 J k x k = 0 , j = 0 , 1 .... and ~--~=0 [ekXk[ < O~. By the given property of {2k}, ekxk = 0 for all k and therefore x --- 0. Let Q -- A*A. For each n, consider {Qek: k>>.n}. If y is orthogonal to this set then ( A e k , A y ) = 0 for k >~n which implies that A y = 0. Since A is oneto-one, y - 0. Thus {Qek: k>~n} is complete and Nn>~o[Q(I - Pn)JYf] = ~ . Since IIAII < 1, I - Q2 is uniformly positive. Consider its spectral factorization I - Q2 = T*T with T, T -1 E c6. Then Q = (I - T ' T ) 1/2 implies Nn~>0[(I- T * T ) I / 2 ( I - Pn))~] = 0¢{ • {0}, and there exists no B C cK with [Br] an isometry. Note
186
A. Feintuch, A. MarkuslSystems & Control Letters 28 (1996) 181-187
that Ker A = 0 implies Ker Q = 0. Thus (Q2f, f ) = 11Q/112 > 0 for f # 0. It follows that HTfI[ < ]1/[1 for f # 0. Thus T is a strict contraction pointwise but has no isometric extension. []
Suppose we have a solution
.
6. Unitary embeddings In a discussion with our colleague Prof. Moshe Livgic, he asked us what happens in the finite dimensional case. It turns out that this is interesting in and of itself and has significance in the infinite dimensional case as well. All the ideas already appear in the 2 × 2 case. We illustrate what happens in this case and leave the generalization to the reader• In the finite dimensional case isometric operators on ~7n are unitary and this is fundamental. Let
a
0
al
0
b
c
i bl
el
.
.
.
.
.
.
.
.
.
.
.
i
a2
0
i
a3
0
_192
C2
i
b3
c3
•
Note that the matrix
Eaa'] a2 a3
must also be unitary since [a,0,al,0] and [a2,0,a3,0] are unit orthogonal vectors. Then
[aa] ' E:;] are unit orthogonal vectors and b = b2 = bl = b3 =0. []
be a contraction on ~2.
Consider ~ = /2(0, cx)) with entries from ~. If T is a non-diagonal causal contraction on ~¢g, there is no unitary operator S on ~¢~ • ~ of the form
Example 14. A unitary matrix of the form
a
0
i
*
0
b
c
i
*
*
•
0
i
*
0
•
*
i
*
*
IT S=T2T
T1 ] 3
with T,. E cg. The proof is the same as in the example. If T is diagonal then of course a unitary embedding exists and each of the blocks is diagonal. It is also possible to obtain a unitary embedding if one allows the choice of non-identical notions of causality in the various blocks. See [11].
exists if and only if b = 0.
References
Proof. If b = 0 we have a solution
a
0
0
c
i1 - [ a l e ) ~/2
0
0
" (1-1a12) l/2
0
0
(1 - I c l 2 ) m
"
-d
0
(1-1c12) 1/2 "
0
[1] D. Arov, Darlington's method for dissipative systems, Soy. Phys. Dok. 16 (1972) 954-956. [2] W. Arveson, Interpolation problems in nest algebras, J. Funct. Anal 20 (1975) 208-233. [3] K. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Vol. 191 (Longman, New York, 1988). [4] P. Dewilde, Input-output description of roomy systems, SIAM J. Control Optim. 14 (1976) 712-736. [5] P. Dewilde, V. Belevitch and R.W. Newcomb, On the problem of degree reduction of a scattering matrix by factorization, J. Franklin Inst. 291 (1971) 387-401.
A. Feintuch, A. MarkuslSystems & Control Letters 28 (1996) 181 187
[6] R.G. Douglas and J.W. Helton, Inner dilations of analytical matrix functions and Darlington synthesis, Acta Sci. Math. 34 (1973) 61-67. [7] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vol. 1, OTAA Vol. 53 (Birkh/iuser, Basel, 1992). [8] R. Saeks, Causality in Hilbert space, S l A M Rev. 12 (1970) 357-383. [9] R. Saeks, The factorization problem - a survey, Proc. IEEC 64 (1976) 90-95.
187
[10] A. van der Veen, Time-varying system theory and computational modeling, Doctoral Dissertation, Delft University of Technology, 1993. [11] A. van der Veen and P. Dewilde, Embedding of time-varying contractive systems in lossless realizations, Math. Control Signals Systems 7 (1994) 306-330. [12] J. Wermer, On invariant subspaces of normal operators, Proc. A M S 3 (1952) 270-277.
ELSEVIER
Systems & Control Letters 28 (1996) 181-187
The lossless embedding problem for time-varying contractive systems A v r a h a m Feintuch*, Alexander M a r k u s Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Received 9 September 1995; revised 1 May 1996
Abstract
We consider the problem of extending a given causal time-varying contraction T to become the partial input-output operator of a lossless system. We give a necessary and sufficient condition for the existence of such an embedding. In particular, this is the case for T strictly contractive. We show how to construct a large class of extensions for this case and discuss various examples of arbitrary contractions. Keywords." Lossless embedding; Linear time-varying systems; Contractions
1. Introduction
While the idea of embedding a contraction into a 2 x 2 isometric operator matrix is almost a half century old, this problem was solved in the seventies by a number of authors with the added constraints that the entries are no longer simply bounded operators but represent linear time-invariant stable systems [5, 4, 1, 6]. For the time-varying systems the problem was first (as far as we know) formulated in [8, 9] and was related to the causal-anticausal factorization of the non-negative operator I - T* T. This problem has been considered recently for linear time-varying discrete time systems by van der Veen in his doctoral dissertation and subsequently in a joint paper with Dewilde [11]. They show that if the input-output operator T has a state-space representation with certain properties, then T has an embedding in an isometric 2 × 2 operator matrix. We begin as they do with a linear time-varying stable system (a bounded causal operator) in discrete time and give a necessary and sufficient condition for * Corresponding author.
the existence of a 2 × 2 isometric matrix whose entries are from the same class (bounded causal operators) in which T is embedded. We show how to construct a large class of solutions in the case T is a strict contraction using the spectral factorizations of (I - T ' T ) 1/2. We also present examples of contractions for which embeddings do not exist. In the final section we show that if we require the operators to be from the same class as T, then unitary extensions do not in general exist. This fact is crucial in understanding fundamental differences between our results and those of [11]. Van der Veen and Dewilde begin with a causal contraction which has a minimal state-space representation (in a standard infinite-dimensional sense) and seek a unitary causal extension with a minimal state-space representation. In order to accomplish this they interpret causality of the extension in a manner which is conceptually similar but technically quite different from that of the given contraction. As we show, an identical notion of causality does not allow unitary extensions. We, on the other hand, are not concerned with state-space considerations at all.
0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S0167-6911 (96)00026-6
182
A. Feintuch, A. Markusl Systems & Control Letters 28 (1996) 181-187
The existence of isometric extensions is shown in two stages. In the first stage (Lemma 5) we give a necessary and sufficient condition for the existence of a 2 × 1 operator matrix, an isometric enlargement (see [6]) for the given contraction T. In the next stage (Lemma 4 - we stated it earlier for stylistic reasons) we show that the same condition is necessary and sufficient for the 2 × 2 isometric extension. An extension with the same number of inputs and outputs is the classical framework considered in the time-invariant case. But there is an added point here. It is precisely in this lemma where a fundamental difference between the time-invariant theory and the time-varying one is clarified. In the time-invariant case the existence of isometric (unitary) extensions depends on the existence of spectral factorizations. In the time-varying case even where the analogous Cholesky factorizations do not exist, isometric extensions can. The critical issue is the existence of causal isometrics whose ranges are orthogonal to each other. This will be discussed in detail in a future work.
2. Preliminaries We consider the Hilbert space ~vf o f infinite square sequences whose entries are in C k, for k < oc. P, will denote the orthogonal truncation projection
We will make use of the following notion of spectral factorization. If A is an uniformly positive operator on J r , a spectral factorization for A is a factorization A -- B*B with B, B -1 E ~. We will however need to consider the situation where A is only non-negative. For this purpose we must consider the notion of an outer operator introduced by Arveson [2]. Suppose B E ~ and let P8 denote the projection onto the closure [B,~f] o f the range ofB. B is outer if Pe commutes with each Pn and B ( I - Pn)~,uf is dense in (I - P~)~Cf~ A B ~ for each P~. Outer means that the "almost invertible" operator is "almost causally invertible". In particular, if B E cg is invertible, B is outer. We will need the following special case of [3, p. 178, Theorem 14.19]. We state the result for cg. Exactly the same result holds for Mz(Cg). Theorem 1. Every non-negative operator A on gg~ Jactors as B ' B f o r some outer B E c6. Furthermore, B can be chosen so that f o r every operator U, UA E cg i f and only i f UB* E ~g. Note that if UB* E cg so is UA since B E cg and is an algebra. The theorem tells us that there is a particular factorization for which UB*B E c6 only when UB* E c~., We will make use of precisely this factorization.
P,(xO,Xl,. . .,xn,xn+l . . . . ) = (xo . . . . . xn, O. . . . ).
A bounded linear operator T on J f is causal if for all n>~O, PnT = PnTPn. Let 0 Pn
'
acting on ~ ® . ~ in the standard manner,
1
PnY j "
An operator S on ~ ~ ~ is causal if for each n >~O, QnS = Q~SQ~. Now every operator on ~ ® . ~ can be represented as a 2 x 2 operator matrix
S=
[LS21 Sll 322 s121
where Sij is an operator on J r . It is easily checked that S is causal on ~ • off if and only if each Sij is causal on off. ~ will denote the (weakly closed) algebra of causal operators on ~ and it is natural to denote the algebra of causal operators on ~ ® ~ by M2(¢g), the 2 × 2 matrices with entries from ~.
3. The main result We consider T E cg with [IT 11~< 1. Thus ( I - T* T)1/2 is non-negative. Our first result is a special case of [2, Theorem 3.3]. Lemma 2. L e t T be an operator on 9~ such that Nn>~o[T(I-Pn )9 ¢t°] = {0}. Then there exists a partial isometry V which is isometric on the range o f T such that A = VT E c6. The required condition given here is a weak version of invertibility on the nest {(I - P n ) J f } , The next lemma is a converse of the above. Lemma 3. L e t T be a operator on . ~ and V an isometry defined on [TJf]. I f VT E ~ , then (-]n>~o[T(I - P,)gff] = {0}. Proof. Since VT E ~, we have, for n/>0, that [VT(I - P n ) J t ° ] c ( l - P n ) - ~ . Also, since V is an
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A. Feintueh, A. Markusl Systems & Control Letters 28 (1996) 181-187
isometry on [Tgg~],
= IITTII 2 + Ilnf[I 2 + Ilgll 2
[VT(I - P.)9~] = V[T(I - P.)Jeg]
= Ilfll 2 + Iloll =.
for each n~>0. Since ['].~>0(I - P.)o~cf = {0}, it follows that ~.>.oV[T(I - P n ) ~ ] = {0}. But V is oneto-one and therefore this can be rewritten as VNn~o[V(I - P . ) ~ ] = {0}. Again, since V is one-to-one, ('].~>0[T(I - P n ) ~ ] : {o}.
[]
[]
Remarks. (1) The fact that there exist causal isometrics W,/'3 which have orthogonal ranges is a unique characteristic of time-varying systems. This is not possible for time-invariant systems. (2) Taking T1 = 0 is particularly useful. It gives the property that T n can be embedded in S n, for all n > 0, by the same embedding.
The next lemma simplifies the problem. Lemma 4. Suppose T E cd is a contraction. Then
there exist T1,7"2, T3 E ~ such that T
T~
is an isometry if and only if there exists B E ~ such that IV8 ] is an isometry.
Lemma 5. Given T E cg with IITII ~ 1. There exists B E cg such that [~] is an isometry ifandonly if there exists an isometry V defined on [(I - T* T)I/2~Ct~] = [(I - T * T ) ~ ] such that V(I - T ' T ) 1/2 Ccg. Proof. Clearly [Br ] is an isometry if and only if B = V(I - T ' T ) 1/2 where V is an isometry defined on [ ( I - T* T)1/2 ~ ] . We are concerned with exactly these V such that B E c£. []
Proof. If We can now state the main result.
is an isometry then so is [ ~ ] . So we need only show that if there exists B E ~ such that [r] is an isometry then we can find Tb T2, T3 E ~f such that
S = [ T7/'2 7"3T'] is an isometry. Suppose we have B E ~ such that [~] is an isometry. Define W to be an isometry which, for each k, maps (Pk - Pk-1)ocg onto (P2k - P 2 k - 1 ) o~¢t°.(That is, the kth coordinate space onto the 2kth one. Note that these are o f the same dimension. We take P - I = 0.) Also define T3 to be an isometry which maps (Pk -Pk-1)~,ug onto (P2k+l --P2k)9 ¢t°. Clearly W, T3 E o.K. Take T1 = 0 and/'2 = WB. Then
Theorem 6. Given T E cg with IIT[I ~< 1. There exist operators 7'1,/'2,/'3 c c~ such that
is an isometry if and only if N [ ( I - T*T)I/2(I - Pn)get °] = {0}.
n~>0 Proof. Suppose Nn~>0[(I - T*T)I/2(I - Pn)9~] = {0}. By Lemma 2, there exists an isometry V defined on [ ( I - T*T)I/2~] such that V(I - T ' T ) 1/2 E cg. By Lemma 5 there exists B Ccg such that [Br ] is an isometry. Applying L e m m a 4 gives the required result. Conversely, if there exist/'1, T2,/'3 such that
S [ f ] 2 = llTfl[2 + llT2f + T39112. Since T2f ± T39 for all f ,9 C J~ff (by construction), we can rewrite this as
S[
~'1 2=
IlTf[12 + IIT2fII2+I[T39[, 2
= I I T f l l 2 + II WBfll 2 + IIo[I 2
is an isometry, then so is [ ~ ]. By L e m m a 5, there exists an isometry V defined on [(I - T*T)X/29~] such that 7"2 = V(I - T ' T ) 1/2 E cg. By L e m m a 3, [-]n>~0[(I- T * T ) I / 2 ( I - P n ) g ] = {0}. []
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A. Feintuch, A. Markus/Systems & Control Letters 28 (1996) 181 187
Remark. We have formulated the results in terms o f one-sided infinite sequences, and fixed finite dimension for the coordinate spaces. Neither of these assumptions are essential. The changes in the proofs are minor and left to the reader.
where [Uj, U2] is an isometry from ~ ® dot~ into ~4'~. Apply Theorem 1 to (I - A ' A ) 1/2 to obtain a spectral factorization (I - A ' A ) 1/2 = B* B with B,B -1 EMz(CK). As above, we want those [U1, U2] with [Ub U2]B* E ~. This is equivalent to
[UI*] n >P n ~ C~ B - I [OP n ~ c' t ~ OUP n g ~] ] ' 4. Strict contractions We begin by noting that if T E cg satisfies IITI[ < 1, then I - T*T and also (I - T ' T ) 1/2 are invertible. Thus (']n~>o[(l- T * T ) I / 2 ( I - P n ) ~ ] =
(I - T*T)I/2An>~O(I - P,)~'(( = {0}. We state this result formally.
Denote the projection onto B-I[P,,Jg ® P,,.z,~] by S.. Then one way to construct
fine FU [ v ; j l to be an isometry from (P2n+l - P 2 ~ - l ) ~ onto (S~ - S~-I )[~(( ® J r ] for each n and let
v;/:Z® u2 J
Corollary 7. I f T E cg and I - T* T is invertible then
there ex&t T1, T2, T3 E ~ such that T
T~
More generally, any sequence o f indices {mn} such that m, - mn-1/> 2 works equally well. We choose to be a co-isometry from (Pm. - P m . _ , ) ~
cZ
is" isometric. We can describe how to construct a large class of such embeddings. (1) We first construct all T1 E c£ such that II[T, T1]ll < 1. It is easier to work with adjoints. We want TI* E ~* such that
[;i] <' The norm condition implies that TI* = L(I - TT*)I/2 with [[Lll < 1. We are concerned with those L such that T~* E ~*. Use Theorem 1 applied to c£,. This gives a spectral factorization (I - TT* )1/2 = BB* with B*, B *-1 E cg, and such that LBB* E ¢£* if and only if LB E ~*. Equivalently, we want those L such that NLII < 1 and L*(I - P n ) W C B * - a ( I - P , ) ~ , for all n>~0. Now B * - I ( I - P n ) W is a closed subspace, so denote by R, the orthogonal projection onto it. Decompose J¢~ as ~ ®(P, - P n - 1 ) ~ and ~ ®(R,_I Rn)W, and let L* be any strict contraction that maps (P, - P , - 1 ) W into (R,-1 - R n ) ~ for n ~>0. We take P - I = 0, R-1 = L Then L* satisfies the required condition and we take Tl = (I - TT*)I/ZL*. (2) Let A = [T, T1]. We extend A to an isometry S E M2(C£) as follows. The second row [/'2, 7"3] of S must be of the form [T2, T3] = [U1,/-72](1 - A ' A ) 1/2
u~* is as follows: De-
onto (S~ - S~-1 )[~¢~ ® ~ ] . In both cases it is easy to check that [ ~i: ] PnJVt~C B-l[pno~f ® PnJf] for all L
~
J
n~>0.
5. Special cases and examples Here we present some results which lead to examples of causal contractions which cannot be embedded into a causal isometry. All of them follow from our main result. Corollary8. Given T E ~q with [ITIl~
(I
-
Pno ) o~¢gC Ker(l - T* T) 1/2.
Since Ker(I - T ' T ) = Ker(I - T ' T ) 1/2 (this holds for all non-negative operators), we can rewrite this as (I - P n 0 ) ~ c Ker(I - T ' T ) . Taking orthogonal compliments gives the required result. []
A. Feintuch, A. M a r k u s / S y s t e m s & Control Letters 28 (1996) 181-187
Example 9. It is easy to construct examples which either satisfy or do not satisfy the above property. Let dim(P~ - P ~ - i ) = 1 and {ek}~ be the standard basis on the space ~ = 12(0,c~). Suppose T * T = I - (.,f)f (I - T * T is one-dimensional). Then (I - T * T ) ~ C P n o ~ for some no if and only if f has only finitely many non-zero Fourier coefficients
185
Proof. The implication (a)=~ (b) is obvious. If condition (b) holds then the part of the spectrum of T which lies on the unit circle 121 = 1 is at most finite and consists of isolated eigenvalues of finite multiplicities. If we denote by 9¢t~2the spectral subspace of T corresponding to this part of its spectrum, then we obtain (a). []
( f , ek).
We now consider the case where (I - T*T)~Uf is infinite dimensional. We recall that for a self-adjoint operator A, A is Fredholm if and only i f A ~ is closed and dim Ker A < oo. In particular for a contraction T, I - T * T is Fredholm if and only if the point 2 -- 1 is either in the resolvent set of T * T or is an isolated eigenvalue of finite multiplicity. Also, if I - T * T is Fredholm so is (I - T* T) 1/2.
Corollary 10. I f T Ccg & a contraction and I - T * T is Fredholm, then T has an embeddin9 in an isometry S ~ M2(~).
Proof. By [7, Corollary 4.2.1], ( I - T* T ) U 2 ( I - P n ) ~ is a closed subspace for each n/> 0. Also, since I - T* T is Fredholm, K e r ( l - T * T) 1/2 is finite dimensional and thus there exists a number no such that Ker(l - T ' T ) '/2 A (I - P n o ) ~ = {0}. Thus N [(I - T*T)I/2(I - P n ) ~ ] n >~no
= N (I - T*T)I/2(I - P n ) ~ n >~no
= (I - T ' T ) t/2
(I -Pn),)~ = {0}. n>~no
Since [(1 - T*T)I/2(I - Pn)Jf] = {0}, n>~0
the result follows from Theorem 6.
[]
L e m m a 11. For a contraction T the follow&g are equivalent: (a) ~ = 9ffl ~ 9¢g2 where o~ffi, o~2 a r e invariant subspaces o f T , dimJ~2 < c~, IITl ,ll < 1 and T [ ~ 2 is a n i s o m e t r y . (b) T = T, + T2 where pact operator.
IIZ, II < 1 andT2 i s a c o m -
This lemma gives the following special case of Corollary 10.
Corollary 12. L e t T E ~ be a contraction which can be represented as T = T1 + T2 with IIT~ tl < 1 and T2 compact (T1, T2 are not necessarily causal). Then T has an isometric extension S E M2(cK). We conclude with the following example which is "almost" a strict contraction and yet has no isometric extension.
Example 13. Define A on ~ = 12(0, c¢) by its matrix representation of the form A =
ILek
3j, k=O
oc2
where ek > 0, )-~0 ek < c~, 1 > 20 > 21 > ' " > 0. We can assume HAll < 1. The hypothesis on {2k} gives the following property [12]: For any sequence {wk}k~0 with ~ o Iwkl < ~ , ~ = o 2~wk = 0 (n = O, 1 . . . ) implies wk = Of o r allk. Now fix n and suppose (Aek, y ) = 0 for all k>>.n. This can be expressed as ~k ~-~j~--02~fij = 0 for k ~>n. Thus the function f ( 2 ) = ~-]~j=035j2J, which is analytic in the unit circle, vanishes at 2k, k ~>n. Thus f - 0 and yj -- 0 for all j. Thus {Aek}k>~n is complete in ~¢f. (This construction of an "overcomplete" system is due to Yu. Lyubich.) Note that KerA = {0}: If x E 0~' such that Ax=0, then ~ - - 0 e k 2 J k x k = 0 , j = 0 , 1 .... and ~--~=0 [ekXk[ < O~. By the given property of {2k}, ekxk = 0 for all k and therefore x --- 0. Let Q -- A*A. For each n, consider {Qek: k>>.n}. If y is orthogonal to this set then ( A e k , A y ) = 0 for k >~n which implies that A y = 0. Since A is oneto-one, y - 0. Thus {Qek: k>~n} is complete and Nn>~o[Q(I - Pn)JYf] = ~ . Since IIAII < 1, I - Q2 is uniformly positive. Consider its spectral factorization I - Q2 = T*T with T, T -1 E c6. Then Q = (I - T ' T ) 1/2 implies Nn~>0[(I- T * T ) I / 2 ( I - Pn))~] = 0¢{ • {0}, and there exists no B C cK with [Br] an isometry. Note
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A. Feintuch, A. MarkuslSystems & Control Letters 28 (1996) 181-187
that Ker A = 0 implies Ker Q = 0. Thus (Q2f, f ) = 11Q/112 > 0 for f # 0. It follows that HTfI[ < ]1/[1 for f # 0. Thus T is a strict contraction pointwise but has no isometric extension. []
Suppose we have a solution
.
6. Unitary embeddings In a discussion with our colleague Prof. Moshe Livgic, he asked us what happens in the finite dimensional case. It turns out that this is interesting in and of itself and has significance in the infinite dimensional case as well. All the ideas already appear in the 2 × 2 case. We illustrate what happens in this case and leave the generalization to the reader• In the finite dimensional case isometric operators on ~7n are unitary and this is fundamental. Let
a
0
al
0
b
c
i bl
el
.
.
.
.
.
.
.
.
.
.
.
i
a2
0
i
a3
0
_192
C2
i
b3
c3
•
Note that the matrix
Eaa'] a2 a3
must also be unitary since [a,0,al,0] and [a2,0,a3,0] are unit orthogonal vectors. Then
[aa] ' E:;] are unit orthogonal vectors and b = b2 = bl = b3 =0. []
be a contraction on ~2.
Consider ~ = /2(0, cx)) with entries from ~. If T is a non-diagonal causal contraction on ~¢g, there is no unitary operator S on ~¢~ • ~ of the form
Example 14. A unitary matrix of the form
a
0
i
*
0
b
c
i
*
*
•
0
i
*
0
•
*
i
*
*
IT S=T2T
T1 ] 3
with T,. E cg. The proof is the same as in the example. If T is diagonal then of course a unitary embedding exists and each of the blocks is diagonal. It is also possible to obtain a unitary embedding if one allows the choice of non-identical notions of causality in the various blocks. See [11].
exists if and only if b = 0.
References
Proof. If b = 0 we have a solution
a
0
0
c
i1 - [ a l e ) ~/2
0
0
" (1-1a12) l/2
0
0
(1 - I c l 2 ) m
"
-d
0
(1-1c12) 1/2 "
0
[1] D. Arov, Darlington's method for dissipative systems, Soy. Phys. Dok. 16 (1972) 954-956. [2] W. Arveson, Interpolation problems in nest algebras, J. Funct. Anal 20 (1975) 208-233. [3] K. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Vol. 191 (Longman, New York, 1988). [4] P. Dewilde, Input-output description of roomy systems, SIAM J. Control Optim. 14 (1976) 712-736. [5] P. Dewilde, V. Belevitch and R.W. Newcomb, On the problem of degree reduction of a scattering matrix by factorization, J. Franklin Inst. 291 (1971) 387-401.
A. Feintuch, A. MarkuslSystems & Control Letters 28 (1996) 181 187
[6] R.G. Douglas and J.W. Helton, Inner dilations of analytical matrix functions and Darlington synthesis, Acta Sci. Math. 34 (1973) 61-67. [7] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vol. 1, OTAA Vol. 53 (Birkh/iuser, Basel, 1992). [8] R. Saeks, Causality in Hilbert space, S l A M Rev. 12 (1970) 357-383. [9] R. Saeks, The factorization problem - a survey, Proc. IEEC 64 (1976) 90-95.
187
[10] A. van der Veen, Time-varying system theory and computational modeling, Doctoral Dissertation, Delft University of Technology, 1993. [11] A. van der Veen and P. Dewilde, Embedding of time-varying contractive systems in lossless realizations, Math. Control Signals Systems 7 (1994) 306-330. [12] J. Wermer, On invariant subspaces of normal operators, Proc. A M S 3 (1952) 270-277.