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- 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. The following theorem gives a characterization of the regular extensions for a regular closed Hermitian operator A (see [4]). Theorem 1. I. For each closed Hermitian extension A of a regular operator A there exists a (lJ-isometric operator V = V(A) on 9)11 with the properties: aJ 1'(V) is (+)• closed and belongs to 91 EEl 91;, ~(V) c 91 EEl 91~i; b) Vh = h only for h = 0, and 1'(A) = 1'(A) ffi (I + V)1'(V). Conversely, for each (lJ-isometric operator V with the properties a) and b) there exists a closed Hermitian extension A in the sense indicated. II. The extension A is regular if and only if the manifold ~(I + V) is (l)-closed. III. The operator A is self-adjoint if and only if1'(V) = 91 ffi 91; , ~(V) = 91 ffi 91~i . The following theorem can be found in [16J . Theorem 2. Let A be a regular self-adjoint extension of a regular Hermitian operator A, that is determined by an admissible operator U and let (10) !.hi = {Ii E 91i, (U - Ilfi E .f:Jo} . Then (11) Bi-extensions. Denote by [.f:Jl, .f:J2J the set of all linear bounded operators acting from the Hilbert space .f:Jl into the Hilbert space .f:J2. Definition. An operator A E [.f:J+ ,.f:J- J is a bi-extension of A if both A :;, A and A* :;, A. If A = A *, then A is called a self-adjoint bi-extension of the operator A. We write 6(A) for the class of bi-extensions of A. This class is closed in the weak topology and is invariant under taking adjoints. The following theorem from [4], [25J gives a description of 6(A). 60 s. V. BeJyi and E. R. Tsekanovskii Theorem 3. Every bi-extension A of a regular Hermitian operator A has the form: (12) A -_ APD(A) + + [*A + R1-1( Q - -P . . (13) A* = AP~(A) + [A* + R11(Q* - ::'P;Ji, + ::'P;Ji, )]P9t 2 ' 2 -, where Q is an arbitrary operator in [9Jt,9Jt] and Q* is its adjoint with respect to the (1 )-metric. Corollary 1. Every self-adjoint bi-extension A of the regular Hermitian operator A is of the form: (14) A -_ APD(A) + + [*A + Rl-1 (5 - -P Let A be a bi-extension of a Hermitian operator A. The operator Af = Af, 1)(A) = U E Sj , Af E Sj} is called the quasi-kernel of A. If A = A* and A is a quasi-kernel of A such that A i- A, A* = A then A is said to be a strong self-adjoint bi-extension of A. Classes SlA and AA. (* )-extensions. Let A be a closed Hermitian operator. Definition. We say that a closed densely defined linear operator T acting on the Hilbert space Sj belongs to the class SlA if: (1) T:::> A and T* :::> A; (2) (-i) is a regular point ofT.l It was mentioned in [4] that sets 1)(T) and 1)(T*) are (+)-closed, the operators T and T* are (+,. )-bounded. The following theorem [25] is an analogue of von Neumann's formulae for the class SlA. Theorem 4. I. To each operator of the class SlA there corresponds an operator M on the space 9Jt l with the following properties: (1) 1)(M) = 91; EEl 91, and !J\(M) = 91~i EEl 91; (2) Mx + x = 0 only for x = 0, and M*x + x = 0 only for x = O. Moreover, the following hold: (15) 1)(T) = 1)(A) EEl (M + I)(91~ EEl 91) , (16) 1)(T*) = 1)(A) EEl (M* + I) (91'-i EEl 91). 1 The condition, that (-i) is a regular point in the definition of the class n A is not essential. It is sufficient to require the existence of some regular point for T . Realization theorems for operator-valued R-functions 61 II. Conversely, for each pair of (1)-adjoint operators M and M* in [9J1 1, 9J1d satisfying (1) and (2) above. formulas (15) and (16) give a corresponding operator T in the class f2 A . Moreover, if f = 9 + (M + I)cp , 9 E 1)(A), and cp E 1)1: tfJ 1)1 then (17) T f = Ag + A*(I + M)cp + iRl1 P;Ji(I - M)'P (f E 1)(T)). Similarly, if f = 9 + (M* + I)'1jJ, 9 E 1)(A), and '1jJ E l)1~i EEl 1)1, then (18) T* f = Ag + A*(I + M*) '1jJ + iRl1 P;Ji(M* - I) '1jJ (f E 1)(T)), Definition. An operator A in [SJ+,SJ-] is called a (*)-extension of an operator T from the class f2A if both A J T and A* J T*. This (* )-extension is called correct, if an operator AR = ! (A + A*) is a strong self• adjoint bi-extension of an operator A. It is easy to show that if A is a (* )-extension of T , then T and T* are quasi-kernels of A and A* , respectively. Definition. We say that the operator T of the class f2A belongs to the class AA if (1) T admits a correct (*)-extension; (2) A is the maximal common Hermitian part of T and T*. Theorem 5. Let an operator T belong to f2A and let M be an operator in [9J1.9J1] that is related to T by Theorem 4. Then T belongs to AA if and only if there exists either (I)-isometric operator or a C)-isometric operator U in [1)1:, l)1~i] such that (U + I)1)1: + (M + I) (1)1: EEl 1)1) = 9J1, (19) { (U + I)1)1: + (M + I)(I)1: ffi 1)1) = 9J1. Corollary 2. If a closed Hermitian operator A has finite and equal defect indices, then the class f2A coincides with the AA . Extended Resolvents and Extended Spectral Functions of a Hermitian Oper• ator. Let A be a closed Hermitian operator on SJ and ~ be a Hilbert space such that SJ is a subspace of~. Let A be a self-adjoint extension of A on ~, and E(t) be the spectral function of A. An operator function R>. = PYj (A - AI) -l1Yj is called a generalized resolvent of A , and E(t) = PYjE(tJlYj is the corresponding generalized spectral function. Here 00 (20) R = dE(t) (ImA =1= 0). >. J t - A -00 If ~ = SJ then 'R>. and E(t) are called canonical resolvent and canonical spectral function, respectively. According to [19] we denote by R>. the (-, ' )-continuous operator from SJ• into SJ which is adjoint to R>.: (21) 62 s. V. Belyi and E. R . Tsekanovskii It follows that R>..f = R>..f for f E h, so that R>.. is an extension of R>.. from 5) to 5)_ with respect to (-, . )-continuity. The function R>.. of the parameter A, (ImA i=- 0) is called the extended generalized (canonical) resolvent of the operator A. We write N for the family of all finite intervals on the real axis. It is known [19] that if L1 E N then E(L1)5) C 5)+ and the operator E( L1) is (., + )-continuous. We denote by E( L1) the (-, . )-continuous operator from 5)_ to 5) that is adjoint to E(L1) E [5),5)+]. Similarly, (22) (E(L1)f, g) = (f,E(L1)g) (f E 5)-, 9 E 5)), One can easily see that E(L1)f = E(L1)f, "If E 5) , so that E(L1) is the extension of E(L1) by continuity. We say that E(L1), as a function of L1 E N, is the extended generalized (canonical) spectral function of A corresponding to the self-adjoint extension A (or to the original spectral function E(L1)). It is known [19] that E(L1) E [5)_,5)+], VL1 E N, and (E(L1)f, f) ~ 0 for aU f E 5)_. It is also known [19] that the complex scalar measure (E(L1)f,g) is a complex function of bounded variation on the real axis. However, (E(L1)f,g) may be unbounded for f,g E 5)_ . Now let R>.. be an extended generalized (canonical) resolvent of a closed Hermitian operator A and let E(L1) be the corresponding extended generalized (canonical) spectral function. It was shown in [19] that for any f , 9 E 5)- , +00 _ Id(E(L1)f,g)1 (23) 2 < 00, J l+t -00 and the following integral representation holds _ _ +00 (24) R - Ri + R-i = J (_1__ _t -) dE(t). >.. 2 t - A 1 + t 2 -00 Lemma 6. Let A = A* +n-1(S - ~p,;t + ~P;_.)P9t be a strong self-adjoint bi-extension of a regular Hermitian operator A with the quasi-kernel A and let E(L1) be the extended canonical spectral function of A. Then for every f E 5) EB L, f i=- 0, and for every 9 E 5)_ there is an integral representation - J+oo (1 t) - 1 - - (25) • (R>..f,g) = t-A -1+t2 d(E(t)f,g)+2((Ri +R_i)f,g). -00 Realization theorems for operator-valued R-functions 63 Theorem 7. Let A = A* + n-1(S - ~P~i + ~P~_.)P9t be a strong self-adjoint bi• extension of a regular Hermitian operator A with the quasi-kernel A and let E(.:1) be the extended canonical spectral function of A. Also, let F = n+ e !>(A) and L = n-l(S• ~P~, + ~P~_,)F . Then for every f E L + £ with f =1= 0 and f E ryt(A - AI), we have +cxo (26) J d(E(t)f, f) = 00, if f rJ. £, -00 and +cxo (26') Jd(E(t)f, f) < 00, if f E £. -cxo Moreover, there exist real constants band c such that +cxo , (27) cllfll: ::; J d(~~{~ f) ::; bllfll:, -cxo for all f E L + £ . Corollary 3. In the settings of Theorem 7 for all f , gEL + £ + cxo d(E(t)g,g) (28) 1 2 -cxo 1 + t ' where a > 0 is a constant (see [2]). 3. LINEAR STATIONARY CONSERVATIVE DYNAMIC SYSTEMS In this section we consider linear stationary conservative dynamic systems (l. s. c. d. s.) e of the form (A - zl) = KJ'P_ (29) { (1m A=KJK*). 'P+ = 'P- - 2iK* x In a system e of the form (29) A, K and J are bounded linear operators in Hilbert spaces, 'P- is an input vector, 'P+ is an output vector, and x is an inner state vector of the system e. For our purposes we need the following more precise definition: 64 S. V. Belyi and E. R. Tsekanovskii Definition. The array (30) A. e-(- n+ c n c n- K El) is called a linear stationary conservative dynamic system or Brodskil-Livsic rigged operator colligation if (1) A. is a correct (* )-extension of an operator T of the class AA. (2) 1 = J* = 1-1 E [E,E], dimE < 00 (3) A. - A.* = 2iK1K*, where K E [E,n-] (K* E [n+,E]) In this case, the operator K is called a channel operator and 1 is called a direction operator. A system e of the form (30) will be called a scattering system (dissipative operator colligation) if 1 = I . We will associate with the system e the operator-valued function (31) Wo(z) = 1- 2iK*(A. - zI)-l K 1 which is called the transfer operator-valued function of the system e or the characteristic operator-valued function of BrodskiI-Livsic rigged operator colligation. According to The• orem 7, ~(K) c ~(A. - AI) and therefore Wo(z) is well-defined. It may be shown [10], [25] that the transfer operator-function of the system e of the form (30) has the following properties: W;(z)lWo(Z) - 1 ::::: 0 (1m z > 0, z E p(T)), (32) W;(z)lWo(z ) - 1 = 0 (1m z = 0, z E p(T)), W;(z)lWo(z) - 1 :S 0 (1m z < 0, z E p(T)), where p(T) is the set of regular points of an operator T. Similar relations take place if we change Wo( z ) to W;(z) in (32). Thus, the transfer operator-valued function of the system e of the form (30) is l-contractive in the lower half-plane on the set of regular points of an operator T and l-unitary on real regular points of an operator T . Let e be a l.s.c.d.s. of the form (30). We consider the operator-valued function (33) The transfer operator-function Wo(z) of the system e and an operator-function Vo(z) of the form (33) are connected with the relation (34) Vo(z) = i[Wo(z) + I]-l[WO(Z) - 1]1. As it is known [11] an operator-function V(z) E [E, E] is called an operator-valued R• function if it is holomorphic in the upper half-plane and 1m V(z) ::::: 0 whenever 1m z > o. Realization theorems for operator-valued R-functions 65 It is known [11,17] that an operator-valued R-function acting on a Hilbert space E (dim E < 00) has an integral representation +00 (35) V(z) = Q + F . z + (1 - - --2t) dG(t), 1-00 t - z 1 + t where Q = Q*, F 2: 0 in the Hilbert space E, and G(t) is a non-decreasing operator• function on (-00, +00) for which +00 dG(t) [E E]. 1-00 1 + t2 E , Definition. We call an operator-valued R-function V(z) acting on a Hilbert space E, (dim E < 00) realizable if in some neighborhood of the point (-i), the function V (z) can be represented in the form (36) V(z) = i[We(z) + It1 [We(z) - I]J, where We(z) is the transfer operator-function of some l.s.c.d.s. e with the direction oper• ator J (J = J* = J-1 E [E, ED. Definition. An operator-valued R-function V(z) E [E , E], (dimE < 00) is said to be a member of the class N(R) if in the representation (35) we have i) F=O, +00 t ii) Qe = 1 --2dG(t)e, -00 1 + t for all e E E with 00 1: (dG(t)e, e)E < 00. We now establish the next result. Theorem 8. Let e be a l.s.c.d.s. of the form (30) with dim E < 00. Then the operator• function Ve(z) of the form (33), (34) belongs to the class N(R). Proof. Let G-i be a neighborhood of (-i) and ).., /1 E G-i . Then, Ve()..) - Ve(/1) = K*(AR - )..1)-1 K - K*(AR - /11)-1 K (37) = (/1- )")K*(AR - )..1)-I(AR - /11)-1 K , and (38) 66 s. V. Belyi and E. R. Tsekanovskii for all ).., J1- E G -i. Therefore, letting).. -+ J1- we can say that Vo (z) is holomorphic in G _; . Without loss of generality (see [25]) we can conclude that Vo(z) is holomorphic in anyone of the half-planes. It is obvious that Vo*(z) = Vo(z) = Vo(z) . Furthermore, (39) Since (-i) is a regular point of the operator T in the system (30) then (see [10]) 1+ iV()")J is invertible in G-i . Let now Dz = (AR - zI)-1 K, then it is easy to see that the adjoint operator D; is given by D; = K*(AR - zI)-I. Therefore, we have ImVo(z) = ImzD;Dz which implies that ImVo(z) ~ 0 when Imz > O. Hence we can conclude that Vo(z) is an operator R-function and admits representation (35). Let now B = K*(AR + iI)-I{AR - iI)-1 K. It follows from (39) that B = t(Vo(i) - Vo*(i)). Using Theorem 7 and representation (35) one can show that 00 ( 40) Bf = J dG(t) f fEE 1 + t2 ' -00 and B E [E, E]. Let £(6.) be the canonical extended spectral function of the quasi-kernel A of the operator AR = ~(A + A*) . Then relying on Lemma 6 for all f, gEE we have +OO( 1 t) A A (41) (Vo()..)f,g)E = J t _).. - 1 + t2 d(G(t)f,g)E + (Qf,g)E, -00 where 6(6.) = K* £{L1)K, 6. E Nand (42) Q = ~K*[(AR - iI)-1 + (AR + iI)-I]K = ~[Vo( -i) + Vo*( -i)]. From Theorem 7 (see also [19]), we have for all fEE with K f E £ , 00 (43) Jd(6(t)f , f)E < 00, -ex:> and (44) -00 Realization theorems for operator-valued R-functions 67 Moreover, (28) implies that (45) By (41) we have for any J,g E E , J+=( 1 t)' (46) (Ve(>.)J,g)E = (QJ,g)E + t _ >. - 1 + t 2 d(G(t)J,g)E· -= On the other hand (35) implies += (47) (Ve(>.)J,g)E = (QJ,g)E + >.(FJ,g)E + J C~ >. - 1 ~ t2) d(G(t)J,g)E. -= Comparing (46) and (47) we get (QJ, g)E = (QJ, g)E, (FJ,g)E = 0, and (G(6.)f, g) = (O(6.)J,g) (6. EN), for all J,g E E. Taking into account the continuity and positivity of F, G(6.), and 0(6.), we find that F = 0 and G(6.) = 0(6.) (6. EN). Thus, +00 ( 48) V(>') = Q + J (~-t-A -t-2)l+t dG(t) , -00 holds. Let Eoo = K- 1 £ , Eoo C E. Since E( 6.) coincides with E( 6.) on £, then for any e E Eoo , we have +00 (49) Jd(O(t)e, e)E < 00. -00 If e tJ. Eoo, then K e tJ. £ (see Theorem 7) and +00 (50) Jd(O(t)e, e)E = 00. -00 Further, since (51) 68 S. V. Belyi and E. R. 1Sekanovskii we have ry{(Q) ~ ry{(K*) ~ E. Now formula (45) yields (52) f,g E E. On the other hand, if e E Eoo then +00 t /+00 t " = K* / --2 dE(t)Ke = -2- dG(t)e. 1+t t +1 -00 -00 This completes the proof. Next, we establish the converse. 2 Theorem 9. Let an operator.-valued function V(z) act on a finite-dimensional Hilbert space E and belong to the class N(R). Then V(z) admits a realization by the system () of the form (30) with a preassigned direction operator J for which I + iV ( -i) J is invertible. Proof. We will use several steps to prove this theorem. STEP 1. Let Coo(E, (-00, +00)) be the set of continuous compactly supported vector• valued functions f (t) (-00 < t < +00) with values in a finite dimensional Hilbert space E. We introduce an inner product (-,.) defined by +00 (53) (j,g) = / (G(dt)f(t),g(t))E -00 for all f, 9 E Coo(E, (-00, +00)). In order to construct a Hilbert space, we identify with zero all functions f(t) such that (j, f) = O. Then we make the completion and obtain the new Hilbert space L't;(E). Let us note that the set Coo(E, (-00, +00)) is dense in L't;(E). Moreover, if f (t) is continuous and +00 (54) 1-00 (G(dt)f(t), f(t))E < 00, then f(t) belongs to L't;(E). Let 1'0 be the set of the continuous vector-valued (with values in E) functions f(t) such that in addition to (54), we have +00 2 (55) 1-00 t (G(dt)f(t), f(t))E < 00. 2The method of rigged Hilbert spaces for solving inverse problems in the theory of characteristic operator-valued functions was introduced in [23] and was developed further in [2] . Realization theorems for operator-valued R-functions 69 Since Coo C :Do, it follows that :Do is dense in Lb(E). We introduce an operator A on:Do in the following way: (56) Af(t) = tf(t). Below we denote again by A the closure of the Hermitian operator A (56). It is easy to see that this operator is Hermitian. Now A is a self-adjoint operator in Lb(E) (see [9]) . Let j,+ = :D(A) and define the inner product (57) (f,g)iJ+ = (f,g) + (Af, Ag) for all f ,g E j,+. It is clear that j,+ is a Hilbert space with norm II· iliJ+ generated by the inner product (57). We equip the space Lb(E) with spaces j,+ and j,_: - 2 - (58) ..\)+ C La (E) C 5)_. Let us denote by R the corresponding Riesz-Berezanskii operator, R E [j,-,j,+l. Consider the following subspaces of the space E: Eoo = {e E E : [:00 d(G(t)e, e)E < oo} (59) Foo = E~. If e E E= , then (54) implies that the function e(t) = e is an element of the space Lb(E). On the other hand, if e E E and e 1. Eoo then e(t) does not belong to Lb(E). It can be shown that any function e(t) = e E E can be identified with an element of j,_ . Indeed, since for all e E E +00 d(G(t)e,e)E (60) 1-00 1 + t2 < 00, the function (61) e(t) -_ v"f+'t2e belongs to the space Lb(E). Letting f(t) E :Do , we have +00 2 (62) 1-00 (1 + t )(G(dt)f(t), f(t))E < 00. Therefore, the function j(t) = v"f+'t2 f(t) belongs to the space Lb(E) and hence +00 (j(t), e(t)) = 1-00 (G(dt)j(t) , e(t))E. 70 s. V. Belyi and E. R. Thekanovskii Furthermore, I(}(t), e(t))1 s IIf(t)II·lIe(t)11 +00 +00 d(G(t)e(t),e(t)) e (63) 1-00 (1 + t 2 )(G(dt)f(t), f(t))E· 1-00 1 + t 2 = IIfllfJ+ . lIellE. Also, +00 1+00 ( e ) 1 (G(dt)f(t), e(t))E = vfl+t2G(dt)f(t), vIf+t2 -00 -00 1 + t E +00 = 1-00 (G(dt)f(t), e(t))E = (}(t), e(t)). Therefore, +00 (64) e(j) = 1-00 (G(dt)f(t) ,e(t))E is a continuous linear functional over 5)+ , for f E 1)0. Since 1)0 is dense in 5)+, e(t) = e belongs to .f)_. We calculate the Riesz-Berezanskii mapping on the vectors e(t) = e, e E E. By the definition of R , for all f E 5)+ we have (j, e) = (j, Re) fJ+. Hence, for all f E 1)0 (see also [2]) 00 00 (j, e) = 1+ (G(dt)f(t) , e(t))E = 1+ (1 + t2) (G(dt)f(t), e(t) 2) -00 -00 1 + t E = (f' 1e~~2) fJ+ = (j, Re)fJ+· Thus (65) Re=~ e E E. 1 + t2 ' Let us note some properties of the operator A. It is easy to see that for all g E 5)+, we have that IIAgl1 S Ilgll fJ +· Taking this into account we obtain (66) Hence, the operator A is (., - )-continuous. Let A be the extension of the operator A to.f) with respect to (., - )-continuity. Now, Realization theorems for operator-valued R-functions 71 holds for all 9 E iL. Note in particular that (68) and (69) for all gin fJ-. It follows from (60) that the element f (70) f(t) = t _ )..' fEE belongs to the space Lb(E). It is easy to show that, for all e E E, ...". -1 e (71) (A - )..1). e = - (1m).. #- 0). t -)..' STEP 2. Now let 5)+ be the Hilbert space constructed in Step 1 and let (72) 1>(A) = fJ+ 8 RE, where by 8 we mean orthogonality in fJ+. We define an operator A on 1>(A) by the following expression: (73) A=AI . ll(A) Obviously, A is a closed Hermitian operator. Let us note that if Eoo = 0 then 1>(A) is dense in Lb(E). Define 5)0 = 1>(A) and let P be the orthogonal projection of 5) = Lb(E) onto 5). We shall show that P A and PA are closed operators in 5). Let (74) The following obvious inclusions hold: A C Ai C A. It is easy to see that 1>(Ad 1>(A) EEl RFoo , 1>(Ad = 5)0 and Ai is a closed Hermitian operator. Indeed, if we identify the space E with the space offunctions e(t) = e, e E E we would obtain Lb(E)8 5)0 = Eoo . Since +00 d(G(t)e, h)E = 0 / 2 -00 1 + t and -ij_ e '"l-e =-- 1+ t 2 ' 72 s. V. Belyi and E. R. Tsekanovskii for all e E Eex" h E Fex" we find that Eoo is (. )-orthogonal to RF00 and hence :D (A l) = flo. We denote by Ai the adjoint of the operator A l . Now we are going to find the defect subspaces l)1i and 1)1-.:£.. of the operator A. Since the subspace E E fl- is (. )-orthogonal to :D(A), we have that (A ± iI)-l E = l)1±i. Moreover, by (71) we have ...". e (75) (A ± il)-le = - e E E. t ± i ' Therefore (76) l)1±i = {f(t) E L~(E), f(t) = t: i' e E E}. Similarly, the defect subspaces of the operator Al are (77) l)1~i = {f(t) E L~(E), f(t) = t: i ' e E Eoo}. Obviously, 1)1~ c :Do because +00 +00 J It ~ AI2 (G(dt)e , e)E ~ K(A) J(G(dt)e , e)E < 00, e E Eoo· -00 - 00 Taking into account that (78) we can conclude that :D(Ai) ~ :D(A). At the same time, the inclusion Al C A implies that :D(Ai) :::) :D(A). Combining these two we obtain :D(Ai) = :D(A) and PA = Ai. Since Ai is a closed operator, PA is also closed. Consequently, A is the regular self-adjoint extension of the operator A which implies A is a regular Hermitian operator. Since A is the self-adjoint extension of operator A we find by (10) that (79) :D(A) = :D(A) + (I - U)l)1i for some admissible isometric operator U acting from l)1i into l)1-i . It is easy to check that U(A - iJ)-le = (A + iJ)-le, for all e in E. Consequently, the operator U has the form: (80) U = E E. (_et-i ) _et+i' e Straightforward calculations show that A(I-U), ( -e) . =t-.e -t-. e = --.2ite t - t t - t t + t t 2 + 1 Realization theorems for operator-valued R-functions 73 Let A' be the adjoint of the operator A. In the space ~(A*) = -D+ we introduce an inner product (81) (j,g)+ = (j,g) + (A* j, A*g), and construct the rigged space -D+ C -D C -D- with corresponding Riesz-Berezanskii oper• ator R. Since PAis a closed Hermitian operator, f,+ is a subspace of -D+. By Theorem 2, -D+ = ~(A) + (U - qYt;, where Taking into account that (u - J) (_e ) - -2ie E .V- t2 + 1 ' eE , we can conclude that Therefore, (82) ~(A*) = ~(A) . {~} + t 2 + 1 ' STEP 3. In this Step we will construct a special self-adjoint bi-extension whose quasi• kernel coincides with the operator A. Then applying (7), we will have where 1)1±; are semidefect spaces of the operator A, 1)1 = RE",,» and ~(A) EB Eoo = -D = L~(E). We begin by setting (83) Here P;j; is an orthoprojection of -D+ onto 1)1. Obviously, the norm II . IiI is equivalent to II . 11+· We denpte by -D+l the space -D+ with the norm II . Ill, so that -D+l C -D C -D-1 is the corresponding rigged space with Riesz-Berezanskii operator R l . By Theorem 1 there exists a (I)-isometric operator V such that (84) ~(A) = ~(A) EB (V + 1)(1)1: EB 1)1) , 74 S. V. Belyi and E. R. Tsekanovskii where 1J(V) = 1)1; ffi 1)1, 9't(V) = l)1~i ffi 1)1 and (-1) is a regular point for the operator V. Moreover, 'P = i(I + P~, )(A* + i1)-l Ii, { (85) V'P = i(I + ;~,-yA* - iI)-lUJ;, where 'P E 1J(V), Ii E 1)1;. Here U is the isometric operator described in Step 2. Consequently we obtain I; = ~(A* + i1)(I + P~)'P, (86) { UI; = -~(A* - i1)(I + P~)V'P, where 'P E 1J(V), I; E l)1i. It follows that Ii - U Ii = 'P + V'P + iA * P~ (V - 1)'P A(Ji - Uli) = i(I + U)/i = A*('P+ V'P) +iP~(I - V)'P Ii + Uli = 'P - V'P - iA* P~(I - V)'P Applying formula (11) we get Since Ii - Uli = 'P + V'P + iA*P~(V -1)'P, we find that Ii - Uli E 5) if and only if P~(V + 1)'P = O. (This follows from the fact that A*P~(V -1)'P E 1J(A) C 5) and from the formula 5) = 5)0 + 1)1 (see [4])). Let us note that if P~(V + 1)'P = 0 then Ii + Uli = 'P - V'P. Thus, (87) &ti = {f = (A* + i1)(I + P~)'P, P~(V + I)'P = O}. Let N = KerP~(I + V) . Then we have (88) 5)+ = 1J(A) + (I - V)N. We denote by Po the projection operator of 5)+ onto 1J(A) along (I - V)N, Pi = I - Po· Since 1J(A) = 5',+, we have Po E [5)+,5',+] . We will denote by Po E [5',-,5)-] the adjoint operator to Po, i.e. (Pol,g) = (J,Pog), for all I E 5)+, g E 5)_. If ii E &t;, then ii + uii ~ (I - V)'P, for 'P E N, and A*(I - V)'P = iP~:'P + iP~'-i V'P + AP~(I - V)'P = i(V + 1)'P + A* P~(I - V)'P = i[(I + V)'P - iA* P~(I - V)'P]. ReaIization theorems for operator-valued R-functions 75 This implies Hence (89) A' ( te) e - F t2 + 1 = - t2 + 1 ' e E 00· Let Q E [E, E] be the operator in the definition of the class N(R). We introduce a new operator Ro acting in the following way: (90) In order to show that Ro E [5)+, EJ, we consider the following calculation for 1 E 5)+: II R 111 I(Rol,g)EI I(Qft-1A'Pd,g)EI o E = ~~~ IlgilE = ~~~ IIgllE l(ft-1 A' Pd, Qg)EI < IIft- 1 A' PdllE ·IIQgIIE = ~~~ IlgilE - ~~~ IlgilE :::; cIIA'Pdll5',+ :::; bIIA'PdllfJ+' b,c - constants. Here we used that Pd c 1)(A), for all 1 E h+, formulas (65) and (89), and the equivalence of the norms II . 115',+ and II . 11+· For 1 E 5)+, we have Pd = (I - V) A' Pd = i(V + I) We now have IIA' P~(V - I) (91) 76 S. V. Belyi and E. R. Tsekanovskii Therefore, for some constant d > 0 we have IIRofll:::;dllfll+, Vf E 5)+ . Thus, Ro E [5)+, EJ. Let Ro be the adjoint operator to Ro, i.e. Ro E [E,5)- J and for all f E 5)+, e E E. (Rof,elE = (j,Roe). Since Ro(:D(Al) = 0, !R(Ro) is (·)-orthogonal to 1:>(A). Lptting !m = l)1'--i ttl 1)1; ttl 1)1, we obtain from (88) (92) !m = (V + 1)(1)1; ttl 1)1) + (1 - V)N. In the space !m we define an operator S in the following way i S(({J + V({J) = 2(1 - V)({J, ({J E 1)1; ttl 1)1 , (93) S(({JN - V({JN) = [-R1(R(j + PO')R.- 1A* + ~(P~: - P~'.)] (({IN - V({JN) , where ({IN E N. In order to show that S is a (I)-self-adjoint operator on !m, we first check that (94) (S(({J+V({J) , ({J+V({Jh = (({J+V({J,S(({J+V({J)h, ({JEI)1; ttl l)1. It is easy to see that This follows from the definition of the space N and the fact that ({J N belongs to 1)1;. Furthermore, since ({IN E 1)1;, and V({JN E l)1'--i we have that P~'-i({JN = ({IN, P~'- i V({JN = ({IN, and P~: V({JN = P~'-i ({IN = o. Consequently, ((({IN + V({JN), ({IN - V({JN h = II({JNlli -11V({JNlli (95) = 11P~: ({JNlli -IIP~,-y({JNlli = o. Since Po(1 - V)N = 0, we have This allows us to consider only the R(j-containing part of (93), i.e. (S(({JN - V({JN),({JN - V({JNh = (-R1R(jR.- 1A*(({JN - V({JN),(({JN - V({JNl! = (R.- 1 A*(({JN - V({JN) , -RO(({JN - V({JN))E = (R.- 1 A*(({JN - V({JN),iQR.- 1 A*P1(({JN - V({JN))E = (-iQR. -1 A * (({J N - V ({J N ), R. -1 A * (({J N - V ({J N )) E = ((({IN - V({JN),R(jR- 1 A*(({JN - V({JN))E = ((({IN - V({JN), R 1R(jR-1 A*(({JN - V({JN)h = ((({IN - V({JN),S(({JN - V({JN)h· Realization theorems for operator-valued R-functions 77 Now we will show that Let us note that P;Ji(<.pN + V<.pN) = 0 implies P;Ji<.pN = -P;JiV<.pN. Also, (<.p, z (S(<.p+ V<.p),<.pN - V<.pNh = 2((1 - V)<.p,<.pN - V<.pNh = i(<.p , <.pNh - ~( = i(<.p,<.pNh - ~(<.p, V<.pNh + ~(<.p,P;JiV<.pNh = i(<.p,<.pNh + ~(P;Ji(1- V)<.p,<.pNh · Also, note that and (<.p + V<.p, S(<.pN - V<.pN h = (<.p + V<.p , -Rl(R(j + PQ')ft-1A*(<.pN - V<.pN)h i - 2( Next, recall that 9t(Ro) is (·)-orthogonal to 1>(A) and <.p + V<.p E 1>(A) = 1>(A) ED (V + 1)(91~ ED 91). It follows that (<.p+ V<.p,R1R(jft-1A*(<.pN - V<.pN)h = (<.p+ V<.p,R(jft-1A*(<.pN - V<.pN)) = 0, (<.p+ V<.p,-R1PQ'ft-1A*(<.pN - V<.pN)h = -(<.p+ V<.p,A*(<.pN - V<.pN))Sh = -(<.p+ V<.p,A*(<.pN - V<.pN)) - (A(<.p+ V<.p) , AoA*(<.pN - V<.pN)). Applying Theorem 1 we obtain: A(<.p + V<.p) = A*(<.p + V <.p) + ~R-l P;Ji(I - V)<.p, 78 s. V. Belyi and E. R. lSekanovskii A*( . i 1 AA*( A*(I - V) E :D(A), AUi - UI;) = A*( Ii - U Ii = A( Thus, ( + i(P;jj(I - V) = i( + ~( = i( This shows that S is a (1 )-self-adjoint operator in 9)1. By Corollary 2, a self-adjoint bi-extension of the operator A is defined by the formula (98) Realization theorems for operator-valued R-functions 79 where S is defined by (97). Obviously, if I = I A + (V + J) STEP 4. In this Step we will construct a (* )-extension of some operator of the class AA. First, we introduce the bounded linear operator K acting from the space E into the space jj_ as follows: (99) where PF= and PE= are orthogonal projections of the space E onto Foo and Eoo respec• tively, and j is an embedding of Eoo in jj_. Let K* E [jj+, E] be an adjoint of the operator K, i.e. (KI,g) = (J,K*g), lEE, 9 E jj+. Let (100) C=K*JK, where J E [E, E] satisfies J = J* = J- 1. Since !R(K) is orthogonal to 1:l(A), C(1:l(A)) = o. Moreover, (CI,g) = (J,Cg) for all I E jj+, 9 E jj+. We define an operator A by (101) We now show that A is a (* )-extension of some operator T of the class AA. Let A be a regular point of the operator A and let R>. = (B - A1)-I. Also, note that As it was shown in Step 1 (see (71)) (A - AI)-1 = _e_ \Ie E E, t - A' where E is considered as a subspace of sL. Clearly, (R>.poe , g) = (Poe, (A - 5.1)-lg) = (e, (A - 5.1)-lg) = ((A - A1)-le,g), \Ie E E, 9 E jj = L~(E). It follows that A e (102) R>.poe = --" \Ie E E. t-/\ 80 S. V. Belyi and E. R. Tsekanovskii It follows that - e (102) R>.poe = --" "Ie E E. t - A Since Ro(1)(A)) = 0, Ro(A - )'1)-lg = 0, for all 9 E Sj, and we have - - * el R>.Kel = R>.POel = --" el E Foc" t - A - - e2 - R>.Ke2 = R>.e2--, E Sj+, e2 E E rx» t-A This implies that the operator K is invertible. Indeed, if Ke = 0, then (Pri+RO)el = -ie2 and R>.Ke = O. Hence, R>.(Pri+ Ro)e = -R>.e2. That is, which implies that e = O. We should also note that R>.K E [E, Sj+j, since R>. maps ~(K) into Sj+ continuously. Let us consider now the operator-valued function V defined by (103) V(A) = K* R>.K, ImA =1= O. Obviously, (V(A)e, h)E = (R>.Ke, Kh) for e E E, h E E, e = el + e2, h = hI + h2. Therefore, (R>.Ke, Kh) = (R>.(Po + RO)el + R>.e2, (Po + Ro)h l + ih2) = (R>.POel + R>.e2, (Po + Ro)h l + ih2) = (R>.POel,Pohl ) + (R>.POel, Rohd + (R>.POel,h2) + (R>.e2,pohd + (R>.e2, Roh2) + (R>.e2, h2) = (PoR>.POel, hI) + (PoR>.POel, h2) + (R>.POel, h2) + (R>.e2, h2) + (RoR>.e2' h2)E + (R>.e2, h2)' We also have Consider an element (t - A) (t 2 + 1) , Realization theorems for operator-valued R-functions 81 Clearly -00 and hence ~-~E1)(A)o t - A t 2 + 1 Moreover, tel -2-- E (I - V)N, e1 E Fx;, o t + 1 This implies Consequently, -00 We also have that (RoR)..Po,hl)E = _(QR-I A* PIR)..POel, hdE = _(R- I A* PI R).. POe] , QhdEo From (65) and (89) we obtain from which it follows that Furthermore we obtain +00 (R)..Pael, h2) = J C~ A) d(G(t)e2' h2)E -00 +00 = J C~ A) d(G(tle2, h2 )E - (Qe], h2 )E + (Qe], h2 )E -00 +00 = Jt2: 1 d(G(t)el, h2)E + (Qel, h2)E -00 -00 82 S. V. Belyi and E. R. Tsekanovskii Since RoR)..e2 = 0, we have +00 (R)..e2, hil = J C~ .J d(G(t)e2 ' hilE - (Q e2, hdE + (Q e2, h1)E -00 -00 Thus, (104) -00 These calculations imply +00 (R)..e, h) = J (t ~ A - t2 ~ 1) d(G(t)e, h)E + (Qe, h)E, -00 hence, +00 (105) (V(A)e, h) = (~ -2-t -) d(G(t)e, h)E + (Qe , h)E J t-A -t +1 -00 Next, we show that (lffi + iI)R±iKe = K e, for all e E E, where lffi is the strong self• adjoint bi-extension defined by (98). By Theorem 7, the equation (lffi - AI)X = f has a unique solution x for any E [ -1 ( 5 - -i P'11'+ i P'11'+ . )] Eoo· f ~ n 1 2 ' + - 2 -, + We will now show that in fact If 'PN E N, then 5 - ip;};, + ip;};, .) ('PN - V 'PN) = n 1(Ro+ pnn-1A*('PN - V 'PN). ( . 2 ' 2 -, Using (89) we can conclude that n- 1(I - V)N = Foo , and hence ip;};, + ip;};, .)] (I - V)N = (PO' + Ro)Foo . ~ [nil (5 - 2 ' 2 -, Realization theorems for operator-valued R-functions 83 Letting P+ = P;Ji: + P;Ji,-" we have Therefore, Eoo 9't --I( S - -P'J!'i+ -P'J!'i+)]_ - 9't(K). + [n 2 ' + 2 -, Since R).. = (Jffi - >.I) -I, the above calculations imply (106) for all e E E. For 1m>. =I 0 we have that R)..K E = l)1).. is the defect space of the operator A . Therefore (Jffi + iI)R±iKe = Ke .and R±iKE = l)1±i. Taking into account (105) we get +00 V( -i) = (_1_ -_t-)dG(t) + Q J t + i t 2 + 1 -00 (107) +00 = -i J dG(t) + Q 1 + t 2 -00 =-iB+Q. Therefore, (108) iV( -i)J + 1 = BJ + iQJ + 1. The operator iV(-i)J + 1 is invertible and so is the right hand side of (108). Since 1 + B J + iQ J = J (I + J B + iJQ) J, where J is a unitary self-adjoint operator in the space E, 0 is a regular point for the operator 1 +BJ +iJQ. At the same time 0 is a regular point for the operators 1 + J B - iJQ = (BJ + iQJ + 1)* and 1 + BJ - iQJ = (1 + JB + iJQ)* . Let Z=(I+BJ-iQJ)-I, ZE[E,E], (109) Z* = (I + J B + iJQ)-I, Z* E [E, E], and let r = (1 + JB + iJQ)-l. Clearly Kerr = o. We will show that for any J E E, the equation (110) (A + i1)g = K J, 84 S. V Belyi and E. R. Tsekanovskii has a unique solution 9 = fLiKf I, where fLi = (JB + iI)-I and A = JB + ie. Moreover, AfLiKf1= JB fLiKf 1+ iK J K* fLiKf I, fEE. As shown above (see also [2)) K' fLif 1= V( -i)fl = (Q - iB)f I, iK JK* fLiKfl = K(JB + iJQ)fl = K(I + JB + iJQ)(I + JB + iJQ)-If - Kfl = KI - Kff, lEE. Also, (A + iI)R_iKf f = (JB + iI)R_iKf f + iK J K' R_iKf I = KI, lEE. If there exists agE 5)+ such that Ag = -ig, then 9 E l)1-i. Since 9't(f) = E, we find that R_iKf E = l)1-i. Therefore 9 = R_iKfe, e E E, and (A + iI)R_iKfe = 0, K e = 0, e = 0, and 9 = 0. It follows that the equation (A + iI)g = K I has a unique solution given by 9 = fLiKff and (A + iI)-IKE = l)1-i. Similarly, °is the regular point for the operator 1+ J B - iJQ in E. Let (111) In the same way as above, we can show that the equation (A * - iI) 9 K f, lEE, has a unique solution of the form 9 = R;Kfd and (A' - iI)-I K E = l)1i . If Ii E l)1i, then f; = fA + f'JR , where IA E :D(A), I'JR E 9J1 = 1)1; EB I)1'-i EBI)1. Therefore, A * Ii = P AfA + A' f'JR = iP J;, A*f'JR = iPf; - PAfA, and (A + iI)f; = (A + iI)IA + iP fi - P AfA + if'JR -1 ~. + ~ . + ) ., RI 5 - -Pm' -Pm' . f'JR ~K JK + ( 2 ' + 2 -, + J;, = (I - P)(A + iI)fA + i(P - I)f; E Eoo C 9't(K). This implies that (A + iI)fi - 2ifi = (A + iI)k That is 2if; = (A + iI)(Ji - f-i)' (J-i E l)1-i). Hence (A + iI)5)+ C l)1i. Since (A + iI):D(A) = (A + iI):D(A), Realization theorems for operator-valued R-functions 85 and (A+iI)~(A)EBSJt; = jj, we have (A+iI)jj+ C jj. Similarly, (A" -iI)jj+ C jj. Therefore we can conclude that the operators (A + iI)-l and (A" - iI)-l are (-, ·)-continuous (see [25]). Let ~(T) = (A + iI)-ljj, (112) ~(Td = (A" - iI)-ljj. It is easy to see that ~(T) and ~(Tl) are dense in jj and that the operators (A + iI)-l l5) and ( A" - iI) -115) are (.,. )-continuous. Let us define T-AI- :D(T) ' (113) Tl = A"I :D(Td . The points (i) and (-i) are regular points for the operators T and Tl respectively. This implies that Tl = T* . Since T and T" are quasi-kernels of operators A and A" respectively, and ReA = lffi is a strong self-adjoint bi-extension of the operator A we find that T E AA (the fact that PT and PT" are closed follows from the (+, . )-continuity of T and T"). STEP 5. Let us construct a linear stationary conservative dynamical system B. Let K E [E, jj -l be the operator defined in the Step 4. It is easy to see that ;i(A-A") =KJK". Therefore, 8- ( A - jj+ C jj C jj_ K EJ) is a l.s.c.d.s. In particular, B is a scattering system if J = I. Since Ve(z) is a linear• fractional transformation of We(z) then Ve(z) = V(z) whenever z is in some neighborhood G -i of the point (-i). This completes the proof of the theorem. Remark. It can be seen that when J = I the invertibility condition for I + iV(A)J is satisfied automatically. Theorem 10. Let an operator-valued function V(z) belong to the class N(R). Then V(z) can be realized' by the scattering (J = I) system (dissipative operator colligation) B of the form (30). The following theorem deals with the realization of two realizable operator-valued R• functions differing from each other only by the constant terms in the representation (48). 86 S. V. Belyi and E. R. Tsekanovskii Theorem 11. Let the operator-valued functions +00 (114) VI ().) = QI + _t-2) dG(t) J (~-t-A l+t -00 and +00 (115) V2 ().) = + _t dG(t) Q2 J (_1_t-). - l+t-2) -00 belong to the class N(R). Then they can be realized by systems (116) () _ ( Al I - -D+ C -D C -D- ~) and (117) respectively, so that the operators TI and T2 acting on the Hilbert space -D are both extensions of the Hermitian operator A defined in this Hilbert space. Proof. Applying Theorem 9 to the function VI ().), we obtain a l.s.c.d.s. ()I of the type (116). The corresponding Hermitian operator Al constructed in the Steps 1 and 2 of the proof of Theorem 9 satisfies the formulas (72) and (73). The construction of Al doesn't involve the operator QI from (114) . It is easy to see that the corresponding rigged Hilbert space -D~) C -D(1) C -D~) was built without the use of the operator QI too. Similarly, if we apply Theorem 9 to the function V2 ().) we get the corresponding Her• mitian operator A2 = Al and the same rigged Hilbert space. This occurs because the operator-functions VI ().) and V2 ().) differ from each other only by the constant terms QJ and Q2. Setting A = Al = A 2, we can conclude that TI and T2 are both extensions of the Hermitian operator A. A closed Hermitian operator A is called a prime operator [25] if there exists no reducing invariant subspace on which it induces a self-adjoint operator. Definition. A l.s.c.d.s. () of the form (30) is said to be a prime system if its Hermitian operator A is a prime operator. Theorem 12. Let the operator-valued function V(z) belong to the class N(R) . Then it can be" realized by the prime system () of the form (30) with a preassigned direction operator 1 for which 1+ iV( -ill is invertible. Proof. Theorem 9 provides us with a possibility of realization for a given operator-valued function V(z) from the class N(R). Let us assume that its Hermitian operator A has a Realization theorems for operator-valued R-functions 87 reducing invariant subspace 5)1 C 5) on which it generates the self-adjoint operator AI. Then we can write the following (. )-orthogonal decomposition (118) where AD is an operator induced by A on 5)0. Now let us consider an operator T :J A as in the definition of the system B. We have (119) where To :J Ao . Indeed, since Al is a self-adjoint operator it can not be extended any further. Clearly, :D(Ad = 5)1 . Similarly, (120) where TO' :J Ao. Furthermore, We now show that the same holds in the (+ l-orthogonality sense. Indeed, if io E 5)~ , il E 5)~ = :D(Ad then (fo, ill+ = (fo, ill + (A* io, A* ill = (fo, il) + (AciD, Adl) = 0 + 0 = O. Consequently, we have 5)+ C 5) C 5)- = 5)~ EEl 5)~ C 5)0 EEl 5)1 C 5)~ EEl 5)~ = 5)~ EEl :D(A1 ) C 5)0 EEl :D(A1 ) C 5)~ EEl 5):. Similarly, we obtain A = AD EEl Al and A* = Ao EEl AI. Therefore, A - A* (AD EEl AI)- (A(j EEl Ad 2i 2i AD - A(j Al - Al 2i EEl 2i _Ao-A(j 0 - 2i EEl, where 0 is the zero operator. This implies that KJK* =KoJK~ EEl O . 88 s. V. Belyi and E. R. Tsekanovskii Let p~ be an orthoprojection operator of.!j+ onto.!j~ and set K = Ko. Now K* = KoP~ , since for all fEE, 9 E .!j+ we have: (Kf,g) = (Kof,g) = (Kof,go + gl) = (Kof,go) + (Kof, gt) = (Kof, go) = (1, Kogo) = (1, KoP~g). Next, consider e E E and x = x O + Xl in .!j+ such that (A - )..1)P~x = K e . Then (An ffi Al - )..1)P~x = Koe, Aoxo - )..xo = Koe, (A - )..1)xo = Koe, x O = (Ao - )..J)-IKoe. On the other hand, x O = (A - )..1) -1 K e. Therefore and This means that the transfer operator-functions of our system e and of the system e _ ( Ao Ko J) o - .!j+ C .!j C .!j_ E coincide. This proves the statement of the theorem. 4. EXAMPLE Let Tx = ~ dx i dt' with :D(T) = {x(t) : x' (t) ELfo,lj ' x(O) = O} , be a differential operator in .!j = Lfo,lj (I > 0). Obviously, T* _ ~ dx x - i dt' with :D(T*) = { x (t) : x'(t) ELfo ,lj ' x(l) = O}, Realization theorems for operator-valued R-functions 89 is the adjoint operator of T. Consider the Hermitian operator A (see also [1]) defined by Ax = ~ dx i dt' 1)(A) = {x(t) : x' (t) EL[o ,1] , x(O) = x(l) = O} , where its adjoint A * is given by A*x = ~ dx i dt' 1)(A*) = {x(t): x'(t) E L[o ,ll}' Then SJ+ = 1)(A*) = wi is a Sobolev space with scalar product (x, y)+ = 11 x(t)y(t) dt + l x'(t)Y'(t) dt. We construct the rigged Hilbert space [9] and consider the operators Ax = ; ddX + ix(O) [o(x -I) - o(x)] , 1 t A*x = ; ddx + ix(l) [o(x -l) - o(x)] , 1 t where x (t) E Wi , o(x), o(x -I) are delta-functions in (Wil-. It is easy to see that A::J T::J A, A* ::J T* ::J A, and __ (+ ~~ + ix (0)[ 0(x - I) - 0( x) ] K -1 ) e (J = -1) W{l c Lro,ll c (Wil- C1 is a BrodskiI-Livsic rigged operator colligation where 1 Kc=c· J2[o(x-I)-O(x)], (cECI ) K*x = (x, ~[O( x -I) - O(X)]) = ~[x(l) - x(O)], for x(t) E wi- Also -.-A - A* = -(1 . -[o(x -I) - o( x )] ) -[o(x 1 -I) - o(x)]. 21 'J2 J2 90 S. V. Belyi and E. R. 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DEPARTMENT OF MATHEMATI CS TROY STATE UNIVERSITY TROY , AL 36082 E-mail address: sbelyi@trojan .troyst.edu DEPARTMENT OF MATHEMATICS UNIVERSITY OF MISSOURI-COLUMBIA COLUMBIA, MO 65211 E-mail address: [email protected] 1991 MATH EMATICS SUBJ ECT CLASSIFICATION. PRIMARY 47 AlD, 47B44; SECONDARY 46E20, 46F05