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Constant J-Unitary Factor and Operator-Valued Transfer Functions

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Constant J-Unitary Factor and Operator-Valued Transfer Functions PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 – 27, 2002, Wilmington, NC, USA pp. 48–56 CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS Yury Arlinski˘ı Department of Mathematics East Ukrainian National University Lugansk, 91034, Ukraine Eduard Tsekanovski˘ı Department of Mathematics POB 2044 Niagara University, NY, 14109, USA Abstract. We study linear, conservative, stationary dynamical systems (l.c.s.d.s) with unbounded main operator and their transfer operator-valued functions. A new class of transfer operator-valued functions is introduced and shown that this class is invariant with respect to multiplication (from the left, from the right or both) by a constant J-unitary operator. The problem how to realize the product W (z)B as a transfer function of some l.c.s.d.s with an arbitrary J-unitary operator B and given realization of W (z) is considered. 1. Introduction. We keep the following notations: L(H1; H2) denotes the Ba- nach space of all continuous linear operators acting from the Hilbert space H1 into the Hilbert space H2, L(H) = L(H; H) and D(Q); R(Q), Ker Q, ½(Q) denote the domain, the range, the null-space and the resolvent set of a linear operator Q, respectively. Let H be a separable Hilbert space with the inner product (¢; ¢) and let H+ ½ H ½ H¡ be a rigged Hilbert space [5]. If E is a Hilbert space and F is a bounded operator £ acting from E into H+ (H¡), then by F we will denote the adjoint operator with respect to inner product (¢; ¢) which is acting from H¡ (H+) into E. Let A be a £ bounded operator from H+ into H¡. The adjoint operator A we also consider as operator from H+ into H¡. Consider a linear, conservative, stationary dynamical system Θ of the form ½ (A ¡ zI)x = KJ'¡ ¤ '+ = '¡ ¡ 2iK x where '¡ 2 E is an input vector, '+ 2 E is an output vector, and x is a state space vector in H, K is a bounded operator from a Hilbert space E into H¡ and J is a signature operator in E (J = J ¤ = J ¡1) and 2iK£JK = A ¡ A£: For the systems Θ we use also the following notation in the form of Brodskii-Livsic rigged operator colligations [1], [4] [6], [7], [9] [10] : µ ¶ A KJ Θ = (1) H+ ½ H ½ H¡ E 1991 Mathematics Subject Classification. Primary: 47A63, 47B25; Secondary: 47B65. 48 CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 49 The importance of such type of systems was pointed out in [7], [8]. In this paper we deal with the special class of systems with the main operator A which satisfies the condition A ⊃ T; A£ ⊃ T ¤; (2) where T is some densely defined unbounded operator acting on Hilbert space H with nonempty resolvent set ½(T ) and the additional property: the linear manifold D(A˚) := ff 2 D(T ) \D(T ¤); T f = T ¤fg is dense in H and the symmetric operator A˚ := T ¹ D(A˚) has finite and equal defect numbers hn; ni. ¤ The Hilbert space H+ in this case is the linear manifold D(A˚ ) equipped by the inner product ¤ ¤ (u; v)+ = (u; v) + (A˚ u; A˚ v); the Hilbert space E is finite-dimensional with dim E = n. and the operator A apart from (2) is chosen as satisfying the condition the linear manifold Ker (A ¡ A£) is the domain (3) of some selfadjoint operator: The transfer operator-valued function of the system takes the form £ ¡1 WΘ(z) = I ¡ 2iK (A ¡ zI) KJ; z 2 ½(T ); (J) is well defined and belongs to the following class ΛE [1], [2], [9]: 1. Wθ(z) is holomorphic function on ½(T ); 2. the fractional linear transformation ¡1 VΘ(z) = i (WΘ(z) + I) (WΘ(z) ¡ I) J initially defined on ½(T ) admits analytical continuation to a Nevanlinna op- erator valued function which has the integral representation Z µ ¶ +1 1 t VΘ(z) = ∆ + ¡ 2 dΣ(t); ¡1 t ¡ z t + 1 where ∆ is selfadjoint operator in E and Σ(t) is nondecreasing operator-valued function such that Z +1 Z +1 d(Σ(t)e; e)E 2 < 1; d(Σ(t)e; e)E = +1; e 2 E n f0g: ¡1 1 + t ¡1 It is known [9] that £ ¡1 VΘ(z) = K (AR ¡ zI) K; £ where AR = (A + A )=2 is the real part of A. Recall that a bounded operator B acting on Hilbert space E is called J-unitary if BJB¤ = J and B¤JB = J: In this paper we prove that if an operator-valued function W (z) acting on finite- (J) dimensional Hilbert space E belongs to the class ΛE , then for an arbitrary J- unitary operator B the functions W (z)B and BW (z) belong again to the same (J) class ΛE and can be realized as transfer operator-valued functions of the same type of l.c.s.d.s as W (z) and with the same unbounded operator T and different main operators A. 50 YURY ARLINSKI˘I AND EDUARD TSEKANOVSKI˘I 2. Construction of the main operator of a system. Let H be a separable Hilbert space and let T be a densely defined on D(T ) linear operator with nonempty resolvent set ½(T ). Consider HT = D(T ) as a Hilbert space HT with the inner 0 product (u; v)T := (u; v) + (T u; T v); u; v 2 HT . Let HT ½ H ½ HT be the rigged Hilbert space. Since T is a bounded operator from HT into H, the adjoint operator £ 0 £ T is bounded from H into HT and satisfies the condition (T u; f) = (u; T f) for ¤ £ ¤ all u 2 HT and all f 2 H. Let T be the adjoint of T in H. Then T ⊃ T . The ¡1 resolvent R»(T ) = (T ¡ »I) of T is bounded operator from H onto D(T ) = HT : It ¤ ¤ ¯ ¡1 ¤ ¤ follows that the resolvent R»¯(T ) = (T ¡ »I) of T has the continuation R»¯(T ) 0 0 ¤ £ ¯ ¡1 on HT and maps HT onto H. Moreover, R»¯(T ) = (T ¡ »I) : £ We suppose that the imaginary part (T ¡ T )=2i which is defined on HT is singular with respect to H. This means that Ker (T ¡ T£) is dense linear manifold 0 in H or equivalently the subspace in HT defined by £ 0 Φ := R(T ¡ T ) (the closure in HT ) (4) satisfies the condition Φ \ H = f0g: Let us define the following operator in H £ D(A˚) := Ker (T ¡ T ) = fh 2 HT :(h; ') = 0 for all ' 2 Φg ; (5) A˚ = T ¹ D(A˚) The operator A˚ is a densely defined, closed symmetric operator and T ⊃ A˚, T ¤ ⊃ A˚. ¤ Let Nz = Ker (A˚ ¡ zI) be the defect subspaces of A˚. Then the following identity holds: £ ¡1 ¤ (T ¡ zI) Φ = Nz; z 2 ½(T ) (6) Our main assumptions on A˚ defined by (5): 1. A˚ has equal defect numbers, 2. there exists a selfadjoint extension Ae of A˚ in H such that D(T ) + D(Ae) = D(A˚¤): (7) ¤ Denote by H+ the Hilbert space D(A˚ ) equipped by the inner product ¤ ¤ (u; v)+ = (u; v) + (A˚ u; A˚ v): Then (+)-orthogonal decomposition holds: H+ = D(A˚) © Ni © N¡i: By the von Neumann formula [3] the domain of every selfadjoint extension Ae of A˚ can be represented in the form D(Ae) = D(A˚) © (I + Ve)Ni; where Ve is an isometry from Ni onto N¡i. Let Ae be a selfadjoint extension of A˚ satisfying the condition (7). Then the direct decomposition takes place H+ = HT +(˙ I + Ve)Ni: (8) ¤ ¤ ¤ ¤ Let HT ¤ = D(T ), (u; v)T ¤ = (u; v) + (T u; T v), u; v 2 D(T ) and let HT ¤ ½ H ½ 0 0 HT ¤ be the corresponding rigging. The operator T : H ! HT ¤ which is defined as the adjoint to T ¤: (T ¤v; f) = (v; Tf) 0 is the continuation of the operator T . Note that the subspace in HT ¤ ¤ 0 Φ¤ := R(T ¡ T ) ( the closure is taken in HT ¤ ) (9) CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 51 satisfies the condition Φ¤\H = f0g: The Hilbert spaces HT and HT ¤ are subspaces of ¤ the Hilbert space H+ and one can easily show that if (7) holds then D(T )+D(Ae) = D(A˚¤). Therefore, H+ = HT ¤ +(˙ I + Ve)Ni: (10) We denote by Pe and Pe¤ the skew projections in H+ onto HT and HT ¤ with respect to the decompositions (8) and (10), respectively. Let H+ ½ H ½ H¡ e£ 0 e£ 0 be the rigged space and let P : HT ! H¡ and P¤ : HT ¤ ! H¡ be the adjoint operators. These operators are defined by the relations e e£ 0 (Pu; ') = (u; P '); u 2 H+;' 2 HT and e e£ 0 (P¤v; Ã) = (v; P¤ Ã); u 2 H+;à 2 HT ¤ : Observe that in view of (8) and (10) the operators Pe and Pe¤ are isomorphisms from HT ¤ onto HT and HT onto HT ¤ , respectively. Moreover the relations hold PePe¤fT = fT ; fT 2 HT ; (11) Pe¤PefT ¤ = fT ¤ ; fT ¤ 2 HT ¤ and e£ e£ e P Φ = P¤ Φ¤ = Ψ; where n o Ψe := ' 2 H¡ :('; h) = 0 for all h 2 D(Ae) : Theorem 1. Let the operator APe : H+ ! H¡ be defined by the formula ˚¤ e£ £ ˚¤ e APe = A + P (T ¡ A )(I ¡ P): (12) Then its adjoint A£ : H ! H acts by the rule Pe + ¡ A£ = A˚¤ + Pe£(T ¡ A˚¤)(I ¡ Pe ): (13) Pe ¤ ¤ Moreover, T ½ A , T ¤ ½ A£ and the following relations hold Pe Pe A = A˚¤ + Pe£(T ¡ A˚¤); A£ = A˚¤ + Pe£(T£ ¡ A˚¤); Pe ¤ Pe A ¡ A£ = Pe£(T ¡ T£)Pe; A ¡ A£ = Pe£(T ¡ T ¤)Pe ; Pe Pe Pe Pe ¤ ¤ ¡1 e£ ¡1 (APe ¡ zI) P¤ ' = (T ¡ zI) '; ' 2 Φ¤ (A£ ¡ zI¯ )¡1Pe£' = (T£ ¡ zI¯ )¡1'; ' 2 Φ; Pe ¡ ¢ ³ ´ (A ¡ zI)¡1Ã; à = Ã; (A£ ¡ zI¯ )¡1à ;à 2 Ψe; z 2 ½(T ): Pe Pe The resolvents (A ¡ zI)¡1, (A£ ¡ zI¯ )¡1 map H+˙ Ψe onto H for all z 2 ½(T ): Pe Pe + The real part A = (A + A£)=2 satisfies the condition A ⊃ Ab, where Ab is a PeR Pe Pe PeR ˚ e ¡1 ˙ e selfadjoint extension of A transversal to A.
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