Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators

A class of characteristic functions corresponding to commutative systems of unbounded nonselfadjoint operators is studied. The theorem on unitary equivalence is proved. The class of functions corresponding to these commutative systems of unbounded nonselfadjoint operators is described. There is obta...

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1. Verfasser:	Zolotarev, V.A.
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spelling	irk-123456789-1066402016-10-02T03:02:52Z Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators Zolotarev, V.A. A class of characteristic functions corresponding to commutative systems of unbounded nonselfadjoint operators is studied. The theorem on unitary equivalence is proved. The class of functions corresponding to these commutative systems of unbounded nonselfadjoint operators is described. There is obtained an analogue of the Hamilton{Caley theorem demonstrating that in the case of finite dimensionality of deficient subspaces there exists such a polynomial P (λ₁, λ₂) that annihilates the resolvents Rk = (Ak - αI)⁻¹; P (R₁, R₂) = 0. 2010 Article Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 2. — С. 192-228. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106640 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description	A class of characteristic functions corresponding to commutative systems of unbounded nonselfadjoint operators is studied. The theorem on unitary equivalence is proved. The class of functions corresponding to these commutative systems of unbounded nonselfadjoint operators is described. There is obtained an analogue of the Hamilton{Caley theorem demonstrating that in the case of finite dimensionality of deficient subspaces there exists such a polynomial P (λ₁, λ₂) that annihilates the resolvents Rk = (Ak - αI)⁻¹; P (R₁, R₂) = 0.
format	Article
author	Zolotarev, V.A.
spellingShingle	Zolotarev, V.A. Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators Журнал математической физики, анализа, геометрии
author_facet	Zolotarev, V.A.
author_sort	Zolotarev, V.A.
title	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators
title_short	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators
title_full	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators
title_fullStr	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators
title_full_unstemmed	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators
title_sort	properties of characteristic function of commutative system of unbounded nonselfadjoint operators
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/106640
citation_txt	Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 2. — С. 192-228. — Бібліогр.: 11 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT zolotarevva propertiesofcharacteristicfunctionofcommutativesystemofunboundednonselfadjointoperators
first_indexed	2025-07-07T18:48:21Z
last_indexed	2025-07-07T18:48:21Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2010, vol. 6, No. 2, pp. 192–228 Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators V.A. Zolotarev Department of Higher Mathematics and Informatics V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv, 61077, Ukraine Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv, 61103, Ukraine E-mail:Vladimir.O.Zolotarev@univer.kharkov.ua Received April 1, 2009 A class of characteristic functions corresponding to commutative systems of unbounded nonselfadjoint operators is studied. The theorem on unitary equivalence is proved. The class of functions corresponding to these commu- tative systems of unbounded nonselfadjoint operators is described. There is obtained an analogue of the Hamilton–Caley theorem demonstrating that in the case of finite dimensionality of deficient subspaces there exists such a polynomial P (λ1, λ2) that annihilates the resolvents Rk = (Ak − αI)−1; P (R1, R2) = 0. Key words: commutative system, unbounded operators, characteristic function. Mathematics Subject Classification 2000: 47A45. In [1], M.S. Livs̆ic introduced an effective method of study of unbounded nonselfadjoint operators which was further developed by A.V. Kuzhel [2, 3], A.V. Shtraus, E.R. Tsekanovsky, and Yu.L. Shmul’yan [5]. Another approach to the studying of unbounded nonselfadjoint operators based on the analysis of the boundary value space was developed in the works by V.A. Derkach and M.M. Malamud which resulted in the analytic formalism for studying the pro- perties of Weyl functions. The dissipative Srödinger operator and its functional model was studied by B.S. Pavlov [7] and his disciples. In the previous work [11] the author suggested a method of study of commutative system of non- selfadjoint unbounded operators which was based on the concepts of commuta- tive colligation and open system associated with it. The paper consists of three parts. The first one includes the necessary facts on the commutative systems c© V.A. Zolotarev, 2010 Characteristic Function of Commutative System Operators of unbounded nonselfadjoint operators. In Section 2 the main properties of the characteristic function of commutative colligations are studied, the complete set of invariants of commutative system of unbounded nonselfadjoint operators is defined and the theorem on the unitary equivalence is proved. It turned out that the characteristic function, besides the traditional J-properties, must satisfy three additional relations, which are the corollary of the commutative property of the initial operator system. Section 3 is dedicated to the description of the class of functions that are characteristic for commutative colligations. An analogue of the Hamilton–Caley theorem is proved, namely, it is proved that in the case of the finiteness of the outer spaces there exists the polynomial P (λ1, λ2) such that P (R1, R2) = 0, where Rk = (Ak − αI)−1 is the resolvent of Ak, k = 1, 2. It is determined that the polynomial P (λ1, λ2) has the “involution” generated by the inversion with respect to some circle. 1. Preliminary Information I. Recall the main definitions and statements about commutative systems of nonselfadjoint unbounded operators given in [11]. Definition 1 [11]. Let a system of the linear unbounded operators {A1, A2} be defined in a Hilbert space H such that: a) the domain D (Ap) of the operator Ap is dense in H, D (Ap) = H, p = 1, 2; b) every operator Ap is closed in H, p = 1, 2; c) there exists the nonempty domain Ω ⊂ C\R such that the resolvents Rp(λ) = (Ap − λI)−1 are regular for all λ ∈ Ω, p = 1, 2; d) at least in one point α ∈ Ω, the resolvents R1 (= R1(α)), R2 (= R2(α)) commute. And let the Hilbert spaces E± and the linear bounded operators ψ−: E− → H, ψ+: H → E+ and { σ−p }2 1 , { τ−p }2 1 , {Np}2 1, Γ: E− → E−; { σ+ p }2 1 ; { τ+ p }2 1 , { Ñp }2 1 , Γ̃: E+ → E+ be given, { σ±p }2 1 and { τ±p } be selfadjoint. The family ∆ = ∆(α) = ( { σ−p }2 1 ; { τ−p }2 1 ; {Np}2 1 ; Γ; H ⊕E−; {[ Ap ψ− ψ+ K ]}2 1 ; H ⊕ E+; Γ̃; { Ñp }2 1 ; { τ+ p }2 1 ; { σ+ p }2 1 ) (1.1) is said to be a commutative colligation if there exists such α ∈ Ω that: 1) 2 Imα ·N∗ p ψ∗−ψ−Np = K∗σ+ p K −σ−p ; 2 Im α · Ñpψ+ψ∗+Ñ∗ p = Kτ−p K∗− τ+ p ; 2) the operators ϕp + = ψ+ (Ap − αI) : D (Ap) → E+, ( ϕp − )∗ = ψ∗− ( A∗p − ᾱI ) : D ( A∗p ) → E− Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 193 V.A. Zolotarev are such that: 3) K∗σ+ p ϕp + + N∗ p ψ∗− (Ap − ᾱI) = 0; Kτ−p ( ϕp − )∗ + Ñpψ+ ( A∗p − αI ) = 0; 4) 2 Im 〈Aphp, hp〉 = 〈 σ+ p ϕp +hp, ϕ p + 〉 ; ∀hp ∈ D (Ap); −2 Im 〈 A∗ph̃p, h̃p 〉 = 〈 τ−p ( ϕp − )∗ h̃p, ( ϕp − )∗ h̃p 〉 ; ∀h̃p ∈ D ( A∗p ) , (1.2) where p = 1, 2. And, moreover, the relations: 5) R2ψ−N1 −R1ψ−N2 = ψ−Γ; Ñ1ψ+R2 − Ñ2ψ+R1 = Γ̃ψ+; 6) Γ̃K −KΓ = i ( Ñ1ψ+ψ−N2 − Ñ2ψ+ψ−N1 ) ; 7) KNp = ÑpK; are true, where Rp = Rp(α), p = 1, 2. It is easy to show [11] that for every operator system satisfying the assump- tions a)–d) there always exist Hilbert spaces E± and corresponding operators ψ±; K; { σ±p }2 1 , { τ±p }2 1 ; {Np}2 1; { Ñp }2 1 ; Γ; Γ̃; such that the relations 1)–7) (1.2) hold. In the studying of nonselfadjoint operators the open systems associated with the corresponding colligations play an important role [8, 10]. Denote a rectangle in R2 + by D = [0, T1]× [0, T2], 0 < Tp < ∞, p = 1, 2, and let u−(t) be a vector function in E− defined as t = (t1, t2) ∈ D. The system of the relations R∆ :    i∂1h1(t) + A1y1(t) = αψ−N1u−(t); y1(t) = h1(t) + ψ−N1u−(t) ∈ D (A1) ; i∂2h2(t) + A2y2(t) = αψ−N2u−(t); y2(t) = h2(t) + ψ−N2u−(t) ∈ D (A2) ; h1(0) = h1; h2(0) = h2; t = (t1, t2) ∈ D, (1.3) where ∂p = ∂/∂tp, p = 1, 2, is said to be the open system F∆ = {R∆, S∆} associated with the colligation ∆ (1.1) and, moreover, the vector functions y1(t) and y2(t) are such that there exists y(t) from H, and y1(t) = R1y(t); y2(t) = R2y(t). (1.4) Thus, the functions {yp(t)}2 1 have a common generator y(t), and (1.4) implies R1y2(t) = R2y1(t). (1.5) As for the initial data h1 and h2 in (1.3), we suppose hp = Rpy(0)− ψ−Npu−(0), p = 1, 2. (1.6) The mapping S∆ is given by S∆ : u+(t) = Ku−(t)− iψ+y(t). (1.7) 194 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators Consider the differential operators Lp = i∂p + α, p = 1, 2. (1.8) Then the main equations (1.3) can be written in the following form:    L1h1(t) + y(t) = 0; R1y(t) = h1(t) + ψ−N1u−(t) ∈ D (A1) ; L2h2(t) + y(t) = 0; R2y(t) = h2(t) + ψ−N2u−(t) ∈ D (A2) . (1.9) Thus, L1h1(t) = −y(t) = L2h2(t). Therefore, taking into account (1.8) and (1.3), we obtain    R1L1y(t) + y(t) = ψ−N1L1u−(t); R2L2y(t) + y(t) = ψ−N2L2u−(t); y(0) = y0; t = (t1, t2) ∈ D; u+(t) = Ku−(t)− iψ+y(t). (1.10) If y(t) satisfies relations (1.10), then h1(t), h2(t) (1.9) and, correspondingly, y1(t) and y2(t) (1.4) are uniquely found by this function. Theorem 1.1 [11]. The equation system (1.3) is consistent if the vector function u−(t) is the solution of the equation {N1L1 −N2L2 + ΓL1L2}u−(t) = 0 (1.11) on condition that (1.4), (1.6) hold and Lp are given by (1.8), p = 1, 2. Theorem 1.2 [11]. If (1.10) takes place for the vector function y(t) and u−(t) is the solution of (1.11), then u+(t) (1.7) satisfies the equation { Ñ1L1 − Ñ2L1 + Γ̃L1L2 } u+(t) = 0. (1.12) Theorem 1.3 [11]. For the open system F∆ = {R∆, S∆} (1.3), (1.7) associ- ated with the colligation ∆ (1.1), the conservation laws hold: 1) ∂1 ‖hp(t)‖2 = 〈 σ−p u−(t), u−(t) 〉− 〈 σ+ p u+(t), u+(t) 〉 , p = 1, 2; 2) ∂2 {〈 σ−1 L1u−(t), L1u−(t) 〉− 〈 σ+ 1 L1u+(t), L1u+(t) 〉} = ∂1 {〈 σ−2 L2u−(t), L2u−(t) 〉− 〈 σ+ 2 L2u+(t), L2u+(t) 〉} . (1.13) Along with the open system F∆ = {R∆, S∆} (1.3), (1.7), describing the evo- lution generated by {A1, A2}, consider also the dual situation corresponding to the dynamics set by the adjoint operator system {A∗1, A∗2}. Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 195 V.A. Zolotarev Let the vector function ũ+(t) in E+ be specified in the rectangle D = [0, T1]× [0, T2] from R2 +, t = (t1, t2) ∈ D, 0 < Tp < ∞; p = 1, 2. The equation system R+ ∆ :    i∂1h̃1(t)−A∗1ỹ1(t) = −ᾱψ∗+Ñ∗ 1 ũ+(t); ỹ1(t) = ψ∗+Ñ∗ 1 ũ+(t)− h̃1(t) ∈ D (A∗1) ; i∂2h̃2(t)−A∗2ỹ2(t) = −ᾱψ∗+Ñ∗ 2 ũ+(t); ỹ2(t) = ψ∗+Ñ∗ 2 u+(t)− h̃2(t) ∈ D (A∗2) ; h̃1(T ) = h̃1; h̃2(T ) = h̃2; t = (t1, t2) ∈ D, (1.14) where, as usual, ∂p = ∂/∂tp, p = 1, 2, and ỹ1(t) and ỹ2(t) are such that ỹ1(t) = R∗ 1ỹ(t); ỹ2(t) = R∗ 2ỹ(t); (1.15) is said to be the dual open system F+ ∆ = { R+ ∆, S+ ∆ } , F+ ∆ = { R+ ∆, S+ ∆ } associated with the colligation ∆ (1.1). The vector functions {ỹp(t)}2 1 have the common generator ỹ(t) ∈ H, besides, R∗ 1ỹ2(t) = R∗ 2ỹ1(t). (1.16) The initial data h̃1, h̃2 of problem (1.14) are found from the equalities h̃p = ψ∗+Ñ∗ p u+(T )−R∗ pỹ(T ), p = 1, 2. (1.17) The mapping S+ ∆ is given by S+ ∆ : ũ−(t) = K∗ũ+(t) + iψ∗−ỹ(t). (1.18) Consider the differential operators L+ p = i∂p + ᾱ, p = 1, 2. (1.19) Similarly to (1.10), we obtain that the vector function ỹ(t) satisfies the relations    R∗ 1L + 1 ỹ(t) + ỹ(t) = ψ∗+Ñ∗ 1 L+ 1 ũ+(t); R∗ 2L ∗ 2ỹ(t) + ỹ(t) = ψ∗+Ñ∗ 2 L+ 2 ũ+(t); ỹ(T ) = ỹT ; t = (t1, t2) ∈ D; ũ−(t) = K∗ũ+(t) + iψ∗−ỹ(t). (1.20) Using (1.20), it is easy to obtain the analogues of Theorems 1.1–1.3. Theorem 1.4 [11]. The equation system (1.20) is consistent if ũ+(t) satisfies the equation { Ñ∗ 1 L+ 1 − Ñ∗ 2 L+ 2 + Γ̃∗L+ 1 L+ 2 } ũ+(t) = 0 (1.21) on condition that (1.15) and (1.17) take place. 196 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators Theorem 1.5 [11]. Let ỹ(t) be the solution of (1.20) and let ũ+(t) satisfy equation (1.21), then for the vector function ũ−(t) (1.18) { N∗ 1 L+ 1 −N∗ 2 L+ 2 + Γ∗L+ 1 L+ 2 } ũ−(t) = 0 (1.22) takes place. Theorem 1.6 [11]. For the dual open system F+ ∆ = { R+ ∆, S+ ∆ } (1.14)–(1.18), the conservation laws are true: 1) ∂p ∥∥∥h̃p(t) ∥∥∥ 2 = 〈 τ−p ũ−(t), ũ−(t) 〉− 〈 τ+ p ũ+(t), ũ+(t) 〉 , p = 1, 2; 2) ∂2 {〈 τ−1 L+ 1 ũ−(t), L+ 1 ũ−(t) 〉− 〈 τ+ 1 L+ 1 ũ+(t), ũ+(t) 〉} = ∂1 {〈 τ−2 L+ 2 ũ−(t), L+ 1 ũ−(t) 〉− 〈 τ+ 2 L+ 2 ũ+(t), L+ 2 ũ+(t) 〉} . (1.23) O b s e r v a t i o n 1.1. The “external parameters” of the commutative colligation ∆ (1.1) are not independent. Moreover, it is easy to show [11] that Ñ∗ p = σ+ p Ñ−1 p τ+ p ; Np = τ−p ( N∗ p )−1 σ−p , p = 1, 2, (1.24) take place, besides, Ñp and N∗ p are boundedly invertible on the images of τ+ p E+ and σ−p E−, p = 1, 2, respectively. 2. Main Properties of Characteristic Functions I. Suppose that the function u−(t) in (1.10) is a plane wave, u−(t) = ei〈λ,t〉u−(0), where 〈λ, t〉 = λ1t1 +λ2t2, t = (t1, t2) ∈ D = [0, T1]× [0, T2], and λ = (λ1, λ2) ∈ C2. And let u+(t) and y(t) in (1.10) depend on t in a similar way, u+(t) = ei〈λ,t〉u+(0), y(t) = ei〈λ,t〉y(0). Then (1.10) yields    y(0) = − (λ1 − α) Tλ1,αψ−N1u−(0); y(0) = − (λ2 − α) Tλ2,αψ−N2u−(0); u+(0) = Ku−(0)− iψ+y(0), (2.1) where Tλp,α = I + (λp − α) Rp (λp), and Rp (λp) = (Ap − λpI)−1 is the resolvent of Ap, λp ∈ Ω, p = 1, 2. The concordance of two different presentations for y(0) (2.1) means that (λ1 − α) Tλ1,αψ−N1u−(0) = (λ2 − α) Tλ2,αψ−N2u−(0). Multiplying this equality by Tα,λ1 , Tα,λ2 and using 5) (1.2), we obtain the relation {(λ1 − α) N1 − (λ2 − α) N2 − (λ1 − α) (λ2 − α) Γ}u−(0) = 0, (2.2) Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 197 V.A. Zolotarev which also follows from the consistency condition (1.11) with the function u−(t) depending on t in the chosen way. To every operator Ap of the commutative colligation ∆ (1.1) there corresponds the characteristic function [11] Sp (λp) def= K + i (λp − α) ψ+Tλp,αψ−Np (p = 1, 2). (2.3) Theorem 2.1. Let a point λ = (λ1, λ2) ∈ C2 be such that for u−(0) (2.2) takes place, then S1 (λ1) u−(0) = S2 (λ2) u−(0). (2.4) The proof of the theorem follows from the last equation of (2.1). If u−(0) satisfies equality (2.2), then (1.12) implies that the function u+(0) = S1 (λ1) u−(0) has the similar property { (λ1 − α) Ñ1 − (λ2 − α) Ñ2 − (λ1 − α) (λ2 − α) Γ̃ } u+(0) = 0. (2.5) Theorem 2.2. If the operators N1 and Ñ1 of the commutative colligation ∆ (1.1) are invertible, then for the characteristic function S1 (λ1) (2.3) the inter- twining condition is true S1 (λ1) N−1 1 [(λ1 − α) Γ + N2] = Ñ−1 1 [ (λ1 − α) Γ̃ + Ñ2 ] S1 (λ1) . (2.6) P r o o f. Equalities (2.2) and (2.5) imply (λ2 − α)−1 (λ1 − α)u−(0) = N−1 1 [(λ1 − α) Γ + N2] u−(0); (λ2 − α)−1 (λ1 − α) u+(0) = Ñ−1 1 [ (λ1 − α) Γ̃ + Ñ2 ] u+(0). Multiplying the first equality by S1 (λ1) and taking into account that u+(0) = S1 (λ1) u−(0), we obtain relation (2.6). If dimE± < ∞, then the existence of non-trivial u−(0) and u+(0) satisfying (2.2) and (2.5), respectively, is possible if only λ = (λ1, λ2) ∈ C2 belongs to the algebraic curves Q = { λ = (λ1, λ2) ∈ C2 : Q (λ1, λ2) = 0 } ; Q̃ = { λ = (λ1, λ2) ∈ C2 : Q̃ (λ1, λ2) = 0 } (2.7) given by the polynomials Q (λ1, λ2) def= det [(λ1 − α) N1 − (λ2 − α) N2 − (λ1 − α) (λ2 − α) Γ] ; Q̃ (λ1, λ2) def= det [ (λ1 − α) Ñ1 − (λ2 − α) Ñ2 − (λ1 − α) (λ2 − α) Γ̃ ] . (2.8) 198 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators The intertwining condition (2.6) yields that the characteristic function S1 (λ1) (2.3) maps the root subspaces of the linear bundles N−1 1 [(λ1 − α) Γ + N2] and Ñ1 [ (λ1 − α) Γ̃ + Ñ2 ] one into another. If S1 (λ1) is invertible at least in one point of the holomorphy from Ω, then dimE− = dimE+ < ∞ and the polynomials (2.8) coincide, Q (λ1, λ2) = Q̃ (λ1, λ2). II. For the dual open system F+ ∆ = { R+ ∆, S+ ∆ } (1.14)–(1.18), consider the case of ũ+(t) = ei〈λ̄,t−T〉u+(T ), where λ = (λ1, λ2) ∈ C2, t = (t1, t2) ∈ D,〈 λ̄, t− T 〉 = λ̄1 (t1 − T1) + λ̄2 (t2 − T2). Suppose that ỹ(t) = ei〈λ̄,t−T〉ỹ(T ), ũ−(t) = ei〈λ̄,t−T〉ũ−(T ), then from (1.20) we obtain    ỹ(T ) = − ( λ̄1 − ᾱ ) T ∗λ1,αψ∗+Ñ∗ 1 ũ+(T ); y(T ) = − ( λ̄2 − ᾱ ) T ∗λ2,αψ∗+Ñ∗ 2 ũ+(T ); ũ−(T ) = K∗ũ+(T ) + iψ∗−ỹ(T ). (2.9) Double representation of ỹ(T ) in (2.9) signifies that {( λ̄1 − ᾱ ) Ñ∗ 1 − ( λ̄2 − ᾱ ) Ñ∗ 2 − ( λ̄1 − ᾱ ) ( λ̄2 − ᾱ ) Γ̃∗ } ũ+(T ) = 0, (2.10) which is the corollary of the consistency condition (1.21). Similarly to the statement of Theorem 2.1, + S1 (λ1) ũ+(T ) = + S2 (λ2) ũ+(T ) (2.11) takes place on condition that ũ+(T ) satisfies relation (2.10), where + Sp (λp) equals + Sp (λp) def= K∗ − i ( λ̄p − ᾱ ) ψ∗−T ∗λp,αψ∗+Ñ∗ p , p = 1, 2. (2.12) The functions Sp(λ) (2.3) and + Sp (λp) (2.12) are linked to each other by the relations N∗ p + Sp (λp) = S∗p (λp) Ñ∗ p , p = 1, 2. (2.13) The equality (1.22) implies that the vector function ũ−(T ) = + S1 (λ1) ũ+(T ) sat- isfies the equality {( λ̄1 − ᾱ ) N∗ 1 − ( λ̄2 − ᾱ ) N∗ 2 − ( λ̄1 − ᾱ ) ( λ̄2 − ᾱ ) Γ∗ } ũ−(T ) = 0. (2.14) For + S1 (λ1) (2.12), the intertwining property also holds + S1 (λ1) (N∗ 1 )−1 [( λ̄1 − ᾱ ) Γ̃∗ + Ñ∗ 2 ] = (N∗ 1 )−1 [( λ̄1 − ᾱ ) Γ∗ + N∗ 2 ] + S1 (λ1) , Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 199 V.A. Zolotarev which follows from (2.6) if one takes into account (2.13). The algebraic curves corresponding to (2.10), (2.14) are the complex adjoints of the curves (2.7). III. 1) (1.13) and 1) (1.23) imply σ−1 − S∗1 (w1) σ+ 1 S1 (λ1) i (λ1 − w̄1) = N∗ 1 ψ∗−T ∗w1,αTλ1,αψ−N1; ( + S1 (w1) )∗ τ−1 + S1 (λ1)− τ+ 1 i ( λ̄1 − w1 ) = Ñ1ψ+Tw1,αT ∗λ1,αψ∗+Ñ∗ 1 ; (2.15) S1 (λ1)− S1 (w1) i (λ1 − w1) = ψ+Tw1,αTλ1,αψ−N1. Define the operator function K(λ,w) in E− ⊕E+ K(λ, w) def=   σ−1 − S∗1(w)σ+ 1 S1(λ) i (λ− w̄) N∗ 1 + S1 (λ)− + S1 (w) i ( w̄ − λ̄ ) Ñ1 S1(λ)− S1(w) i(λ− w) ( + S1 (w) )∗ τ−1 + S1 (λ)− τ+ 1 i ( λ̄− w )   . (2.16) It is obvious that the kernel K(λ,w) (2.16) is positively defined [10, 11] as λ, w ∈ Ω since K(λ,w) = Π∗(w)Π(λ), where Π(λ) = [ Tλ,αψ−N1, T ∗ λ,αψ∗+Ñ∗ 1 ] and Tλ,α = I + (λ− α)R1(λ). A subspace H1 ⊆ H is said to be a reducing one [3, 5, 6] for a commuta- tive system of the linear unbounded operators {A1, A2} if there exists nonempty common domain of holomorphy Ω of resolvents Rp(λ) = (Ap − λI)1 such that in every point λ ∈ Ω Rp(λ)P1 = P1Rp(λ), p = 1, 2, where P1 is the orthoprojector on H1. For the commutative colligation ∆ (1.1), define the subspace H1 in H: H1 =span { R2(w)R1(λ)ψ−u− + R∗ 2 (w̃) R∗ 1 ( λ̃ ) ψ∗+u+ : u± ∈ E±;λ,w, λ̃, w̃ ∈ Ω } . (2.17) Theorem 2.3. Let the operators N1 and Ñ1 be invertible. Then the subspace H1 (2.17) reduces the commutative operator system {A1, A2} of the colligation ∆ (1.1), besides, the restriction of {A1, A2} to H0 = H ª H1 is the commutative system of selfadjoint operators. 200 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators P r o o f. The analyticity of the resolvents Rp(λ) = ∞∑ k=0 (λ− α)kRk+1 p , p = 1, 2, (2.18) in the neighborhood Uδ(α) = {λ ∈ Ω : \|λ−α\| < δ} of the point α ∈ Ω yields that the subspace H1 (2.17) generates the vectors Rm 2 Rn 1ψ−u− + (R∗ 2) p (R∗ 1) q ψ∗+u+, where u± ∈ E+; m, n, p, q ∈ Z+. The equalities R2ψ− = R1ψ−N2N −1 1 + ψ−ΓN−1 1 ; R∗ 2ψ ∗ + = R∗ 1ψ ∗ +Ñ∗ 2 ( Ñ∗ 1 )−1 + ψ∗+Γ̃∗ ( Ñ∗ 1 )−1 (2.19) (taking into account 5) (1.2)) imply that the subspace H1 (2.17) is given by H1 = span { Rn 1ψ−u− + (R∗ 1) m ψ∗+u+ : u± ∈ E±; n,m ∈ Z+ } . (2.20) It is easy to ascertain [11] that the subspace H1 (2.20) reduces A1, besides, the restriction of A1 to H0 = HªH1 is a selfadjoint operator. Therefore the operator T1 = I + i2 Im αR1, being the Caley transform of the operator A1 restricted to H0, is a unitary operator. It remains to prove that the subspace H1 (2.20) also reduces the operator A2 and that the restriction A2\|H0 is a selfadjoint operator. The equalities (2.19) imply that to prove the reducibility of the A2 by the subspace H1 (2.20), it is necessary to make sure that R2 (R∗ 1) m ψ∗+u+ ∈ H1; R∗ 2R n 1ψ−u− ∈ H1 for all u± ∈ E± and all n, m ∈ Z+. For instance, prove that R∗ 2R n 1ψ−u− ∈ H1, ∀u−, n ∈ Z+. (2.21) To do this, consider the following subspaces: Ln = span {R∗ 2R n 1ψ−u− : u− ∈ E−} , n ∈ Z+, and let Ln = Ln 1 ⊕ Ln 0 , where Ln q = PqL n and Pq is the orthoprojector on Hq, q = 0, 1. Since H1 (2.20) reduces A1, then R∗ 1L n q ⊂ Hq and T ∗1 Ln q ⊂ Hq, q = 0, 1, for all n ∈ Z+. Prove that Ln 0 = {0} for all n ∈ Z+, which signifies that inclusion (2.21) is true. The point 3) (1.2) yields that T ∗1 ψ− = ψ∗+σ+ 1 KN−1 1 , therefore T ∗1 R∗ 2ψ−u− = −R∗ 2ψ ∗ +σ+ 1 KN−1 1 u− ∈ H1 and thus T ∗1 L0 0 ⊆ L0 1. This implies that L0 0 = {0}, in view of the unitarity of the restriction of T ∗1 on H0. Let, in view Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 201 V.A. Zolotarev of the mathematical induction principle, Lk 0 = {0} be proved as k = 0, 1, . . . , n, now prove that Ln+1 0 = {0}. Really, T ∗1 R∗ 2R n+1 1 ψ− = R∗ 2T ∗ 1 1 i2 Imα (T1 − I) Rn 1ψ− = 1 i2 Imα R∗ 2T ∗ 1 T1R n 1ψ− − 1 i2 Imα T ∗1 R∗ 2R n 1ψ− = 1 i R∗ 2B1R n 1ψ− + 1 i2 Imα R∗ 2R n 1ψ− − 1 i2 Imα T ∗1 R∗ 2R n 1ψ−, where Bp = iRp− iR∗ p +2 ImαR∗ pRp, B̃p = iRp− iR∗ p +2 ImαRpR ∗ p, p = 1, 2 [11]. Using B1 = ψ∗+σ+ 1 ψ− (see [11]), we obtain T ∗1 Ln+1 0 ⊂ span {Ln 0 + T ∗1 Ln 0} = {0}, and thus Ln+1 0 = {0} in view of unitarity of T ∗1 \|H0 . To complete the proof of the theorem, it remains to determine that the re- striction of A2 to H0 is a selfadjoint operator. Prove that the operator T2 = I + i2 Imα · R2 restricted to H0 is unitary. Since B2H = ψ∗+σ+ 2 ψ+H and B̃2H = ψ−τ−2 ψ∗−H (see [11]) belong to H1 (2.20), then H0 ⊂ KerB2 and H0 ⊂ Ker B̃2, what guarantees that the restriction of T2 to H0 is unitary. The colligation ∆ (1.1) is said to be simple if H = H1, where H1 is given by (2.17). Consider two commutative colligations ∆ and ∆̂ (1.1) such that E± = Ê±; σ±p = σ̂±p ; τ±p = τ̂±p ; Np = N̂p; Ñp = ˜̂ pN , p = 1, 2; Γ = Γ̂; Γ̃ = ˜̂Γ; K = K̂; and, moreover, α = α̂ ∈ Ω ∩ Ω̂ 6= {∅}. These colligations are said to be unitarily equivalent if there exists such a unitary operator U : H → Ĥ that UAp = ÂpU ; UD (Ap) = D ( Âp ) ; UA∗p = Â∗pU ; UD ( A∗p ) = D ( Â∗p ) (p = 1, 2); Uψ− = ψ̂−; ψ̂+U = ψ+. (2.22) It is easy to show that the characteristic functions of unitarily equivalent colliga- tions ∆ and ∆̂ coincide, S1 (λ1) = Ŝ1 (λ1) (2.3) for all λ ∈ Ω ∩ Ω̃. Theorem on the unitary equivalence 2.4. Let ∆ and ∆̂ (1.1) be two simple commutative colligations such that E± = Ê±; σ±p = σ̂±p ; τ±p = p̂±; Np = N̂p; Ñp = {̃N̂p}, p = 1, 2; Γ = Γ̂; Γ̃ = ˜̂Γ; and α = α̂ ∈ Ω ∩ Ω̂ (6= ∅). Then if the operators N1 and Ñ1 are invertible and in some neighborhood Uδ(α) of point α the characteristic functions coincide, S1 (λ1) = Ŝ1 (λ1) (2.3), the colligations ∆ and ∆̂ are unitarily equivalent. 202 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators P r o o f. The coincidence of characteristic functions, S1 (λ1) = Ŝ1 (λ1), as λ1 ∈ Uδ(α) and N1, Ñ1 are unitary, and (2.15) imply that ψ∗−Tw1,αTλ1,αψ− = ψ̂∗−T̂ ∗w1,αT̂λ1,αψ̂−; ψ+Tw1,αT ∗λ1,αψ∗+ = ψ̂+T̂w1,αT̂ ∗λ1,αψ̂∗+; ψ+Tw1,αTλ1,αψ− = ψ̂+T̂w,αT̂λ1,αψ̂− for all λ1, w1 ∈ Uδ(α). Taking into account holomorphy (2.18) of the resolvents R1 (λ1) and R̂1 (λ1) in the neighborhood Uδ(α), we can rewrite these equalities in the equivalent form ψ∗− (R∗ 1) m Rn 1ψ− = ψ̂∗− ( R̂∗ 1 )m R̂n 1 ψ̂−; ψ+Rm 1 (R∗ 1) n ψ∗+ = ψ̂+R̂m 1 ( R̂∗ 1 )n ψ̂∗+; ψ+Rm 1 Rn 1ψ− = ψ̂+R̂m 1 R̂n 1 ψ̂− (2.23) for all n, m ∈ Z+. Define the linear operator U : H → Ĥ URn 1ψ−u− def= R̂n 1 ψ̂−u−; u (R∗ 1) m ψ∗+u+ def= ( R̂∗ 1 )m ψ̂∗+u+, (2.24) where u± ∈ E± and n, m ∈ Z+. The simplicity of the colligations ∆, ∆̂ and the invertibility of N1, Ñ1 yield that the spaces H and Ĥ are given by (2.20), and thus the operator U (2.24) is unitary in view of (2.23). It is easy to prove (see [11]) that UR1 = R̂1U ; Uψ− = ψ̂−; ψ̂+U = ψ+. It remains to prove that UR2 = R̂2U . And since the application of the resolvent R2 (R∗ 2) to the vectors Rn 1ψ−u (correspondingly, to (R∗ 1) m ψ∗+u+) is expressed also in terms of these vectors, then it is obvious that ( UR2 − R̂2U ) Rn 1ψ−u− = 0; ( UR∗ 2 − R̂∗ 2U ) (R∗ 1) m ψ∗+u+ = 0, (2.25) for all u± ∈ E+ and all n, m ∈ Z+. Thus, it is necessary to prove that ( UR2 − R̂2U ) (R∗ 1) m ψ∗+u+ = 0 (2.26) for all u+ ∈ E+ and all m ∈ Z+. It is easy to see that when m = 0 T̂1 ( UR2 − R̂2U ) ψ∗+u+ = ( UR2 − R̂2U ) T1ψ ∗ +u+ = − ( UR2 − R̂2U ) ψ−τ−1 K∗ ( Ñ∗ 1 )−1 u+ = 0 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 203 V.A. Zolotarev in view of (2.25) and T1ψ ∗ +Ñ∗ 1 + ψ−τ−1 K∗ = 0 (see 3) (1.2)). Prove that this implies (2.26) when m = 0. One can see that 0 = T̂ ∗1 T1 ( UR2 − R̂2U ) ψ∗+ = 2 ImαB̂1 ( UR2 − R̂2U ) ψ∗+ + ( UR2 − R̂2U ) ψ∗+, and to prove (2.26) (m = 0), it is necessary to establish that ψ̂∗+σ+ 1 ψ̂+ ( UR2 − R̂2U ) ψ∗+ = 0. And the last, ψ̂+ ( UR2 − R̂2U ) ψ∗+ = ψ+R2ψ ∗ + − ψ̂+R̂2ψ̂+ = 0, follows easily from the definition of U (2.24) and formulas (2.19), (2.23). Thus, relation (2.26) for m = 0 is proved. Using the principle of mathematical induction, suppose that equality (2.26) is already proved for m = n; prove that it is also true for m = n + 1. It is easy to see that T̂1 ( UR2 − R̂2U ) (R∗ 1) n+1 ψ∗+u+ = ( UR2 − R̂2U ) T1R ∗ 1 (R∗ 1) n ψ∗+u+ = 1 i2 Imα ( UR2 − R̂2U ) T1 (T ∗1 − I) (R∗ 1) n ψ∗+u+ = i ( UR2 − R̂2U ) B̃1 (R∗ 1) n ψ∗+u+ = i ( UR2 − R̂2U ) ψ−τ−1 ψ∗− (R∗ 1) n ψ∗+u+ = 0 in view of the induction supposition and of (2.25). And since 0 = T̂ ∗1 T̂1 ( UR2 − R̂2U ) (R∗ 1) n+1 ψ∗+u+ = 2 ImαB̂1 ( UR2 − R̂2U ) (R∗ 1) n+1 ψ∗+u+ + ( UR2 − R̂2U ) (R∗ 1) n+1 ψ∗+u+, then to prove (2.26) for m = n + 1, it is sufficient to prove that ψ̂+ ( UR2 − R̂2U ) (R∗ 1) n+1 ψ∗+ = ψ+R2 (R∗ 1) n+1 ψ∗+ − ψ̂+R̂2 ( R̂∗ 1 )n+1 ψ̂∗+ = 0, and this obviously follows from (2.23) and (2.19). 204 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators Thus, the characteristic function S1 (λ1) (2.3) and the “external set of para- meters” { σ±p }2 1 ; { τ±p }2 1 ; {Np}2 1; { Ñp }2 1 ; Γ; Γ̃, on condition that the operators N1 and Ñ1 are invertible, define the simple commutative colligation ∆ (1.1) up to the unitary equivalence. IV. Since S1 (λ1) (2.3) is the main analytic object, in terms of which the simple commutative colligation ∆ (1.1) is characterized, here we describe the main properties of the function S1(λ) = K + i(λ− α)ψ+Tλ,αψ−N1, (2.27) where for simplicity we denote Tλ,α = I + (λ− α)R1(λ). Consider the generating vector function (2.1) y = −(λ− α)Tλ,αψ−N1u−, (2.28) where λ, α ∈ Ω, and u− ∈ E−. As in (1.4), using y (2.28) construct y1 = R1y = −(λ− α)R1(λ)ψ−N1u1 ∈ D (A1) . (2.29) Then the colligation relation 4) (1.2), 2 Im 〈A1y1, y1〉 = 〈 σ+ 1 ϕ1 +y1, ϕ 1 +y1 〉 , (2.30) when y1 (2.29) is chosen in this way, signifies that 1 i 〈A1R1(λ)ψ−N1u−, R1(w)ψ−N1û−〉 − 1 i 〈R1(λ)ψ−N1u−, A1R1(w)ψ−N1û−〉 = 〈 σ+ 1 ψ+Tλ,αψ−N1u−, ψ+Tw,αψ−N1û− 〉 , for all u−, û− ∈ E− and all λ, w, α ∈ Ω. Using A1R1(λ) = λR1(λ)+ I, we obtain 1 i N∗ 1 ψ∗− {R∗ 1(w) [λR1(λ) + I]− [w̄R∗ 1(w) + I]R1(λ)}ψ−N1 = 1 (λ− α) (w̄ − ᾱ) [S∗1(w)−K∗] σ+ 1 [S1(λ)−K] . And, since [S∗1(w)−K∗] σ+ 1 [S1(λ)−K] = S∗1(w)σ+ 1 S1(λ) + K∗σ+ 1 [K − S1(λ)] + [K∗ − S∗1(w)]σ+ 1 K −K∗σ+ 1 K, then, taking into account (2.27) and 1., 3. (1.2), we obtain the equality [S∗1(w)−K∗]σ+ 1 [S1(λ)−K] = S∗1(w)σ+ 1 S1(λ)− σ−1 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 205 V.A. Zolotarev +iN∗ 1 ψ∗− { (λ− α)Tλ,ᾱ − (w̄ − ᾱ) T ∗w,ᾱ + (α− ᾱ) I } ψ−N1. Therefore, 1 (λ− α) (w̄ − ᾱ) ( S∗1(w)σ+ 1 S1(λ)− σ−1 ) = iN∗ 1 ψ∗− { (w̄ − λ) R∗ 1(w)R1(λ) +R1(λ)−R∗ 1(w)− 1 w̄ − ᾱ (I + (λ− ᾱ) R1(λ)) + 1 λ− α (I + (w̄ − α) R∗ 1(w)) − α− ᾱ (λ− α) (w̄ − ᾱ) I } ψ−N1. After elementary calculations, we obtain the relation S∗1(w)σ+ 1 S1(λ)− σ−1 = i (w̄ − λ) N∗ 1 ψ∗−T ∗w,αTλ,αψ−N1 which exactly coincides with the first equality of (2.15). Lemma 2.1. If (2.30) holds for the operator A1 of the commutative colliga- tion ∆ (1.1) on the vector functions y1 (2.29), then the first formula in (2.15) is true for the characteristic function S1(λ) (2.27). Thus, the observance of the conservation law 1) (1.13), p = 1, is adequate to the colligation relation (2.32) for the operator A1. Using (1.4), construct the vector function y2 by the generating function y (2.28) y2 = R2y = −(λ− α)R2Tλ,αψ−N1u− ∈ D (A2) , (2.31) where u− ∈ E−, and λ, α ∈ Ω. Write the colligation relation 4. (1.2) for y2 2 Im 〈A2y2, y2〉 = 〈 σ+ 2 ϕ2 +y2, ϕ 2 +y2 〉 , (2.32) which is equivalent to 1 i 〈A2R2Tλ,αψ−N1u−, R2Tw,αψ−N1û−〉 −1 i 〈R2Tλ,αψ−N1u−, A2R2Tw,αψ−N1û−〉 = 〈 σ+ 2 ψ+Tλ,αψ−N1u−, ψ+Tw,αψ−N1û− 〉 for all u−, û− ∈ E− and all λ, w, α ∈ Ω. Since A2R2 = αR2 + I, the last equality yields 1 (λ− α) (w̄ − ᾱ) [S∗1(w)−K∗]σ+ 2 [S1(λ)−K] = N∗ 1 ψ∗− { α− ᾱ i R∗ 2T ∗ w,αTλ,αR2 + 1 i R∗ 2T ∗ w,αTλ,α − 1 i T ∗w,αTλ,αR2 } ψ−N1. 206 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators It is obvious that Tλ,αR2ψ−N1 = 1 λ− α [Tλ,αψ−N1Lλ − ψ−N2] , (2.33) where Lλ is the linear bundle of the operators Lλ = N−1 1 [(λ− α)Γ + N2] . (2.34) Using the form of function S1(λ) (2.3) and (2.35), we obtain S∗1(w)σ+ 2 S1(λ)−K∗σ+ 2 S1(λ)−K∗σ+ 2 K − i(λ− α)K∗σ+ 2 ψ+Tλ,αψ−N1 +i (w̄ − ᾱ) N∗ 1 ψ∗−T ∗w,αψ∗+σ+ 2 K = α− ᾱ i { L∗wN∗ 1 ψ∗−T ∗w,α −N∗ 2 ψ∗− } {Tλ,αψ−N1Lλ − ψ−N2} + λ− α i { L∗wN∗ 1 ψ∗−T ∗w,α −N∗ 2 ψ∗− } · Tλ,αψ−N1 − w̄ − ᾱ i N∗ 1 ψ∗−T ∗w,α {Tλ,αψ−N1Lλ − ψ−N2} . Denote by K1,1(λ,w) the left upper block of the kernel K(λ,w) (2.16) K1,1(λ,w) = σ−1 − S∗1(w)σ+ 1 S1(λ) i (λ− w̄) = N∗ 1 ψ∗−T ∗w,αTλ,αψ−N1. (2.35) Rewrite condition 3) (1.2) as K∗σ+ 2 ψ+ + N∗ 2 ψ− (I + (α− ᾱ) R2) = 0. Then, taking into account (2.33), we have K∗σ2ψ+Tλ,αψ−N1 = −N∗ 2 ψ∗− (I + (α− ᾱ) R2)Tλ,αψ−N1 = −N∗ 2 ψ∗−Tλ,αψ−N1 − (α− ᾱ) N∗ 2 ψ∗−Tλ,αR2ψ−N1 = −N∗ 2 ψ∗−Tλ,αψ−N1 − α− ᾱ λ− α N∗ 2 ψ∗−Tλ,αψ−N1Lλ + α− ᾱ λ− α N∗ 2 ψ∗−ψ−N2. Therefore, using 1) (1.2), (2.35), after simple calculations we obtain the equality i α− ᾱ { S∗1(w)σ+ 2 S1(λ)− σ−2 } = L∗wK1,1(λ,w)Lλ −λ− α ᾱ− α L∗wK1,1(λ,w)− w̄ − ᾱ α− ᾱ K1,1(λ,w)Lλ. (2.36) The fact that this relation follows easily from the conservation law 2) (1.13) is an important observation. Really, let u±(t) = ei〈λ,t〉u±(0) and û±(t) = ei〈w,t〉û±(0), then 2) (1.13) implies (λ2 − w̄2) (λ1 − α) (w̄1 − ᾱ) {〈 σ−1 u−(0), û−(0) 〉− 〈 σ+ 1 u+(0), û+(0) 〉} Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 207 V.A. Zolotarev = (λ1 − w̄1) (λ2 − α) (w̄2 − ᾱ) {〈 σ−2 u−(0), û−(0) 〉− 〈 σ+ 2 u+(0); û+(0) 〉} . And since Lλ1u−(0) = λ1 − α λ2 − α u−(0); Lw1 û(0) = w1 − α w2 − α û−(0); (2.37) in view of (2.2), then, taking into consideration u+(0) = S1(λ)u−(0); û+(0) = S1(w)û−(0), we obtain (λ2 − w̄2) 〈{ σ−1 − S∗1 (w1) σ+ 1 S1 (λ1) } Lλ1u−(0), Lw1 û−(0) 〉 = (λ1 − w̄1) 〈{ σ−2 − S∗1 (w1)σ+ 2 S1 (λ1) } u−(0), û−(0) 〉 . (2.37) implies (λ2 − α) Lλ1u−(0) = (λ1 − α) u−(0); (w2 − α) Lw1 û−(0) = (w1 − α) û−(0). Therefore, taking into consideration (2.35), we have 〈{ (α− ᾱ)L∗w1 K1,1 (λ1, w1)Lλ1 + (λ1 − α) L∗wK1,1 (λ1, w1) − (w̄1 − ᾱ) K1,1 (λ1, w1) } u−(0), û−(0) 〉 = i 〈{ S∗1 (w1) σ+ 2 S1 (λ1)− σ−2 } u−(0), û−(0) 〉 , which, in view of the arbitrariness of u−(0), û−(0) ∈ E−, gives us (2.36). Lemma 2.2. Let the commutative colligation ∆ (1.1) be given and N1 be invertible. Then the colligation relation (2.32) for the operator A2, where y2 is given by (2.31), implies that the characteristic function S1(λ) (2.27) satis- fies equality (2.36), besides, K1,1(λ,w) and Lλ are given by formulas (2.35) and (2.34), respectively. Moreover, relation (2.36) is equivalent to the conservation law 2) (1.13). Proceed to the consideration of colligation relations 4. (1.2) for the adjoint operators A∗1 and A∗2 of the commutative colligation ∆ (1.1). Specify now the generating function (2.9) ỹ = − ( λ̄− ᾱ ) T ∗λ,αψ∗+Ñ∗ 1 ũ+, (2.38) where ũ+ ∈ E+, and λ, α ∈ Ω. According to (1.15), construct the vector function by ỹ, ỹ1 = R∗ 1ỹ = − ( λ̄− ᾱ ) R∗ 1(λ)ψ∗+Ñ∗ 1 ũ+ ∈ D (A∗1) . (2.39) As in the previous case (see Lemma 1.1), it is easy to show that the colligation relation −2 Im 〈A∗1ỹ1, ỹ1〉 = 〈 τ−1 ( ϕ1 − )∗ ỹ1, ( ϕ1 − )∗ ỹ1 〉 , (2.40) 208 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators for ỹ1 (2.39), implies that the block K2,2(λ, w) of the kernel K(λ,w) (2.16) is given by K2,2(λ,w) = ( + S1 (w) )∗ τ−1 + S1 (λ)− τ+ 1 i(λ− w) = Ñ1ψ+Tw,αT ∗λ,αψ∗+Ñ∗ 1 , (2.41) besides, (2.12), (2.13) + S1 (λ) = (N∗ 1 )−1 S∗1(λ)Ñ∗ 1 = K∗ − i ( λ̄− ᾱ ) ψ∗−T ∗λ,αψ∗+Ñ∗ 1 . Lemma 2.3. If for the operator A∗1 of the commutative colligation ∆ (1.1) re- lation (2.40) is true on the vector functions ỹ1 (2.31), then for the block K2,2(λ,w) of the kernel K(λ,w) (2.16) representation (2.41) takes place. By the generating function ỹ (2.38) according to (2.9), define ỹ2 ỹ2 = R∗ 2ỹ = − ( λ̄− ᾱ ) R∗ 2T ∗ λ,αψ∗+Ñ∗ 1 ũ+ ∈ D (A∗2) (2.42) and consider the colligation relation 4) (1.2) for A∗2 −2 Im 〈A∗2ỹ2, ỹ2〉 = 〈 τ−2 ( ϕ2 − )∗ ỹ2, ( ϕ2 − )∗ ỹ2 〉 , (2.43) where ỹ2 is given by (2.42). Applying similar considerations (see the proof of Lemma 2.2), it is easy to prove that i α− ᾱ {( + S1 (w) )∗ τ2− + S1 (λ)− τ+ 2 } = ( L+ w )∗ K2,2(λ, w)L+ λ −λ− ᾱ α− ᾱ ( L+ w )∗ K2,2(λ,w)− w − α ᾱ− α K2,2(λ, w)L+ λ , (2.44) where L+ λ is the linear bundle, L+ λ = ( Ñ∗ 1 )−1 [( λ̄− ᾱ ) Γ̃∗ + Ñ∗ 2 ] , (2.45) and K2,2(λ,w) are given by (2.41). Lemma 2.4. Let ∆ (1.1) be a commutative colligation and Ñ1 be invertible. Then relation (2.43) for the operator A∗2 on the vectors ỹ2 (2.42) implies that (2.44) holds for the characteristic function + S1 (λ) (2.12), where K2,2(λ,w) and L+ λ are given by formulas (2.41) and (2.45), and + S1 (λ) is constructed by S1(λ) (2.3) by rule (2.13). Moreover, equality (2.44) is equivalent to the conservation law 2) (1.23). Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 209 V.A. Zolotarev Note that the structure and properties of other blocks of the kernel K(λ,w) (2.16) are also determined by the properties of the commutative system {A1, A2} of the colligation ∆ (1.1). Consider the obvious equality 〈A1y1, ỹ1〉 = 〈y1, A ∗ 1ỹ1〉 , (2.46) assuming that y1 and ỹ1 are given by (2.29) and (2.39), respectively. This implies (λ− α)(w − α) 〈 A1R1(λ)ψ−N1u−, R∗ 1(w)ψ∗+Ñ∗ 1 ũ+ 〉 = (λ− α)(w − α) 〈 R1(λ)ψ−N1u−, A∗1R ∗ 1(w)ψ∗+Ñ∗ 1 ũ1 〉 , and since A1R1(λ) = λR1(λ) + I, then λÑ1ψ+ (Tw,α − I) (Tλ,α − I) ψ−N1 + (λ− α)Ñ1ψ+ (Tw,α − I) ψ−N1 = wÑ1ψ+ (Tw,α − I) (Tλ,α − I) ψ−N1 + (w − α)Ñ1ψ+ (Tλ,α − I) ψ−N1. Therefore, (λ−w)Ñ1ψ+Tw,αTλ,αψ−N1 = (λ−α)Ñ1ψ+Tλ,αψ−N1− (w−α)Ñ1ψ+Tw,αψ−N1. Taking into account the form of S1(λ) (2.27), we have K2,1(λ,w) = Ñ1 S1(λ)− S1(w) i(λ− w) = Ñ1ψ+Tw,αTλ,αψ−N1. (2.47) Lemma 2.5. If (2.46) holds for the operator A1 of the commutative colliga- tion ∆ (1.1), then the block K2,1(λ,w) of the kernel K(λ,w) (2.16) has repre- sentation (2.47). Study the similar to (2.46) equality for A2 〈A2y2, ỹ〉 = 〈y2, A ∗ 2ỹ2〉 , (2.48) assuming that y2 and ỹ2 are given by formulas (2.31) and (2.42). In view of A2R2 = αR2 + I, it is easy to see that (2.48) leads to the relation Ñ1ψ+R2Tw,αTλ,αψ−N1 = Ñ1ψ+Tw,αTλ,αR2ψ−N1. We can write this equality in the following way: (λ− α) [ L̃wψ+Tw,α − Ñ−1 1 N2ψ+ ] · Tλ,αψ−N1 = (w − α)ψ+Tw,α [Tλ,αψ−N1Lλ − ψ−N2] , 210 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators where L̃λ = Ñ−1 1 [ (λ− α)Γ̃ + Ñ2 ] . (2.49) It is obvious that L̃λ (2.49) and L+ λ (2.45) satisfy the relation L+ λ = ( Ñ∗ 1 )−1 L̃∗λÑ∗ 1 . (2.50) Using the definition of S1(λ) (2.27) and the last formula of (2.15), we obtain λ− α λ− w L̃w [S1(λ)− S1(w)]− Ñ−1 1 Ñ2S1(λ) = w − α λ− w [S1(λ)− S1(w)]Lλ − S1(w)N−1 1 N2. This easily implies that (w − α) [ L̃λS1(λ)− S1(λ)Lλ ] = (λ− α) [ L̃wS1(w)− S1(w)Lw ] , and the above implies that the relation 1 λ− α [ L̃λS1(λ)− S1(λ)Lλ ] = C is constant and does not depend on λ. Thus, L̃λS1(λ)− S1(λ)Lλ = (λ− α)C, which is impossible when C 6= 0, because the coefficient of the expression L̃λS1(λ)− S1(λ)Lλ when (λ− α) is equal to zero Ñ−1 1 Γ̃K + iÑ−1 1 Ñ2ψ+ψ−N1 −KN−1 1 Γ− iψ+ψ−N2 = 0 in view of 6) and 7) (1.2). Thus, C = 0, and we again come to the intertwining condition (2.6). Lemma 2.6. Let ∆ (1.1) be a commutative colligation and the operators N1, Ñ1 be invertible. Then equality (2.48) for A2 implies the intertwining condition (2.6) for the characteristic function S1(λ) (2.27). Summarizing the statements of Lemmas 2.1–2.6, we obtain the following theorem. Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 211 V.A. Zolotarev Theorem 2.5. Let ∆ (1.1) be a commutative colligation and the operators N1, Ñ1 be invertible. Then the characteristic function S1(λ) (2.27) satisfies the relations: 1) S1(λ)Lλ = L̃λS1(λ); 2) i α− ᾱ { S∗1(w)σ+ 2 S1(λ)− σ−2 } = L∗wK1,1(λ,w)Lλ −λ− α ᾱ− α L∗wK1,1(λ,w)− w̄ − ᾱ α− ᾱ K1,1(λ,w)Lλ; 3) i α− ᾱ {( + S1 (w) ) τ−2 + S2 (λ)− τ+ 2 } = ( L+ w )∗ K2,2(λ,w)L+ λ − λ̄− ᾱ α− ᾱ ( L+ w )∗ K2,2(λ,w)− w − α ᾱ− α K2,2(λ,w)L+ λ , (2.51) where Lλ, L̃λ, and L+ λ are the linear bundles of operators (2.34), (2.49) and (2.45); and Kp,s(λ, w) are the corresponding blocks of the kernel K(λ, w) (2.16). Moreover, + S (λ) is defined from S1(λ) by formula (2.13), and L+ λ and L̃λ (2.49) are linked to each other by relation (2.50). O b s e r v a t i o n 2.1. The colligation relations (2.30), (2.40), and (2.46) for the operators A1 and A∗1 of the commutative colligation ∆ (1.1) have the “metric nature” and give the well-known (2.15) representations for the blocks Kp,s(λ,w) of the positively defined kernel K(λ,w) (2.16). Similar relations (2.32), (2.46), and (2.48) for the operators A2 and A∗2 of the commutative colligation ∆ (1.1) lead to the new nontrivial conditions for the characteristic function S1(λ) (2.27) that should be considered as a corollary of commutativity of the operators A1 and A2. Note that the equalities 2) and 3) of (2.51) follow from the conservation laws 2) (1.13), (1.23) and also have the sensible interpretation in terms of conditions (1.2) of the colligation ∆ (1.1). O b s e r v a t i o n 2.2. Between the “external parameters” of the colligation ∆ (1.1) besides the colligation relations 1)–7) (1.2) there exist additional relations. In particular, assuming in 2) and 3) (2.51) that λ = w = α, we obtain K∗σ+ 2 K − σ−2 = N∗ 2 (N∗ 1 )−1 { K∗σ+ 1 K − σ−1 } N−1 1 N2; Kτ−2 K∗ − τ+ 2 = Ñ2Ñ −1 1 { Kτ−1 K∗ − τ+ 1 }( Ñ∗ 1 )−1 Ñ∗ 2 . Probably, these are not the only possible conditions of dependance between “external parameters” of the colligation ∆ (1.1). 212 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators 3. Theorem of Existence and Analogue of Hamilton–Caley Theorem I. In this section, we prove the theorem of the existence, namely, describe the properties which the operator function S1 (λ1) from E− into E+ must satisfy to be a characteristic function of some colligation ∆ (1.1). Moreover, we prove that in the case of the finite dimension of E− and E+ there exists such a polynomial P (λ1, λ2) that “annihilates” A1 and A2. In H1 (2.17), define the vector functions F (λ, u−) = Tλ,αψ−N1u−; F̃ (λ, u+) = T ∗λ,αψ−+Ñ∗ 1 u+, (3.1) where u± ∈ E±; λ, α ∈ Ω; and Tλ,α = I + (λ − α)R1(λ). Obviously, the linear span of the F (λ, u−) and F̃ (λ, u+), on condition of the invertibility of N1 and Ñ1, generates the whole H1. Theorem 3.1. Let there be given the commutative colligation ∆ (1.1), the operators N1 and Ñ1 of which are invertible. Then the resolvents {R1, R2} and {R∗ 1, R ∗ 2} act on the vector functions F (λ, u−) and F̃ (λ, u+) (3.1) in the following way: 1) R1F (λ, u−) = F (λ, u−)− F (α, u−) λ− α ; 2) R2F (λ, u−) = F (λ, Lλu−)− F (α,Lαu−) λ− α ; 3) R∗ 1F (λ, u−) = F (λ, u−) + F̃ ( α, ( Ñ∗ 1 )−1 σ+ 1 S1(λ)u− ) λ− ᾱ ; 4) R∗ 2F (λ,Qλu−) = F (λ,Lλu−) + F̃ ( α, ( Ñ∗ 1 )−1 σ+ 2 S1(λ)u− ) ; 5) R∗ 1F̃ (λ, u+) = F̃ (λ, u+)− F̃ (α, u+) λ̄− ᾱ ; 6) R∗ 2F̃ (λ, u+) = F̃ ( λ,L+ λ u+ )− F̃ (α, L+ α u+) λ̄− ᾱ ; 7) R1F̃ (λ, u+) = F̃ (λ, u+) + F ( α, N−1 1 τ−1 + S1 (λ)u+ ) λ̄− α ; 8) R2F̃ ( λ,Q+ λ u+ ) = F̃ ( λ,L+ λ u+ ) + F ( α,N−1 1 τ−2 + S1 (λ)u+ ) (3.2) for all u± ∈ E± and all λ, α ∈ Ω, besides, the linear bundles Qλ and Q+ λ are given by Qλ = (λ− α)I + (α− ᾱ) Lλ; Lλ = N−1 1 [(λ− α)Γ + N2] ; Q+ λ = ( λ̄− ᾱ ) I + (ᾱ− α) L+ λ ; L+ λ = ( Ñ∗ 1 )−1 [( λ̄− ᾱ ) Γ̃∗ + Ñ∗ 2 ] , (3.3) S1(λ) and + S1 (λ) are given by (2.3) and (2.12). Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 213 V.A. Zolotarev P r o o f. The proof of 1) (3.2) follows easily from the Hilbert identity for resolvents R1F (λ, u−) = R1Tλ,αψ−N1u− = [R1(α) + (λ− α)R1(α)R1(λ)]ψ−N1u− = R1(λ)ψ−N1u− = Tλ,α − I λ− α ψ−N1u− = F (λ, u−)− F (α, u−) λ− α . Relation 5) (3.2) is proved in a similar way. It is easy to see that the equalities 2) and 6) (3.2) follow from formulas 5) (1.2). Since the relations 3) and 7) (3.2) have the dual nature, it is sufficient to prove one of them, for instance, 7) (3.2). It is obvious that R1F̃ (λ, u+) = R1 ( I + ( λ̄− ᾱ ) R∗ 1(λ) ) ψ∗+Ñ∗ 1 u+ = R1ψ ∗ +Ñ∗ 1 u+ + ( λ̄− ᾱ ) R1R ∗ 1T ∗ λ,αψ∗+Ñ∗ 1 u+ since R1Tλ,α = R1(λ). (α− ᾱ) R1R ∗ 1 = iψ−τ−1 ψ∗− + R1 −R∗ 1 imply that R1F̃ (λ, u+) = R1ψ ∗ +Ñ∗ 1 u+ + λ̄− ᾱ α− ᾱ { iψ−τ−1 ψ∗− + R1 −R∗ 1 } T ∗λ,αψ∗+Ñ∗ 1 u+ = R1ψ ∗ +Ñ∗ 1 u+ + 1 α− ᾱ ψ−τ−1 [ K∗− + S1 (λ) ] u+ + λ̄− ᾱ α− ᾱ R1T ∗ λ,αψ∗+Ñ∗ 1 u+− in view of the definition of + S1 (λ) (2.12) and 5) (3.2). Hence, ( α− λ̄ ) R1F̃ (λ, u+) = −ψ−τ−1 + S1 (λ)u+ − T ∗λ,αψ∗+Ñ∗ 1 u+ since (α− ᾱ) R1ψ ∗ +Ñ∗ 1 + ψ−τ−1 K∗ + ψ∗+Ñ∗ 1 = 0, in view of condition 3) (1.2) of the colligation ∆ (1.1). Thus, formula 7) (3.2) is proved. Prove that formulas 4) and 8) (3.2) are true. Prove, for instance, equality 4). To do this, use the fact that R∗ 2 = R2 − (α− ᾱ) R∗ 2R2 + iψ∗+σ+ 2 ψ+. It is easy to see that R∗ 2Tλ,αψ−N1 = R2Tλ,αψ−N1 − (α− ᾱ) R∗ 2R2Tλ,αψ−N1 + iψ∗+σ+ 2 ψ+Tλ,αψ−N1 = 1 λ− α { Tλ,αψ−N1L2 − ψ−N2 − (α− ᾱ) R∗ 2 [Tλ,αψ−N1Lλ − ψ−N2] +ψ∗+σ+ 2 [S1(λ)−K] } in virtue of 2) (3.2) and the definition of S1(λ) (2.12). Taking now into account ψ∗+σ+ 2 K + [I + (ᾱ− α) R∗ 2] ψ−N2 = 0, we obtain R∗ 2Tλ,αψ−N1 {(λ− α)I + (α− ᾱ) Lλ} = Tλ,αψ−N1Lλ + ψ∗+σ+ 2 S1(λ). This equality exactly coincides with 4) (3.2). 214 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators Corollary 3.1. If the suppositions of Theorem 3.1 hold, then the formulas Tλ,αF (α, u−) = F (λ, u−) ; T ∗λ,αF̃ (α, u+) = F̃ (λ, u+) (3.4) take place for all u± ∈ E± and all λ, α ∈ Ω. II. Proceed now to the description of the class of functions formed by the characteristic functions S1(λ) (2.3) of the commutative colligation ∆ (1.1). Theorem 3.2. Let the commutative colligation ∆ (1.1) be given and the operators N1 and Ñ1 be boundedly invertible. Suppose that the operators ( K∗σ+ 2 K − σ−2 )−1 and ( Kτ−2 K∗ − τ+ 2 )−1 exist and are bounded in E− and E+, respectively. Then there exists a neighborhood Uδ(α) = {λ ∈ C : \|λ − α\| < δ} of the point α such that the linear bundles Qλ, Lλ and Q+ λ , L+ λ (3.3) are invertible for all λ ∈ Uδ(α). P r o o f. Prove that the operators Qλ and Lλ are boundedly invertible in some neighborhood Uδ(α) of the point α (the proof is similar for Q+ λ and L+ λ ). The point 2) (2.54) implies that i { K∗σ+ 2 S1(λ)− σ−2 } = N∗ 2 (N∗ 1 )−1 K1,1(λ, α) ·Qλ. (3.5) Prove that the invertibility of K∗σ+ 2 K−σ−2 necessitates the bounded invertibility of the operator { K∗σ+ 2 S1(λ)− σ−2 } in some neighborhood Uδ(α) of the point α. Since K∗σ+ 2 S1(λ)− σ−2 = K∗σ+ 2 K − σ−2 + i(λ− α)K∗σ+ 2 ψ+Tλ,αψ−N1, then the series { K∗σ+ 2 S1(λ)− σ−2 }−1 = { K∗σ+ 2 K − σ−2 }−1 · ∞∑ p=0 (λ− α)p [ −iK∗σ+ 2 ψ+Tλ,αψ−N1 { K∗σ+ 2 K − σ−2 }−1 ]p converges uniformly when \|λ− α\| ¿ 1 in virtue of holomorphy of S1(λ) (2.3) in the point λ = α. Thus, the operator { K∗σ+ 2 S1(λ)− σ−2 } is boundedly invertible in some neighborhood Uδ(α) of the point α. Let C = i { K∗σ+ 2 S1(λ)− σ−2 } , A = N∗ 2 (N∗ 1 )−1 K1,1(λ, α) and B = Qλ, then equality (3.5) in this notation is C = A·B. The bounded invertibility of C implies n ‖u−‖ ≤ ‖Cu−‖ , 0 < n < ∞, Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 215 V.A. Zolotarev for all u− ∈ E−, and since the operator A is bounded when λ ∈ Uδ(α), then n ‖u−‖ ≤ ‖Cu−‖ ≤ ‖A‖ · ‖Bu−‖ . Therefore, for B the estimation m ‖u−‖ ≤ ‖Bu−‖ is true for all u− ∈ E−, where m = n · ‖A‖−1 > 0. Thus, the invertibility of the linear bundle Qλ (3.3) for λ ∈ Uδ(α) is proved. (3.3) implies that Qλ − (λ− α)I = (α− ᾱ) Lλ; and for the invertibility of Lλ it is necessary to establish that (Qλ − (λ− α)I)−1 exists and it is bounded when λ ∈ Uδ(α). The last obviously follows from the uniform convergence of the series (Qλ − (λ− α)I)−1 = Q−1 λ ( I − (λ− α)Q−1 λ )−1 = ∞∑ p=0 (λ− α)p [ Q−1 λ ]p+1 . The theorem is proved. O b s e r v a t i o n 3.1. The invertibility of the bundles Qλ and Lλ (Q+ λ and L+ λ ) in the point λ = α implies that the operator N2 (Ñ∗ 2 ) is boundedly invertible. Thus, Theorem 3.2 yields that the invertibility of the expressions K∗σ+ 2 K − σ−2 ; Kτ−2 K∗ − τ∗2 ensures the existence of the bounded inverse of the operators N2 and Ñ∗ 2 . Proceed to the definition of the class of operator functions generated by cha- racteristic functions S1(λ) (2.3) of the commutative colligations ∆ (1.1). Class Ωα(σ, τ, N,Γ). Let E± be Hilbert spaces, α ∈ C\R+, and, moreover, suppose that in E−, correspondingly in E+, the linear bounded operators { σ−p }2 1 ; { τ−p }2 1 ; {Np}2 1 ; Γ : E− → E−; { σ+ p }2 1 ; { τ+ p }2 1 ; { Ñp }2 1 ; Γ̃ : E+ → E+ (3.6) are specified, where { σ±p }2 1 and { τ±p }2 1 are selfadjoint, and N1 and Ñ1 are invert- ible. An operator function S(λ): E− → E+ is said to belong to the class Ωα(σ, τ, N,Γ) if: 1) the function S(λ) is holomorphic in some neighborhood Uδ(α) = {λ ∈ C : \|λ− α\| < δ} of a point α and S(α) 6= 0; 216 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators 2) the kernel K(λ,w) (2.16) constructed by the functions S(λ) and S+(λ) = (N∗ 1 )−1 S∗(λ)Ñ∗ 1 is Hermitian positive for all λ, w ∈ Uδ(α); 3) the operator function S(λ) satisfies relations (2.51), where the linear bun- dles Lλ and L+ λ are constructed by using formulas (3.3) and L̃λ = Ñ−1 1 ( L+ λ )∗ Ñ1; 4) the operators { K∗σ+ 2 K − σ−2 } and { Kτ−2 K∗ − τ+ 2 } are boundedly invert- ible, where K = S(α); 5) for the operator family (3.6), (1.24) and S(α)N1 = Ñ1S(α) take place. It is obvious that the characteristic function S1(λ) (2.3) belongs to the class Ωα(σ, τ, N,Γ). Theorem of existence 3.3. Let the operator function S(λ): E− → E+ belong to the class Ωα(σ, τ,N,Γ). Then there exists a commutative colligation ∆ (1.1) such that the characteristic function S1(λ) (2.3) of the operator A1 coincides with S(λ), S1(λ) = S(λ) for all λ ∈ Uδ(α). P r o o f. Consider the family of “δ-functions” eλf assuming that every eλf has the support concentrated in the point λ ∈ Uδ(α) and possesses the value f = (u−, u+) ∈ E− ⊕ E+. The formal linear combinations N∑ k=1 eλk fk, where λk ∈ Uδ(α), fk ∈ E−⊕E+, 1 ≤ k ≤ N , N ∈ Z+, constitute the linear mani- fold L on which we, by means of the kernel K(λ, w) (2.16), define the Hermitian nonnegative bilinear form 〈eλf, ewg〉K def= 〈K(λ, w)f, g〉E−⊕E+ . (3.7) As a result of closure of the linear span L by norm generated by form (3.7) and of factorization by the kernel of this metric, we obtain the Hilbert space HK [9]. Specify the linear operators K: E− → E+, ψ−: E− → HK , ψ∗+: E+ → HK using the formulas K = S(α); ψ−u− = eαN−1 1 u−; ψ∗+u+ = eα ( Ñ∗ 1 )−1 u+ (3.8) and prove that relations 1) (1.2) take place for K, ψ−, ψ+ (3.8). Taking into account the form of the block K1,1(λ,w) of the kernel K(λ,w) (5.16), we have 〈 ψ−N1u−, ψ−N1u ′ − 〉 K = 〈 eαu−, eαu′− 〉 K = 〈 K1,1(α, α)u−, u′− 〉 = 〈 σ−1 −K∗σ+ 1 K i (α− ᾱ) u−, u′− 〉 , Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 217 V.A. Zolotarev which proves that 2 ImαN∗ 1 ψ∗−ψ−N1 = K∗σ+ 1 K−σ−1 . To prove 2 Imα·N∗ 2 ψ∗−ψ−N2 = K∗σ+ 2 K − σ−2 , consider 〈 ψ−N2u−, ψ−N2u ′ − 〉 K = 〈 eαN−1 1 N2u−, eαN−1 1 N2u ′ − 〉 K = 〈 N∗ 2 · (N∗ 1 )−1 K1,1(α, α)N−1 1 N2u−, u′− 〉 = i α− ᾱ 〈( K∗σ+ 2 K − σ−2 ) u−, u′− 〉 in view of 2) (2.51). The relations 2 ImαÑpψ+ψ∗+Ñ∗ p = Kτ−p K∗ − τ+ p , p = 1, 2, are proved in the similar way taking into account the form of the block K2,2(λ,w) of the kernel K(λ,w) and equality 3) (2.51). It is easy to show that ψ∗−eλf = (N∗ 1 )−1 σ−1 −K∗σ+ 1 S(λ) i ( λ− λ̄ ) u− + + S (λ)−K∗ i ( ᾱ− λ̄ ) u+; ψ+eλf = S(λ)−K i(λ− α) u− + Ñ−1 1 · Kτ−1 + S (λ)− τ+ 1 i ( λ̄− α ) u=. (3.9) As in (3.2), define the action of the resolvents {R1, R2} and {R∗ 1, R ∗ 2} in HK by using the formulas: R1eλf = eλ ( u− λ− α , u+ λ̄− α ) + eα ( N−1 1 τ−1 + S (λ) λ̄− α u+ − u− λ− α , 0 ) ; R∗ 1eλf = eλ ( u− λ− ᾱ , u+ λ̄− ᾱ ) + eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 S(λ)u− λ− ᾱ − u+ λ̄− ᾱ ) ; (3.10) R2eλf = eλ ( Lλu− λ− α ,L+ λ ( Q+ λ )−1 u+ ) +eα ( N−1 1 τ−2 + S (λ) ( Q+ λ )−1 u+ − Lαu− λ− α , 0 ) ; R∗ 2eλf = eλ ( Lλ (Qλ)−1 u−, L+ λ u+ λ̄− ᾱ ) + eα ( 0, ( Ñ∗ 1 )−1 σ+ 2 S(λ)Q−1 λ u− − L+ α u+ λ̄− ᾱ ) , where λ ∈ Uδ(α), f = (u−, u+) ∈ E−⊕E+, and the linear bundles Lλ, Qλ, L+ λ , Q+ λ are given by (3.3). Prove that relations 3) (1.2) are true. So, to prove K∗σ+ 1 ϕ1 + + N∗ 1 ψ∗− (A1 − ᾱI) = 0, write this equality in the following way: ψ∗+σ+ 1 K +ϕ−N1 + (ᾱ− α) R∗ 1ψ−N1 = 0. Then, taking into account (3.8) and (3.10), we obtain ψ∗+σ+ 1 Ku− + ψ−N1u− + (ᾱ− α) R∗ 1ψ−N1u− = eα ( Ñ∗ 1 )−1 σ+ 1 Ku− + eαu− +(ᾱ− α) eα u− α− ᾱ + (ᾱ− α) · eα ( Ñ∗ 1 )−1 σ+ 1 K α− ᾱ u− = 0, 218 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators q.e.d. Kτ−1 ( ϕ1− )∗ + Ñ1ψ+ (A∗1 − αI) = 0 is proved in a similar way. Rewrite the equality Kτ−2 ( ϕ2− )∗+Ñ2ψ+ (A∗2 − αI) = 0 as ψ−τ−2 K∗+ψ∗+Ñ∗ 2 +(α− ᾱ) R2ψ ∗ +Ñ∗ 2 = 0. Then, using (3.8), we have eαN−1 1 τ−2 K∗u+ + eα ( Ñ∗ 1 )−1 Ñ∗ 2 u+ + (α− ᾱ) R2eα ( Ñ∗ 1 )−1 Ñ∗ 2 u+ = eαN−1 1 τ−2 K∗u+ + eα ( Ñ∗ 1 )−1 Ñ∗ 2 u+ −R2eαQ+ α u+ = 0 since Q+ α = (ᾱ− α) L+ α = (ᾱ− α) ( Ñ∗ 1 )−1 Ñ∗ 2 in view of the definition of Q+ λ and L+ λ (3.3) and (3.10). K∗σ+ 2 ϕ2 + + N∗ 2 ψ∗− (A2 − ᾱI) = 0 is proved in exactly the same way. It is obvious that the intertwining condition S(λ)Lλ = L̃λS(λ) 1) (2.51) and KN1 = Ñ1K, definition of the class Ωα(σ, τ,N,Γ), yield KN2 = Ñ2K, which proves 7) (1.2). Note that Tλ,αeαu− = eλu−; T ∗λ,αeαu+ = eλu+ (3.11) take place for all λ, α ∈ Ω and all u± ∈ E±. In fact, (3.10) imply (λ−α)R1eλu− = eλu−−eαu−, and thus Tα,λeλu− = eαu−, which proves the first equality of (3.11). To prove the first condition in 5) (1.2), consider Tλ,α [R2ψ−N1u− −R1ψ−N2 − ψ−Γu−] = R2Tλ,αeαu− −R1Tλ,αeαN−1 1 N2u− −Tλ,αeαN−1 1 Γu− = R2eλu− −R1eλN−1 1 N2u− − eλN−1 1 Γu− = 1 λ− α {eλLλu− − eαLαu−} − 1 λ− α { eλN−1 1 N2u− − eαN−1 1 N2u− } −eλN−1 1 Γu− = 1 λ− α eλ { Lλ −N−1 1 N2 − (λ− α)N−1 1 Γ } = 0. Taking into account the invertibility of Tλ,α, Tλ,αTα,λ = I, we obtain the required. The proof of the second equality in 5) (1.2) is of a similar nature, besides, it is necessary to use the second relation of (3.11). Since ApRp = αRp + I, then 1 i 〈ApRpeλf, Rpewg〉K − 1 i 〈Rpeλf, ApRpewg〉K = 1 i 〈(αRp + I) eλf,Rpewg〉K − 1 i 〈Rpeλf, (αRp + I) ewg〉K = 〈{ α− ᾱ i R∗ pRp − iR∗ p + iRp } eλf, ewg 〉 K Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 219 V.A. Zolotarev for p = 1, 2. Therefore, to prove 4. (1.2), we have to prove that Bp = iRp − iR∗ p + α− ᾱ i R∗ pRp = ψ∗+σ+ p ψ+, p = 1, 2, (3.12) where Rp and R∗ p are given by (3.10), and ψ∗+ and ψ+ are given by formulas (3.8) and (3.9), respectively. To prove (3.12) when p = 1, consider B1eλf = iR1eλf − iR∗ 1eλf + α− ᾱ i R∗ 1R1eλf = eλ ( iu− λ− α , iu+ λ̄− α ) + eα ( iN−1 1 τ−1 + S (λ)u+ λ̄− α − iu− λ− α , 0 ) +eλ (−iu− λ− ᾱ , −iu+ λ̄− ᾱ ) + eα ( 0, −i ( Ñ∗ 1 )−1 σ+ 1 S(λ)u− λ− ᾱ + iu+ λ̄− ᾱ ) + α− ᾱ i R∗ 1 { eλ ( u− λ− α , u+ λ̄− α ) + eα ( N−1 1 τ−1 + S (λ)u+ λ̄− α − u− λ− α , 0 )} = eα ( iN−1 1 τ−1 + S (λ)u+ λ̄− α − iu− λ− α , −i ( Ñ∗ 1 )−1 σ+ 1 S(λ)u− λ− ᾱ + iu+ λ̄− ᾱ ) + α− ᾱ i eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 S(λ) λ− ᾱ · u− λ− α − 1 λ̄− ᾱ · u+ λ̄− α ) + α− ᾱ i eα ( 1 α− ᾱ [ N−1 1 τ−1 + S (λ)u+ λ̄− α − u− λ− α ] , 0 ) + α− ᾱ i eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 K α− ᾱ · { N−1 1 τ−1 + S (λ)u+ λ̄− α − u− λ− α }) = eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 S(λ)−K i(λ− α) u− + Ñ∗−1 1 σ+ 1 Ñ−1 1 Kτ−1 + S (λ)− Ñ∗ 1 i ( λ̄− α ) u+ ) . And if one takes into account 5) of the definition of class Ωα(σ, τ, N,Γ), Ñ∗ 1 = σ+ 1 Ñ−1 1 τ+ 1 (1.24), one can get B1eλf = eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 { S(λ)−K i(λ− α) u− + Ñ−1 1 Kτ−1 + S (λ)− τ+ 1 i ( λ̄− α ) u+ }) 220 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators = ψ∗+σ+ 1 ψ+eλf in view of the definition of ψ∗+ (3.8) and (3.9), which proves (3.12) when p = 1. Analogously, (3.12) is proved for B2. Using A∗pR∗ p = ᾱR∗ p + I, we obtain −1 i 〈 A∗pR ∗ peλf,R∗ pewg 〉 + 1 i 〈 R∗ peλf, A∗pR ∗ pewg 〉 = 〈{ α− ᾱ i RpR ∗ p + iRp − iR∗ p } eλf, ewg 〉 , p = 1, 2, and thus to prove the second relation of 4. (1.2), it is sufficient to prove that B̃p = iRp − iR∗ p + α− ᾱ i RpR ∗ p = ψ−τ−p ψ∗−, p = 1, 2, (3.13) where Rp and R∗ p are given by formulas (3.10), ψ− and ψ∗− are given by (3.8), (3.9). The proof of formulas (3.13) is similar to that of (3.12). Since ApRp = αRp + I and A∗pR∗ p = ᾱR∗ p + I, (3.10) implies A1R1eλf = eλ ( λu− λ− α , λ̄u+ λ̄− α ) + eα ( N−1 1 τ−1 + S (λ)u+ λ̄− α − u− λ− α , 0 ) ; A∗1R ∗ 1eλf = eλ ( λu− λ− ᾱ , λ̄u+ λ̄− ᾱ ) + eα ( 0, ( Ñ∗ 1 )−1 σ+ 1 S(λ)u− λ− α − u+ λ̄− ᾱ ) ; A2R2eλf = eλ (( αLλ λ− α + I ) u−, ( αL+ λ ( Q+ λ )−1 + I ) u+ ) +eα ( N−1 1 τ−2 + S (λ) ( Q+ λ )−1 u+ − Lαu− λ− α , 0 ) ; A∗2R ∗ 2eλf = eλ (( αLλQ−1 λ + I ) u−, ( αL+ λ λ̄− ᾱ + I ) u+ ) +eα ( 0, ( Ñ∗ 1 )−1 σ+ 2 S(λ)Q−1 λ u− − L+ α u+ λ̄− ᾱ ) (3.14) for all λ ∈ Uδ(α) and all f = (u−, u+) ∈ E− ⊕ E+, besides, the existence of Q−1 λ and ( Q+ λ )−1 follows from 4) of the definition of class Ωα(σ, τ, N,Γ). Specify the operator A1 in HK A1eλf = eλ ( λu−, λ̄u+ ) . (3.15) Then (3.14) and (3.10) imply that eλf = R1 (A1 − λI) eλf = R1eλ ( (λ− α)u−, ( λ̄− α ) u+ ) Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 221 V.A. Zolotarev = eλf + eα ( N−1 1 τ−1 + S (λ)u+ − u−, 0 ) . Therefore, the domain D (A1) of the operator A1 (3.15) is given by D (A1) = { N∑ p=1 eλpfp ∈ HK : λp ∈ Uδ(α); fp = ( up −, up + ) ∈ E− ⊕E+; up − = N−1 1 τ−1 + S (λp) up +; 1 ≤ p ≤ N ; N ≤ ∞ } . (3.16) Similar considerations show that the adjoint operator A∗1 equals A∗1eλf = eλ ( λu−, λ̄u+ ) , (3.15∗) and its domain D (A∗1) is represented by D (A∗1) = { N∑ p=1 eλfp ∈ HK : λp ∈ Uδ(α); fp = ( up −, up + ) ∈ E− ⊕ E+; up + = ( Ñ∗ 1 )−1 σ+ 1 S (λp) up −; 1 ≤ p ≤ N ;N ≤ ∞ } . (3.16∗) It is easy to establish that the operator A∗1 (3.15∗), (3.16∗) is the adjoint of A1 (3.15), (3.16). By (3.14) specify the operator A2 in HK A2eλf = eλ (( α + (λ− α)L−1 λ ) u−, ( α + ( L+ λ )−1 Q+ λ ) u+ ) , (3.17) besides, the existence of the inverse of Lλ and L+ λ again follows from 4) of the definition of class Ωα(σ, τ,N,Γ). Taking now into account (3.10) and (3.14), we have eλf = R2 (A2 − αI) eλf = R2eλ ( (λ− α)L−1 λ u−, ( L+ λ )−1 Q+ λ u+ ) = eλf + eα ( N−1 1 τ−2 + S (λ) ( L+ λ )−1 u+ − LαL−1 λ u−, 0 ) . Thus, the domain D (A2) of the operator A2 represents D (A2) = { N∑ p=1 eλpfp ∈ HK : λp ∈ Uδ(α); fp = ( up −, up + ) ∈ E− ⊕E+; up − = LλpN −1 2 τ−2 + S (λp) ( L+ λp )−1 up +; 1 ≤ p ≤ N ; N ≤ ∞ } . (3.18) It is easy to show that the adjoint A∗2 of the operator A2 (3.17), (3.18) is given by A∗2eλf = eλ (( ᾱ + L−1 λ Qλ ) u−, ( ᾱ + ( λ̄− ᾱ ) ( L+ λ )−1 ) u+ ) , (3.17∗) 222 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators and its domain equals D (A∗2) = { n∑ p=1 eλpfp ∈ HK : λp ∈ Uδ(α); fp = ( up −, up + ) ∈ E− ⊕ E+; up + = L+ λp ( Ñ∗ 2 )−1 σ+ 2 S (λp) L−1 λp up −; 1 ≤ p ≤ N ; N ≤ ∞ } . (3.18∗) Construct now the commutative colligation ∆K = ({ σ−p } ; { τ−p }2 1 ; {Np}2 1 ; Γ; HK ⊕ E−; {[ Ap ψ− ψ+ K ]}2 1 ; HK ⊕E+; Γ̃; { Ñp }2 1 ; { τ+ p }2 1 ; { σ+ p }2 1 ) , (3.19) where K, ψ−, ψ+, and {A1, A2} are given correspondingly by formulas (3.8), (3.9) and (3.15)–(3.18), Ω = Uδ(α). Finally, prove that the characteristic function S1(λ) of the operator A1 (3.15), (3.16) of the colligation ∆K coincides with S(λ). (3.8) and (3.11) imply Tλ,αψ−N1u− = eλ (u−, 0) . Using the form of the operator ψ+ (3.9), we have ψ+Tλ,αψ−N1u− = S(λ)−K i(λ− α) u− and thus S1(λ) = S(λ). III. Formulas (3.10) imply R1eλu− = eλu− − eαu− λ− α ; R2eλu− = eλN−1 1 Γu− + eλN−1 1 N2u− − eαN−1 1 N2u− λ− α . Therefore, if on the subspace in HK , H− K = span {eλu− : λ ∈ Uδ(α);u− ∈ E−} , (3.20) the linear bounded operators Npeλu− def= eλNpu−, p = 1, 2, and Γeλu− def= eλΓu− are given, then it is obvious that {N1R2 −N2R1 − Γ} f− = 0 (3.21) Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 223 V.A. Zolotarev for all f− ∈ H− K . Consider also the action of the resolvents R1 and R2 on the elements of another subspace in HK , H+ K = span {eλu+ : λ ∈ Uδ(α);u+ ∈ E+} . (3.22) Then (5.69) implies ( λ̄− α ) R1eλu+ = eλu+ + eαN−1 1 τ−1 + S (λ)u+; R2eλQ+ λ u+ = eλL+ λ u+ + eαN−1 1 τ−2 + S (λ)u+. (3.23) Therefore R2eλQ+ λ u+ −R1eλ ( λ̄− α ) L+ λ u+ = eαN−1 1 ( τ−2 + S (λ)− τ−1 + S (λ)L+ λ ) u+. (3.24) Taking into account the form of the linear bundles Q+ λ and L+ λ (3.3), transform the left-hand side of the equality R2eλQ+ λ u+ − ( λ̄− α ) R1eλL+ λ u+ = ( λ̄− α ) R2eλu+ + (α− ᾱ) R2eλu+ +(ᾱ− α) ( λ̄− α ) R2eλ ( Ñ∗ 1 )−1 Γ̃∗u+ + \|α− ᾱ\|2 R2eλ ( Ñ∗ 1 ) Γ̃∗u+ +(ᾱ− α)R2eλ ( Ñ∗ 1 ) Ñ∗ 2 u+ − ( λ̄− α )2 R1eλ ( Ñ∗ 1 )−1 Γ̃∗u+ − ( λ̄− α ) (α− ᾱ) R1eλ ( Ñ∗ 1 )−1 Γ̃∗u+ − ( λ̄− α ) R1eλ ( Ñ∗ 1 )−1 Ñ∗ 2 u+ = ( λ̄− α ){ R2eλu+ + (ᾱ− α) R2eλ ( Ñ∗ 1 )−1 Γ̃∗u+ − (α− ᾱ) R1eλ ( Ñ∗ 1 )−1 Γ̃∗u+ −R1eλ ( Ñ∗ 1 )−1 Ñ∗ 2 u+ } +(α− ᾱ) R2 [( λ̄− α ) R1eλu+ − eαN−1 1 τ−1 + S (λ)u+ ] + \|α− ᾱ\|2 R2 [( λ̄− α ) R1eλ ( Ñ∗ 1 )−1 Γ̃∗u+ − eαN−1 1 τ−1 + S (λ) ( Ñ∗ 1 )−1 Γ̃∗u+ ] +(ᾱ− α)R2 [( λ̄− α ) R1eλ ( Ñ∗ 1 )−1 Ñ∗ 2 u+ − eαN−1 1 τ−2 + S (λ) ( Ñ∗ 1 )−1 Ñ∗ 2 u+ ] − ( λ̄− α ) [ eλ ( Ñ∗ 1 )−1 Γ̃∗u+ + eαN−1 1 τ−1 + S (λ) ( Ñ∗ 1 )−1 Γ̃∗u+ ] in view of the first relation in (3.23). Define the linear operators Ñ∗ 1 , Ñ∗ 2 , and Γ̃∗ in H+ K (3.22), Ñ∗ p eλu+ def= eλÑ∗ p u+, p = 1, 2, and Γ̃∗eλu+ def= eλΓ̃∗u+. 224 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators Then, in view of the invariancy of the subspace H− K (3.20) with respect to the resolvents R1 and R2, we have R2eλQ+ λ u+ − ( λ̄− α ) R1eλL+ λ u+ = ( λ̄− α ) { R2 + (ᾱ− α) ( Ñ∗ 1 )−1 Γ̃∗R2 +(ᾱ− α) ( Ñ∗ 1 )−1 Γ̃∗R1 − ( Ñ∗ 1 )−1 Ñ∗ 2 R1 + (α− ᾱ) R1R2 + \|α− ᾱ\|2 × ( Ñ∗ 1 )−1 Γ̃∗R1R2 + (ᾱ− α) ( Ñ∗ 1 )−1 Ñ∗ 2 R1R2 − ( Ñ∗ 1 )−1 Γ̃∗ } eλu+ + f−, where f− ∈ H− K (3.20). Thus, we can be finally written relation (3.24) as ( λ̄− α ) ( Ñ∗ 1 )−1 { Ñ∗ 1 R2 (I + (α− ᾱ) R1)− Ñ∗ 2 R1 (I + (α− ᾱ) R2) −Γ̃∗ (I + (α− ᾱ) R1) (I + (α− ᾱ) R2) } eλu+ = g−, where g− ∈ H− K (3.20). Taking into account (3.21), we have {N1R2 −N2R1 − Γ} · ( Ñ∗ 1 )−1 { Ñ∗ 1 R2 (I + (α− ᾱ) R1) −Ñ∗ 2 R1 (I + (α− ᾱ) R2)− Γ̃∗ (I + (α− ᾱ) R1) (I + (α− ᾱ) R2) } eλu+ = 0. (3.25) Let dimE± = n± < ∞. By using Q (λ1, λ2) and Q̃ (λ1, λ2) (2.8), define the following polynomials: Q− (λ1, λ2) = (λ1, λ2) n− Q ( 1 λ1 + α, 1 λ2 + α ) = det [N1λ2 −N2λ1 + Γ] ; Q̃+ (λ1, λ2) = (λ1λ2) n+ Q̃ ( 1 + (ᾱ− α) λ̄1 λ̄1 + α, 1 + (ᾱ− α) λ̄2 λ̄2 + α ) = det [ Ñ∗ 1 λ2 (1 + (α− ᾱ) λ1)− Ñ∗ 2 λ1 (1 + (α− ᾱ) λ2) (3.26) −Γ̃∗ (1 + (α− ᾱ) λ1) (1 + (α− ᾱ)λ2) ] . Using Q− (λ1, λ2) and Q̃+ (λ1, λ2) (3.26), construct the polynomial P (λ1, λ2) def= Q− (λ1, λ2) · Q̃+ (λ1, λ2) . (3.27) Formulate an analogue of the Hamilton–Caley theorem for a commutative system of unbounded operators {A1, A2}. Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 225 V.A. Zolotarev Theorem 3.4. Let a simple commutative colligation ∆ (1.1) be given such that dimE± = n± < ∞, and the operators N1, Ñ1 are boundedly invertible. More- over, suppose that relations (1.24) are true, and { K∗σ+ 2 K − σ−2 } , { Kτ−2 K∗ − τ+ 2 } are invertible. Then the resolvents {R1, R2} of the main operators {A1, A2} of the colligation ∆ (1.1) annihilate the polynomial P (R1, R2) = 0, (3.28) where P (λ1, λ2) is given by (3.27) and constructed by the polynomials Q (λ1, λ2) and Q̃ (λ1, λ2) (2.8) using formulas (3.26). P r o o f. Since Q− (λ1, λ2) IE− = B (λ1, λ2) {N1λ2 −N2λ1 − Γ} , where B (λ1, λ2) is a matrix-valued polynomial of λ1, λ2, then (3.21) implies Q− (R1, R2) f− = 0 for all f− ∈ H− K (3.20). Similar considerations, by using (3.25), show that P (R1, R2) f+ = 0 for all f+ ∈ H+ K (3.22). And since the closed linear span H± K generates the whole space HK , we finally obtain P (R1, R2) f = 0 for all f ∈ HK . Application of Theorem 2.4 finishes the proof. Suppose that the characteristic function S1(λ) (2.3) is such that S1(α) is invertible, then the intertwining condition 1) (2.51) implies that n = n− = n+ and Q (λ1, λ2). Therefore polynomial (3.27) in this case is given by P (λ1, λ2) = Q− (λ1, λ2) ·Q+ (λ1, λ2) , (3.29) where Q± (λ1, λ2) are defined by the polynomial Q (λ1, λ2) (2.8) using formulas (3.26). Let w̄p = λp 1 + (α− ᾱ) λp , p = 1, 2. (3.30) Then it is obvious that the inverse transform to (3.30) is equal to λp = w̄p 1 + (ᾱ− α) w̄p = wp 1 + (α− ᾱ) wp , p = 1, 2. (3.31) 226 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 Characteristic Function of Commutative System Operators It is easy to see that P (λ1, λ2) = P ( w̄1 1 + (ᾱ− α) w̄1 , w̄2 1 + (ᾱ− α) w̄2 ) = det [ N1 w̄2 1 + (ᾱ− α) w̄2 −N2 w̄1 1 + (ᾱ− α) w̄2 − Γ ] · det { 1 1 + (ᾱ− α) w̄1 × 1 1 + (ᾱ− α) w̄2 · [N∗ 1 w̄2 −N∗ 2 w̄1 − Γ∗] } = (1 + (ᾱ− α) w̄1) −2n (1 + (ᾱ− α) w̄2) −2n P (w1, w2). Thus, polynomial (3.29) has the “antiholomorphic involution” (1 + (ᾱ− α) w̄1) 2n (1 + (ᾱ− α) w̄2) 2n P ( w̄1 1 + (ᾱ− α) w̄1 , w̄2 1 + (ᾱ− α) w̄2 ) = P (w1, w2) (3.32) relatively to transform (3.30). As known, the broken-linear transformation (3.30) is a holomorphic trans- formation of the generalized circle into the circle and its boundary (circle) is invariant relatively to this transformation. To find this circle, multiply both sides of the equality λ = λ̄ 1 + (ᾱ− α) λ̄ by ᾱ− α. Then, after elementary transformations, we obtain 1 + (α− ᾱ) λ = 1 1 + (ᾱ− α) λ̄ . This signifies that ξ = 1 + (α− ᾱ) λ satisfies the relation ξ = 1 ξ̄ , therefore ξ belongs to the unit circle T. Thus, we obtain the circle Tα = { λ = ξ − 1 α− ᾱ ∈ C : \|ξ\| = 1 } , (3.33) the radius r of which is equal to r = \|2 Im α\|−1 and the center Tα (3.33) is in the point i (when α ∈ C+) or in the point −i (when α ∈ C−). It is obvious that the transformation (3.30) written in the form 1 + (α− ᾱ) w = 1 1 + (ᾱ− α) λ̄ represents the inversion relatively to the circle Tα (3.33). Theorem 3.5. The polynomial P (λ1, λ2) (3.29) has the antiholomorphic involution (3.32) given by inversion (3.30) relative to the circle Tα (3.33). Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 227 V.A. Zolotarev References [1] M.S. Livs̆ic, On One Class of Linear Operators in Hilbert Space. — Mat. Sb. 19/61 (1946), No. 2, 236–260. [2] A.V. Kuzhel, On the Reduction of Unbounded Nonselfadjoint Operators to the Triangular Form. — DAN SSSR 119 (1958), No. 5, 868–871. (Russian) [3] A. Kuzhel, Characteristic Functions and Models of Nonselfadjoint Operator. Kluwer Acad. Publ., Dordrecht, London, 1996. [4] A.V. Shtraus, Characteristic Functions of Linear Operators. — Izv. AN SSSR, Ser. Mat. 24 (1960), No. 1, 43–74. (Russian) [5] E.R. Tsekanovskiy and Yu.L. Shmul’yan, Matters of the Expansion Theory of Unbounded Operators in Framed Hilbert Spaces. — Itogi Nauki. Mat. Analiz VINITI 14 (1977), 59–100. (Russian) [6] V.A. Derkach and M.M. Malamud, Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps. — J. Funct. Anal. 95 (1991), No. 1, 1–95. [7] B.S. Pavlov, Spectral Analysis of Dissipative Shrödinger Operator in Terms of Functional Model. — Itogi Nauki. Sovr. Probl. Mat., Fund. Napravl. VINITI 65 (1991), 95–163. (Russian) [8] M.S. Livs̆ic, N. Kravitsky, A. Markus, and V. Vinnikov, Theory of Commuting Nonselfadjoint Operators. Kluwer Acad. Publ., Dordrecht, London, 1996. [9] V.A. Zolotarev, Time Cones and Functional Model on Riemann Surface. — Mat. Sb. 181 (1990), No. 7, 965–994. (Russian) [10] V.A. Zolotarev, Analitic Methods of Spectral Representations of Nonselfadjoint and Nonunitary Operators. MagPress, Kharkov, 2003. (Russian) [11] V.A. Zolotarev, On Commutative Systems of Nonselfadjoint Unbounded Linear Operators. — J. Math. Phys., Anal., Geom. 5 (2009), 275–279. 228 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2

Properties of Characteristic Function of Commutative System of Unbounded Nonselfadjoint Operators

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