Operator Theory 55 Advances and Applications, Vol. 98 © 1997 Birkhiiuser Verlag Basel/Switzerland REALIZATION THEOREMS FOR OPERATOR-VAL UED R-FUNCTIONS S.V. BELYI AND E.R. TSEKANOVSKII Dedicated to the memory of Professor Israel Glazman In this paper we consider realization problems for operator-valued R-functions acting on a Hilbert space E (dim E < 00) as linear-fractional transformations of the transfer operator-valued functions (characteristic functions) of linear sta­ tionary conservative dynamic systems (BrodskiI-Livsic rigged operator colli­ gations). We give complete proofs of both the direct and inverse realization theorems announced in [6], [7J. 1. INTRODUCTION Realization theory of different classes of operator-valued (matrix-valued) functions as transfer operator-functions of linear systems plays an important role in modern operator and systems theory. Almost all realizations in the modern theory of non-selfadjoint op­ erators and its applications deal with systems (operator colligations) in which the main operators are bounded linear operators [8], [10-14J, [17], [21J. The realization with an unbounded operator as a main operator in a corresponding system has not been investi­ gated thoroughly because of a number of essential difficulties usually related to unbounded non-selfadjoint operators. We consider realization problems for operator-valued R-functions acting on a finite dimensional Hilbert space E as linear-fractional transformations of the transfer operator­ functions of linear stationary conservative dynamic systems (l.s.c.d.s.) e of the form (A - zI) = K J'P­ { (1m A = KJK*), 'P+ = 'P- - 2iK*x or K e- ( A - 5)+ C 5) c 5)- 56 S. V. Belyi and E. R. Tsekanovskii In the system 8 above A is a bounded linear operator, acting from fJ+ into fJ - , where fJ+ C fJ C fJ- is a rigged Hilbert space, A =:J T =:J A, A* =:J T* =:J A, A is a Hermitian operator in fJ, T is a non-Hermitian operator in fJ, K is a linear bounded operator from E into fJ-, J = J* = J- 1 , 'P± E E, 'P- is an input vector, 'P+ is an output vector, and x E fJ+ is a vector of the inner state of the system 8. The operator-valued function Wo(z) = 1- 2iK*(A - zI)-1K J is the transfer operator-function of the system 8. We establish criteria for a given operator-valued R-function V(z) to be realized in the form V(z) = i[Wo(z) + I]-1[Wo(z) - I]J. It is shown that an operator-valued R-function V (z) = Q + F . z + +00 (1- - --t) dG(t) . 1-<Xl t - z 1 + t 2 ' acting on a Hilbert space E (dim E < (0) with some invertibility condition can be realized if and only if +00 F = 0 and Qe = _t-2 dG(t)e, J l+t -00 for all e E E such that 00 1: (dG(t)e, e)E < 00. Moreover, if two realizable operator-valued R-functions are different only by a constant term then they can be realized by two systems 81 and 82 with corresponding non-selfadjoint operators that have the same Hermitian part A. The rigged operator colli gat ion 8 mentioned above is exactly an unbounded version of the well known BrodskiI-Livsic bounded operator colligation Q of the form [11] Q=(~ K ~) (1m T=KJK*), with a bounded linear operator T in fJ (and without rigged Hilbert spaces). To prove the direct and inverse realization theorems for operator-valued R-functions we build ?- functional model which generally speaking is an unbounded version of the BrodskiI-Livsic model with diagonal real part. This model for bounded linear operators was constructed in [11]. When this paper was submitted for publication, an article by D, Arov and M. Nudelman [5] appeared considering realization problem for another class of operator-valued functions Realization theorems for operator-valued R-functions 57 (contractive) but not in terms of rigged operator colligations. At the end of this paper there is an example showing how a given R-function can be realized by a rigged operator colligation. Acknowledgement. The authors express their gratitude to the referees and to G. Androlakis, P. Casazza, M. Lammers, and V. Peller for their valuable suggestions that helped to improve the presentation of this paper. 2. PRELIMINARIES In this section we recall some basic definitions and results that will be used in the proof of the realization theorem. The Rigged Hilbert Spaces. Let.5) denote a Hilbert space with inner product (x, y) and let A be a closed linear Hermitian operator, i.e. (Ax, y) = (x, Ay) ('Ix, y E :D(A)), acting in the Hilbert space .5) with generally speaking, non-dense domain :D(A). Let .5)0 = :D(A) and A * be the adjoint to the operator A (we consider A acting from .5)0 into .5») . Now we are going to equip .5) with spaces .5)+ and .5)- called, respectively, spaces with positive and negative norms [9] . We denote.5)+ = :D(A*) ((:D(A*) =.5») with inner product (1) (J,g)+ = (J,g) + (A*f, A*g) and then construct the rigged Hilbert space .5)+ C .5) C .5)_ . Here.5)_ is the space of all linear functionals over .5)+ that are continuous with respect to 11 · 11+ . The norms of these spaces are connected by the relations Ilxll ::; Ilxll+ (x E .5)+), and Il x ll- ::; Ilxll (x E .5»). It is well known that there exists an isometric operator R which maps .5)- onto.5)+ such that (x, y)_ = (x, Ry) = (nx, y) = (nx, Ry)+ (x, y E .5)-), (2) (u, v)+ = (u, n-1v) = (R-1u, v) = (R-1u, R-1v)_ (u, v E .5)+). The operator n will be called the Riesz-Berezanskii operator. In what follows we use symbols (+), (-), and (-) to indicate the norms 11 · 11+ , 11 · 11 , and 11 · 11_ by which geometrical and topological concepts are defined in .5)+ , .5) , and .5)_ . Analogues of von Neumann's formulae. It is easy to see that for a Hermitian operator A in the above settings :D(A) C :D(A*)(= .5)+) and A*y = PAy (Vy E :D(A)) , where P is an orthogonal projection of .5) onto .5)0 . We put (3) .c := .5) e .5)o !.m,X:= (A - AI):D(A) S)'t,X:= (!.m;;).L The subspace S)'t,X is called a defect subspace of A for the point >. The cardinal number dimS)'t,X remains constant when A is in the upper half-plane. Similarly, the number dimS)'t,X remains constant when A is in the lower half-plane. The numbers dimS)'t,X and dimS)'t;; 58 S. V. Belyi and E. R. Tsekanovskii (1m>.. < 0) are called the defect numbers or deficiency indices of operator A [1] . The subspace 91A which lies in fh is the set of solutions of the equation A"g = >..Pg . Let now PA be the orthogonal projection onto 91A' set (4) It is easy to see that 91~ = 91A nno and 91~ is the set of solutions of the equation A' 9 = >..g (see [25]) , when A" : n -+ no is the adjoint operator to A. The subspace 91~ is the defect subspace of the densely defined Hermitian operator P A on no ([22]). The numbers dim91~ and dim91X (1m>.. < 0) are called semi-defect numbers or the semi-deficiency indices of the operator A [16). The von Neumann formula (5) (1m>.. i= 0), holds, but this decomposition is not direct for a non-densely defined operator A. There exists a generalization of von Neumann's formula [3], [24] to the case of a non-densely defined Hermitian operator (direct decomposition). We call an operator A regular, if PA is a closed operator in no . For a regular operator A we have (6) (1m>.. i= 0) where 91 := R£'. This is a generalization of von Neumann's formula. For>.. = ±i we obtain the (+ )-orthogonal decomposition (7) Let A be a closed Hermitian extension of the operator A. Then :D(A) c n+ and PAx = A"x (Vx E :D(A)). According to [25] a closed Hermitian extension A is said to be regular if :D (A) is (+ )-closed. According to the theory of extensions of closed Her­ mitian operators A with non-dense domain [16]' an operator U (:D(U) ~ 91i, ryt(U) ~ 91-d is called an admissible operator if (U - I)fi E :D(A) (Ii E :D(U)) only for Ii = O. Then (see [4]) any symmetric extension A of the non-densely defined closed Hermitian operator A, is defined by an isometric admissible operator U, :D(U) ~ 91i, ryt(U) ~ 91-i by the formula (8) Ah.=AfA+(-ifi-iUfi), fAE:D(A) where :D(A) = :D(A)+(U -I):D(U). The operator A is self-adjoint if and only if:D(U) = 91i and ryt(U). = 91-i. Let us denote now by P;Ji the orthogonal projection operator in n+ onto 91. We intro­ duce a new inner product (" 'h defined by (9) Realization theorems for operator-valued R-functions 59 for all f , 9 E .f:J+. The obvious inequality Ilfll~ :::; Ilflli :::; 211fll~ shows that the norms II . 11+ and II· III are topologically equivalent. It is easy to see that the spaces l' (A) , 91; , 91~i' 91 are (1 )-orthogonal. We write 9)11 for the Hilbert space 9)1 = 91; + 91~i 8:< 91 with inner product (f, 9 h. We denote by .f:J+l the space .f:J+ with norm 11·lh, and by Rl the corresponding Riesz-Berezanskii operator related to the rigged Hilbert space .f:J+l C .f:J C .f:J-l.