Functional Models in De Branges Spaces of One Class Commutative Operators

For a commutative system of the linear bounded operators T₁, T₂, which operate in the Hilbert space H and none of the operators T₁, T₂ is a compression, the functional model is constructed. The model is built for a circle in de Branges space.

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1. Verfasser:	Syrovatskyi, V.N.
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spelling	irk-123456789-1068082016-10-06T03:02:29Z Functional Models in De Branges Spaces of One Class Commutative Operators Syrovatskyi, V.N. For a commutative system of the linear bounded operators T₁, T₂, which operate in the Hilbert space H and none of the operators T₁, T₂ is a compression, the functional model is constructed. The model is built for a circle in de Branges space. Для коммутативной системы линейных ограниченных операторов T₁, T₂, которые действуют в гильбертовом пространстве H и ни один из операторов T₁, T₂ не является сжатием, построена функциональная модель. Эта модель строится в пространстве де Бранжа для круга. 2014 Article Functional Models in De Branges Spaces of One Class Commutative Operators / V.N. Syrovatskyi // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 430-450. — Бібліогр.: 9 назв. — англ. 1812-9471 DOI: http://dx.doi.org/10.15407/mag10.04.430 http://dspace.nbuv.gov.ua/handle/123456789/106808 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
collection	DSpace DC
language	English
description	For a commutative system of the linear bounded operators T₁, T₂, which operate in the Hilbert space H and none of the operators T₁, T₂ is a compression, the functional model is constructed. The model is built for a circle in de Branges space.
format	Article
author	Syrovatskyi, V.N.
spellingShingle	Syrovatskyi, V.N. Functional Models in De Branges Spaces of One Class Commutative Operators Журнал математической физики, анализа, геометрии
author_facet	Syrovatskyi, V.N.
author_sort	Syrovatskyi, V.N.
title	Functional Models in De Branges Spaces of One Class Commutative Operators
title_short	Functional Models in De Branges Spaces of One Class Commutative Operators
title_full	Functional Models in De Branges Spaces of One Class Commutative Operators
title_fullStr	Functional Models in De Branges Spaces of One Class Commutative Operators
title_full_unstemmed	Functional Models in De Branges Spaces of One Class Commutative Operators
title_sort	functional models in de branges spaces of one class commutative operators
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/106808
citation_txt	Functional Models in De Branges Spaces of One Class Commutative Operators / V.N. Syrovatskyi // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 430-450. — Бібліогр.: 9 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT syrovatskyivn functionalmodelsindebrangesspacesofoneclasscommutativeoperators
first_indexed	2025-07-07T19:03:52Z
last_indexed	2025-07-07T19:03:52Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2014, vol. 10, No. 4, pp. 430–450 Functional Models in De Branges Spaces of One Class Commutative Operators V.N. Syrovatskyi V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv 61077, Ukraine E-mail: VSirovatsky@gmail.com Received March 17, 2013, revised May 26, 2014 For a commutative system of the linear bounded operators T1, T2, which operate in the Hilbert space H and none of the operators T1, T2 is a com- pression, the functional model is constructed. The model is built for a circle in de Branges space. Key words: functional model, de Branges space, commutative systems of operators. Mathematics Subject Classification 2010: 47B32. The functional model of the compression operator T acting in the Hilbert space H was first obtained by B.S. Nagy and C. Foias [5]. The model allows to present the operator T as an operator of multiplication by the independent vari- able in a special space of functions [5, 2]. The study of the spectral characteristics of this model has led to a number of non-trivial problems on either functional analysis or theory of functions including the issues of interpolation, tasks of basis, completeness, etc. [2]. When the Nagy–Foias dilation technique [5] was used, there appeared signif- icant difficulties in the constructing of similar functional models for the commu- tative systems of the operators {T1, T2} defined in the Hilbert space H. Thus, the above problem could not be solved even for T1 and T2 being compressible. The solution was found in [7], which is based on a generalization of the concept node for commutative system operators, and in fact was proposed by Livshits. In [8], a functional model of a pair of commutative operators is built when one of them is compressed. The construction is based on the Fourier transfor- mation technique. If none of the operators {T1, T2} is not a compression, then the given method is not applicable. In this paper, we construct the functional models for a commutative system of the operators {T1, T2} where neither T1 nor T2 is compressed. For this case the functional model is constructed in de Branges space corresponding to the unit circumference obtained in [6]. c© V.N. Syrovatskyi, 2014 Functional Models in De Branges Spaces of One Class Commutative Operators 1. Background Information Let us consider the bounded linear operator T acting in the Hilbert space H. The collection 4 = (J ;H ⊕ E;V = [ T Φ Ψ K ] ; H ⊕ Ẽ; J̃) (1.1) is called a unitary knot [1-4] if the linear operator V = [ T Φ Ψ K ] : H ⊕ E 7→ H ⊕ Ẽ (1.2) satisfies the correlation V ∗ [ I 0 0 J̃ ] V = [ I 0 0 J ] , V [ I 0 0 J ] V ∗ = [ I 0 0 J̃ ] , (1.3) where J and J̃ are involutions in the Hilbert spaces E and Ẽ, respectively, J = J∗ = J−1, J̃ = J̃∗ = J̃−1. Any bounded linear operator T in H can always be included into a unitary knot 4 (1.1) if we set [2], −E = DT ∗H; Ẽ = DT H; Ψ =√ \|DT \|; Φ = √ \|DT ∗ \|; J = signDT ∗ ; J̃ = signDT ; K = −J̃T ∗; where, as usually, DT = I − T ∗T are defective operators of T , and √ \|A\|, signA of the self-adjoint operator A are understood in terms of the corresponding spectral decompositions. The knot 4 (1.1) is called simple [2] if H = H1, where H1 = span{TnΦf + T ∗mΨ∗g; f ∈ E; g ∈ Ẽ; n,m ∈ Z+}. (1.4) The subspaces H1 and H0 = H⊥ 1 = H ªH1 reduce the operator T , and the reducing of T to H0 is a unitary operator [2]. The main invariant of the knot 4 (1), which describes simple knots, is a characteristic operator function introduced by Livshits in 1946, [1], S4 = K + Ψ(zI − T )−1Φ, (1.5) which plays the main role in the theory of triangular [2] and functional models [4, 5] for the operators close to the unitary ones (in terms of definition (1.1)). Suppose that dimE = dim Ẽ = r < ∞ and J = J̃ . Let us choose the orthonormalized bases {eα}r 1 and {e′α}r 1 in E and Ẽ. Then from the results of Potapov [2] it follows that the matrix-function S4(z) = ‖ < S4(z)eα, e ′ β > ‖, in the case when the spectrum σ(T ) of the operator T belongs to the unit circumference T = {z ∈ C; \|z\| = 1}, has the multiplicative structure S4(z) = ←− l∫ 0 exp { eıϕt + z eıϕt − z J dFt } , (1.6) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 431 V.N. Syrovatskyi where ϕt is a non-negative non-decreasing on [0, `] function, and 0 ≤ ϕt ≤ 2π; Ft is a non-decreasing hermitian (r× r) matrix-function on [0, `] for which trFt ≡ t. Using Potapov’s presentation (1.6), it is not difficult to build a triangular model of the operator for S4(z) (5). By L2 r,l(Fx), denote the Hilbert space of the vector functions L2 r,l(Fx) = { f(x) = (f1(x), . . . , fr(x)); l∫ 0 f(x) dFxf∗(x) < ∞ } . (1.7) In L2 r,l(Fx) (1.7), define the linear operator T , Tf(x) = f(x)eıϕx − 2 l∫ x f(t) dFtΦ∗t Φ ∗−1 x Jeıϕx , (1.8) where the matrix Φx is a solution of the integral equation Φx + x∫ 0 Φt dFtJ = I, x ∈ [0, l]. (1.9) Similarly, the matrix-function Ψx is a solution of Ψx + l∫ x Ψt dFtJ = J, x ∈ [0, l]. (1.10) Let us define the operators Φ : E 7→ L2 r,l(Fx) and Ψ : L2 r,l(Fx) 7→ E (here E = Cn) as follows: Φf(x) = √ 2fΨxeıϕx , Ψf(x) = √ 2 l∫ 0 f(x) dFxΦ∗x, (1.11) where f ∈ E and K = S4(∞) (1.6). The collection 4c = (J ; L2 r,l(Fx)⊕ E;V = [ T Φ Ψ K ] ;L2 r,l(Fx)⊕ E; J) (1.12) is a unitary knot (1.1)–(1.3) and is called a triangular model of the simple knot4 (1.1), where L2 r,l(Fx), T , Φ, Ψ are from (1.7), (1.8)–(1.11). The latter means that simple components (1.4) of the knots 4 (1.1) and 4c (1.12), when the spectrum of the operator T is on the unit circumference σ(T ) ⊆ T, are unitarily equivalent 432 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators [2] under the condition that J = J̃ and dimE = dim Ẽ = r < ∞. Let us suppose that dimE = 2 and J = JN , where JN = [ −1 0 0 1 ] . (1.13) According to [6], we introduce Lx(z) = (1− zT )−1Φ(1, 1), (1.14) L̃x(z) = (1− zT ∗)−1Ψ∗(1,−1). (1.15) Definition 1. The de Branges space B(E, G) is a Hilbert space formed by the vector functions F (z) = [F1(z), F2(z)], where Fk(z), (k = 1, 2) are F1(z) = l∫ 0 f(t) dFtL ∗ t (z̄), F2(z) = l∫ 0 f(t) dFtL̃ ∗ t (z̄). (1.16) Present the de Branges space as follows: Bφf = [F1(z), F2(z)]. (1.17) The scalar product in B(E, G) is induced by the prototype mapping Bϕ (1.17), < F (z), F̂ (z) >Bϕ(E,G)=< f(t), f̂(t) >L2 2,l(Ft), (1.18) while F (z) = Bϕf(t), F̂ (z) = Bϕf̂(t), where f(t), f̂(t) ∈ L2 2,l(Ft). The functions Ex(z), Ẽx(z), Gx(z), G̃x(z) are defined by the relations [6] Lx(z) = (e−iφx − z)−1[Ex(z), Ẽx(z)], (1.19) L̃x(z) = (1− ze−iφx)−1[Gx(z), G̃x(z)]. (1.20) Let T1, T2 be a commutative system of the linear bounded operators acting in the Hilbert space H. The collection of the Hilbert spaces E, Ẽ and the operators Φ ∈ [E, H]; Ψ ∈ [H, Ẽ];K ∈ [E, Ẽ];σs, τs, Ns, Γ1 ∈ [E, E]; σ̃s, τ̃s, Ñs, Γ̃1 ∈ [Ẽ, Ẽ](s = 1, 2) is called a commutative unitary metric knot 4, 4 = (Γ1, σs, τs, Ns,H ⊕E, Vs, + V s,H ⊕ Ẽ, Ñs, τ̃s, σ̃s, Γ̃1), (1.21) if for the expansions Vs = [ Ts ΦNs Ψ K ] , + V s = [ T ∗s Ψ∗Ñ∗ s Φ∗ K∗ ] Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 433 V.N. Syrovatskyi the following relations are true: 1) V ∗ s [ I 0 0 σ̃ ] Vs = [ I 0 0 τs ] , + V ∗ s [ I 0 0 σs ] + V s = [ I 0 0 τ̃s ] , 2) T2ΦN1 − T1ΦN2 = ΦΓ1, Ñ1ΨT2 − Ñ2ΨT1 = Γ̃1Ψ, 3) Ñ2ΨΦN1 − Ñ1ΨΦN2 = KΓ1 − Γ̃1K, KNs = ÑsK(s = 1, 2), where σs, τs, (σ̃s, τ̃s) are self-adjoint in E(Ẽ), (s = 1, 2). The operators acting in the spaces E and Ẽ of the knot 4 (1.21) are depen- dent. An arbitrary commutative system of the linear bounded operators T1, T2 can always be included into the knot 4 (1.21) [1]. If the ”defective” operators σ1 and σ̃1 in E and Ẽ are reversible, we can always suppose that N1 and Ñ1 are reversible. Let us introduce N, Ñ,Γ, Γ̃ in the following form: N = N−1 1 N2, Γ = N−1 1 Γ1, Ñ = Ñ1 −1 Ñ2, Γ̃ = Ñ1 −1 Γ̃1. (1.22) Let us set the linear operators T1 and T2 in L2 r,l(Fx) (1.7): T1f(x) = f(x)eıϕx − 2 l∫ x f(t) dFtΦ∗t Φ ∗−1 x Jeıϕx , (1.23) T2f(x) = f(x) (N(x)eıϕx + Γ(x))− 2 l∫ x f(t) dFtΦ∗t Φ ∗−1 x JN(x)eıϕx , (1.24) where N(x) and Γ(x) satisfy the differential Lax equations [7]: N ′(x) = [axJ,N(x)], N(0) = Ñ2, Γ′(x) = [Γ(x), axJ ], Γ(0) = Γ̃2, [axJ,Γ(x) + eıϕxN(x)] = 0, where dFx = axdx. 2. Effect of Operators T1 and T ∗ 1 on Vectors Lx and L̃x Let the knot4 (1.21) corresponds to the commutative system of the operators {T1, T2}. Suppose that E = Ẽ, dimE = dim Ẽ = 2 and σ1 = σ̃1 = JN (1.13), the spectrum of the operator T1 consists of one point {1}, and therefore, ϕx = 0. By Lx(z) and L̃x(z), denote the vector functions (1.14),(1.15) which correspond to the operator T1(T = T1). We also denote the functions Ex(z), Ẽx(z), Gx(z), G̃x(z) by (1.19), (1.20). Lemmas 1–4 were proved in [9]. They define the effect of the operators T1 and T ∗1 on the vectors Lx and L̃x. 434 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators Lemma 1. [9] The operator T1 affects the vector function Lx(z) (1.14) in the following way: T1Lx(z) = Lx(z)− Lx(0) z . (2.1) Lemma 2. [9] The operator T1 affects the vector function L̃x(z) (1.15) in the following way: T1L̃x(z) = zL̃x(z) + G̃l(z)−Gl(z) 2 Lx(0)− G̃l(z) + Gl(z) 2 (1,−1)Ψx. (2.2) Lemma 3. [9] The operator T ∗1 affects the vector function L̃x(z) in the fol- lowing way: T ∗1 L̃x(z) = L̃x(z)− L̃x(0) z . (2.4) Lemma 4. [9] The operator T ∗1 affects the vector function Lx(z) in the fol- lowing way: T ∗1 Lx(z) = zLx(z) + E0(z)− Ẽ0(z) 2 L̃x(0) + E0(z) + Ẽ0(z) 2 (1, 1)Φx. (2.5) Let us prove the lemma below. Lemma 5. If the vector functions Lx(z) and L̃x(z) are set by (1.14) and (1.15), and Φx, Ψx are the solutions of integral equations (1.9) and (1.10), then l∫ 0 (1,−1)ΨtdFtL ∗ t (z) = −1− 1 2 R1J ( E0(z) Ẽ0(z) ) , (2.6) l∫ 0 (1,−1)ΨtdFtL̃ ∗ t (z) = 1 2 R1 ( 1 1 ) + Gl(z)− G̃l(z) 2 , (2.7) l∫ 0 (1, 1)ΦtdFtL ∗ t (z) = 1 2z R2 ( 1 1 ) + E0(z)− Ẽ0(z) 2z , (2.8) l∫ 0 (1, 1)ΦxdFtL̃ ∗ t (z) = 1 2z R2J ( Gl(z) G̃l(z) ) , (2.9) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 435 V.N. Syrovatskyi where R1 and R2 have the forms R1 = ( Gl(0)E0(0)− G̃l(0)E0(0)± 1 Gl(0) + G̃l(0) , (Gl(0)− G̃l(0))(Gl(0) + G̃l(0))± 1 Gl(0) + G̃l(0) ) , (2.10) R2 = ( Gl(∞)E0(∞)−G2 l (∞)−Gl(∞)G̃l(∞)± 1 Gl(∞) + G̃l(∞) , −3Gl(∞)E0(∞) + G̃2 l (∞) + Gl(∞)G̃l(∞)− 2G̃l(∞)E0(∞)± 1 Gl(∞) + G̃l(∞) ) . (2.11) P r o o f. Let us consider the equation for the vector function Lx(z) (1− z)Lx(z) + 2z l∫ x Lt(z)dFtΦ∗t Φ ∗−1 x J = (1, 1)Ψx and differentiate it by x to get (1− z)L′x(z)− 2zLx(z)axΦ∗xΦ∗−1 x J + 2z l∫ x Lt(z)dFtΦ∗t Φ ∗−1 x (Φ∗−1 x )′J = (1, 1)Ψ′ x. Since Ψ′ x = ΨxaxJ and Φ′x = −ΦxaxJ , then Φ∗x ′ = −JaxΦ∗x and (Φ∗−1 x )′ = Φ∗−1 x Jax. By using these statements, we obtain (1− z)L′x(z)− 2zLx(z)axJ + ((1, 1)Ψx − (1− z)Lx(x))axJ = (1, 1)ΨxaxJ, (1− z)L′x(z) = (1 + z)Lx(z)axJ, i.e., L′x(z) = 1+z 1−zLx(z)axJ and L∗x ′(z) = 1+z 1−zJaxL∗x(z). Let us consider the following statements: (ΨxJL∗x(z))′ = ΨxaxJJLx(z) + ΨxJ 1 + z 1− z JaxLx(z) = 2 1− z ΨxaxL∗x(z), ((1,−1)ΨxJL∗x(z))′ = (1,−1) 2 1− z ΨxaxL∗x(z). Since Ψl = J , Φ0 = I, and Ll(z) = (1, 1)J 1 1−z , L∗l (z) = J( 1 1 ) 1 1−z , L0(z) = 1 1−z (E0(z), Ẽ0(z)) and L∗0(z) = 1 1−z (E0(z), Ẽ0(z)), then after integrating the statement (1,−1)ΨxaxL∗x(z) = 1− z 2 ((1,−1)ΨxJL∗x(z)), 436 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators we obtain l∫ 0 (1,−1)ΨxdFxL∗x(z) = 1− z 2 (1,−1) ( JJJ ( 1 1 ) 1 1− z −Ψ0J 1 1− z ( E0(z) Ẽ0(z) )) = −1− 1 2 (1,−1)Ψ0J ( E0(z) Ẽ0(z) ) . (2.12) Similarly, using (ΦxJL∗x(z))′ = 2z 1−zΦxaxL∗x(z), we integrate the following state- ment: l∫ 0 (1, 1)ΦtdFtL ∗ t (z) = 1− z 2z (1, 1) ( ΦlJJ ( 1 1 ) 1 1− z − IJ 1 1− z (E0(z), Ẽ0(z)) ) = 1 2z (1, 1) ( Φl ( 1 1 ) − J(E0(z), Ẽ0(z)) ) = 1 2z (1, 1)Φl ( 1 1 ) + E0(z)− Ẽ0(z) 2z . (2.13) Now we take the equation for the vector function L̃x(z), (1− z)L̃x(z) + 2z l∫ 0 L̃t(z)dFtJΦ−1 t Φx = (1,−1)Φx, and differentiate it by x to get (1− z)L̃′x(z) + 2zL̃x(z)axJΦ−1 x Φx − 2z l∫ 0 L̃t(z)dFtJΦ−1 t ΦxaxJ = −(1,−1)ΦxaxJ, (1− z)L̃′x(z) + 2zL̃x(z)axJ + ((1− z)L̃x(z)− (1,−1)Φx)axJ = −(1,−1)ΦxaxJ, (1− z)L̃′x(z) + (1 + z)L̃x(z)axJ = 0. Thus, L̃′x(z) = −1+z 1−z L̃x(z)axJ and (L̃∗x(z))′ = −1+z 1−zJaxL̃∗x(z). Let us consider the following statements: (ΨxJL̃∗x(z))′ = ΨxaxJJL̃∗x(z)−ΨxJ 1 + z 1− z JaxL̃∗x(z) = −2z 1− z ΨxaxL̃∗x(z). And after integration we obtain l∫ 0 (1,−1)ΨxaxL̃∗x(z) = 1− z −2z (1,−1) ( JJ 1 1− z ( Gl(z) G̃l(z) ) −ΨxJ 1 1− z I ( 1 −1 )) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 437 V.N. Syrovatskyi = 1 −2z (1,−1) ( Gl(z) G̃l(z) ) − 1 −2z (1,−1)Ψ0 ( −1 −1 ) . Hence, l∫ 0 (1,−1)ΨxaxL̃∗x(z) = − 1 2z (1,−1)Ψ0 ( 1 1 ) − Gl(z)− G̃l(z) 2z . (2.14) Similarly, we can get (ΦxJL̃∗x(z))′ = 2z 1− z ΦxaxL̃∗x(z), then l∫ 0 (1, 1)ΦxaxL̃∗x(z) = 1− z 2z (1, 1) ( ΦlJ 1 1− z ( Gl(z) G̃l(z) ) − IJ 1 1− z I ( 1 −1 )) , (2.15) and l∫ 0 (1, 1)ΦxaxL̃∗x(z) = 1 2z ΦlJ ( Gl(z) G̃l(z) ) + 1 z . Write down a characteristic matrix-function S4(z) element-wisely, S4(z) = ( a(z) b(z) c(z) d(z) ) , and find its coefficients. Since N0(z) = −S4(z), Ñ∗ l (z) = S4(z), (1, 1)Nx(z)J = (E0(z), Ẽ0(z)) and (1,−1)Ñl(z) = (Gl(z), G̃l(z)), then Ñ∗ l (z) ( 1 −1 ) = ( Gl(z) G̃l(z) ) . For S4(z), we get the equations −(1, 1) ( a(z) b(z) c(z) d(z) ) J = (E0(z), Ẽ0(z)), ( a(z) b(z) c(z) d(z) )( 1 −1 ) = ( Gl(z) G̃l(z) ) . By solving this system, we obtain the coefficients of the matrix-function S4(z): c(z) = E0(z)− a(z), b(z) = a(z)−Gl(z), d(z) = E0(z)− G̃l(z)− a(z). 438 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators Now we will use the condition \| detS4(z)\|2 = 1, i.e., \| detS4(z)\| = ±1, or a(z)d(z)− b(z)c(z) = ±1, to get the expression for a(z) a(z) = Gl(z)E0(z)1 Gl(z) + G̃l(z) . Now we can find the expression of (1,−1)Ψx: (1,−1)Ψx = N0(0)J = −(1,−1)S4(0)J = ( Gl(0)E0(0)− G̃l(0)E0(0)± 1 Gl(0) + G̃l(0) , (Gl(0)− G̃l(0))(Gl(0) + G̃l(0))± 1 Gl(0) + G̃l(0) ) and the expression of (1, 1)Φl: (1, 1)Φl = (1, 1)Ñl(∞) = ( Gl(∞)E0(∞)−G2 l (∞)−Gl(∞)G̃l(∞)± 1 Gl(∞) + G̃l(∞) , −3Gl(∞)E0(∞) + G̃2 l (∞) + Gl(∞)G̃l(∞)− 2G̃l(∞)E0(∞)± 1 Gl(∞) + G̃l(∞) ) . Having defined these expressions as R1 and R2, respectively, and using integrals (2.12)–(2.15), we obtain the expressions stated in the lemma definition. Lemma 6. The operator T ∗1 affects the vector function Lx(z) (1.14) in the following way: T ∗1 Lx(z) = (z + µ(z))Lx(z) + ν(z)L̃x(z) + E0(z)− Ẽ0(z) 2 L̃x(0), (2.16) where ν(z) = c2(z)c3(z)− c1(z)c4(z) c2(z)− c4(z) , (2.17) µ(z) = c1(z)− c3(z) c2(z)− c4(z) , (2.18) c1(z) = (E0(z) + Ẽ0(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) ( 1 2z R2 ( 1 1 ) + E0(z)− Ẽ0(z) 2z ) , (2.19) c2(z) = (G′ l(z) + G̃′ l(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) , (2.20) c3(z) = E0(z) + Ẽ0(z) E′ 0(z)− Ẽ′ 0(z) ( 1 2z R2J ( Gl(z) G̃l(z) )) , (2.21) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 439 V.N. Syrovatskyi c4(z) = 2(Gl(z)Gl(z)− G̃l(z)G̃l(z)) (E′ 0(z)− Ẽ′ 0(z))(1− z2) , (2.22) and R1, R2 have the forms of (2.10) and (2.11), respectively. P r o o f. According to Lemma 4, T ∗1 Lx(z) = zLx(z) + E0(z)− Ẽ0(z) 2 L̃x(0) + E0(z) + Ẽ0(z) 2 (1, 1)Φx. Now we will show that the vectors Lx(z) and L̃x(z) are linearly independent with each fixed x ∈ [0, l] and any z ∈ C. Assuming the opposite, δ(z)Lx(z) = L̃x(z), let us suppose that δ(z)(1− zT )−1Φ(1, 1) = (1− zT ∗)−1Ψ∗(1,−1). Apply the operator T1 to both parts of the equation δ(z) (1− zT )−1Φ(1, 1)− Φ(1, 1) z = z(1− zT ∗)−1Ψ∗(1,−1)− ΦJS∗ ( 1 z ) (1,−1), (1− zT )−1Φ(1, 1)(δ(z)− z2) = Φ(1, 1)δ(z) + zΦJS∗ ( 1 z ) (1,−1). Let us consider the case where δ(z) = z2. From the previous equation we obtain that (1,−1)S∗(1 z ) = −z(1,−1), which is impossible because S∗(1 z ) = K∗+zΨ∗(1− zT ∗1 )−1Φ∗ and S∗(1 z ) 6= 0 where z = 0. If δ(z) 6= z2, then (1− zT )−1Φ(1, 1) = Φ ( (1, 1)δ(z) + zJS∗(1 z )(1,−1) δ(z)− z2 ) and (1− zT )−1Φ(1, 1) ∈ ΦE for ∀z, but Lx(z) /∈ ΦE for ∀z. Thus the functions Lx(z) and L̃x(z) are linearly independent and form basis in E2 for each fixed x for ∀z. Therefore we present the last term in the form E0(z) + Ẽ0(z) 2 (1, 1)Φx = µ(z)Lx(z) + ν(z)L̃x(z) (2.23) subsequently multiplying (2.23) by L̃∗x(z), E0(z) + Ẽ0(z) 2 l∫ 0 (1, 1)ΦxdFtL̃ ∗ t (z) = µ(z) l∫ 0 Lt(z)dFtL̃ ∗ t (z)+ν(z) l∫ 0 L̃t(z)dFtL̃ ∗ t (z). 440 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators Let us calculate the integrals in the above statement. First we get N0(z)− Ñ∗ l (ω) = 2(ω − z) l∫ 0 Mt(z)dFtM̃ ∗ t (ω), (2.24) Ñl(z)−N∗ 0 (ω) = 2(z − ω) l∫ 0 M̃t(z)dFtM ∗ t (ω). (2.25) We multiply (2.25) on the left by (−1, 1) and on the right by (1, 1)T . Since (−1, 1)Ñl(z) = (Gl(z), G̃l(z)) and (1, 1)N0(ω) = (E0(ω), Ẽ0(ω)), then Gl(z) + G̃l(z) + E0(ω)− Ẽ0(ω) = 2(z − ω) l∫ 0 L̃t(z)dFtL ∗ t (ω). Write the expression in the form (z − ω) l∫ 0 L̃t(z)dFtL ∗ t (ω) = Gl(z) + G̃l(z) 2 + Ẽ0(ω)− E0(ω) 2 . Let us define f(z) = Gl(z)+G̃l(z) 2 and g(ω) = Ẽ0(ω)−E0(ω) 2 . Since f(z) = −g(z), then f(z)− g(ω) z − ω → f ′(z), ω 7→ z, l l∫ 0 L̃t(z)dFtL ∗ t (z) = 1 2 ( dGl(z) dz + dG̃l(z) dz ) . (2.26) Now, if we multiply (2.24) on the left by (1, 1) and on the right by (−1, 1)T , then (E0(z), Ẽ0(z)) ( −1 1 ) − (1, 1) ( Gl(ω) G̃l(ω) ) = 2(ω − z) l∫ 0 Lt(z)dFtL̃ ∗ t (ω), E0(z)− Ẽ0(z) + Gl(ω) + G̃l(ω) = 2(z − ω) l∫ 0 Lt(z)dFtL̃ ∗ t (ω) and, similarly, l∫ 0 Lt(z)dFtL̃ ∗ t (z) = 1 2 ( dEl(z) dz − dẼl(z) dz ) . (2.27) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 441 V.N. Syrovatskyi We also have the expressions for two integrals: l∫ 0 L̃t(z)dFtL̃ ∗ t (ω) = Gx(z)Gx(ω)− G̃x(z)G̃x(ω) 1− zω , (2.28) l∫ 0 Lt(z)dFtL ∗ t (ω) = E0(z)E0(ω)− Ẽ0(z)Ẽ0(ω) 1− zω . (2.29) By using (2.26)–(2.29), we obtain E0(z) + Ẽ0(z) 2 l∫ 0 (1, 1)ΦxdFtL̃ ∗ t (z) = ν(z) ( E′ l(z)− Ẽ′ l(z) 2 ) + µ(z) ( Gx(z)Gx(z)− G̃x(z)G̃x(z) 1− z2 ) . Now we multiply statement (2.23) on the right by L∗x(z), E0(z) + Ẽ0(z) 2 l∫ 0 (1, 1)ΦxdFtL ∗ t (z) = ν(z) l∫ 0 Lt(z)dFtL ∗ t (z)+µ(z) l∫ 0 L̃t(z)dFtL ∗ t (z). By using expressions (2.26)–(2.29), in a similar way, we obtain E0(z) + Ẽ0(z) 2 l∫ 0 (1, 1)ΦxdFtL ∗ t (z) = ν(z) ( E0(z)E0(z)− Ẽ0(z)Ẽ0(z) 1− z2 ) + µ(z) ( G′ l(z) + G̃′ l(z) 2 ) . Now let us calculate ν(z) and µ(z). Taking into account (2.10) and (2.11), we will define the coefficients: c1(z) = (E0(z) + Ẽ0(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) ( 1 2z R2 ( 1 1 ) + E0(z)− Ẽ0(z) 2z ) , c2(z) = (G′ l(z) + G̃′ l(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) , c3(z) = E0(z) + Ẽ0(z) E′ 0(z)− Ẽ′ 0(z) ( 1 2z R2J ( Gl(z) G̃l(z) )) , 442 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators c4(z) = 2(Gl(z)Gl(z)− G̃l(z)G̃l(z)) (E′ 0(z)− Ẽ′ 0(z))(1− z2) . Hence, ν(z) = c2(z)c3(z)− c1(z)c4(z) c2(z)− c4(z) , µ(z) = c1(z)− c3(z) c2(z)− c4(z) , and thus the expression c2(z)− c4(z) is not identically equal to zero. Finally we get T ∗Lx(z) = zLx(z) + E0(z)− Ẽ0(z) 2 L̃x(0) + µ(z)Lx(z) + ν(z)L̃x(z), which proves the lemma. Lemma 7. The operator T1 affects the vector function L̃x(z) (1.15) in the following way: T1L̃x(z) = (z − ν̃(z))L̃x(z) + G̃l(z)−Gl(z) 2 Lx(0)− µ̃(z)Lx(z), (2.30) where ν̃(z) = c2(z)c̃3(z)− c̃1(z)c4(z) c2(z)− c4(z) , (2.31) µ̃(z) = c̃1(z)− c̃3(z) c2(z)− c4(z) , (2.32) c̃1(z) = (E0(z) + Ẽ0(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) ( −1− 1 2 R1J ( E0(z) Ẽ0(z) )) , (2.33) c̃3(z) = E0(z) + Ẽ0(z) E′ 0(z)− Ẽ′ 0(z) ( 1 2 R1 ( 1 1 ) + Gl(z)− G̃l(z) 2 ) , (2.34) and c2(z), c4(z) are (2.20) and (2.22), and R1, R2 are (2.10) and (2.11), respec- tively. P r o o f. According to Lemma 2, T1L̃x(z) = zL̃x(z) + G̃l(z)−Gl(z) 2 Lx(0)− G̃l(z) + Gl(z) 2 (1,−1)Ψx. We can perform the calculations that are similar to those made in Lemma 5. Since the functions Lx(z) and L̃x(z) are linearly independent and form the basis in L2, we can present the latter term of the above statement in the form G̃l(z) + Gl(z) 2 (1,−1)Ψx = µ̃(z)Lx(z) + ν̃(z)L̃x(z). Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 443 V.N. Syrovatskyi Similarly, we multiply it by Lx(z) and L̃x(z) and using the expressions for (2.26)– (2.29), we obtain G̃l(z) + Gl(z) 2 l∫ 0 (1,−1)ΨtdFtL̃ ∗ t (z) = ν̃(z) ( E′ l(z)− Ẽ′ l(z) 2 ) + µ̃(z)l ( Gx(z)Gx(z)− G̃x(z)G̃x(z) 1− z2 ) , G̃l(z) + Gl(z) 2 l∫ 0 (1,−1)ΨtdFtL ∗ t (z) = ν̃(z) ( E0(z)E0(z)− Ẽ0(z)Ẽ0(z) 1− z2 ) + µ̃(z) ( G′ l(z) + G̃′ l(z) 2 ) . By using (2.10) and (2.11) and introducing similar coefficients, we obtain c̃1(z) = (E0(z) + Ẽ0(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) ( −1− 1 2 R1J ( E0(z) Ẽ0(z) )) , (2.35) c̃2(z) = (G′ l(z) + G̃′ l(z))(1− z2) 2(E0(z)E0(z)− Ẽ0(z)Ẽ0(z)) = c2(z), c̃3(z) = E0(z) + Ẽ0(z) E′ 0(z)− Ẽ′ 0(z) ( 1 2 R1 ( 1 1 ) + Gl(z)− G̃l(z) 2 ) , (2.36) c̃4(z) = 2(Gl(z)Gl(z)− G̃l(z)G̃l(z)) (E′ 0(z)− Ẽ′ 0(z))(1− z2) = c4(z). Thus, for ν̃(z) and µ̃(z) we get ν̃(z) = c2(z)c̃3(z)− c̃1(z)c4(z) c2(z)− c4(z) , (2.37) µ̃(z) = c̃1(z)− c̃3(z) c2(z)− c4(z) . (2.38) Finally we obtain the expression T1L̃x(z) = zL̃x(z) + G̃l(z)−Gl(z) 2 Lx(0)− µ̃(z)Lx(z)− ν̃(z)L̃x(z), which proves the lemma. 444 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators 3. De Branges Transformation In [9], the following results were obtained, namely Lemmas 8–10. Lemma 8. [9] De Branges transformation BL (Definition 1) affects T1f in the following way: BL(T1f) = (z + µ(z))F1(z) + ν(z)F2(z) + E0(z)− Ẽ0(z) 2 F2(0), (3.1) where F1 and F2 have the same form as in (1.16). Lemma 9. [9] De Branges transformation B L̃ affects T1f in the following way: B L̃ (T1f) = F2(z)− F2(0) z , (3.2) where F1 and F2 have the same form as in (1.16). Lemma 10. [9] If the vector (1,−1) is latent for Ñ∗ + zΓ̃∗ with each z, then de Branges transformation B L̃ affects T2f , where T2f is from the knot 4 (1.21), in the following way: B L̃ (T2f(z)) = F2(z)n(z)− F (0)n(0) z , (3.3) where F1 and F2 have the form of (1.16), and the function n(z) satisfies the statement (Ñ∗ + zΓ̃∗)(1,−1) = n(z)(1,−1). (3.4) Let us prove the lemma below. Lemma 11. If the vector (1, 1) is latent for (N + zΓ), then de Branges trans- formation BL affects T2f , where T2f is from the knot 4 (1.21), in the following way: BL(T2f(z)) = F1(z) m(z) + µ̃(z) m(z) F1(z) + ν̃(z) m(z) F2(z), (3.5) where F1 and F2 have the form of (1.16), and the function m(z) satisfies the statement (N + zΓ)−1(1, 1) = 1 m(z) (1, 1). (3.6) Therefore the coefficients µ̃(z) and ν̃(z) have the forms µ̃(z) = I1(z)d3(z)− I2(z)d1(z) d2(z)d3(z)− d1(z)d4(z) , (3.7) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 445 V.N. Syrovatskyi ν̃(z) = I1(z)d4(z)− I2(z)d2(z) d1(z)d4(z)− d2(z)d3(z) , (3.8) where I1(z) = 1 2z (1, 1) √ 2S ( 1 z ) σ̃2 ( Φl ( 1 1 ) − J(E0(z), Ẽ0(z)) ) , (3.9) I2(z) = 1 2z (1, 1) √ 2S ( 1 z ) σ̃2 ( ΦlJ ( Gl(z) G̃l(z) ) + ( 1 1 )) , (3.10) d1(z) = E0(z)E0(z)− Ẽ0(z)Ẽ0(z) 1− \|z\|2 , (3.11) d2(z) = G′ l(z) + G̃′ l(z) 2 , (3.12) d3(z) = E′ 0(z)− Ẽ′ 0(z) 2 , (3.13) d4(z) = Gl(z)Gl(z)− G̃l(z)G̃l(z) 1− \|z\|2 . (3.14) P r o o f. Using the expressions for T1 (1.23) and T2 (1.24), we obtain T2Φ = T1ΦN + ΦΓ, zT2Φ = zT1ΦN + zΦΓ = (zT1 − 1)ΦN + Φ(Γz + N), z(zT1 − 1)−1T2Φ = ΦN + (zT1 − 1)−1Φ(Γz + N), T ∗2 T2z(zT1 − 1)−1Φ = T ∗2 ΦN + T ∗2 (zT1 − 1)−1Φ(Γz + N). Due to the knots relations, we get the statement T ∗2 T2 + Ψ∗σ̃2Ψ = I. Then (I −Ψ∗σ̃2Ψ)z(zT1 − 1)−1Φ = T ∗2 ΦN + T ∗2 (zT1 − 1)−1Φ(Γz + N), (Ψ∗σ̃2Ψ− I)z(1− zT1)−1Φ = T ∗2 ΦN + T ∗2 (zT1 − 1)−1Φ(Γz + N), zΨ∗σ̃2Ψz(1− zT1)−1Φ− z(1− zT1)−1Φ = T ∗2 ΦN − T ∗2 (1− zT1)−1Φ(Γz + N). Since the characteristic function has the form S(z) = K + Ψ(z − T1)−1Φ (1.5), then after writing the expressions S( 1 z )−K = Ψ( 1 z − T1)−1Φ = zΨ(1− zT1)−1Φ, T2ΦN + Ψσ̃2K = 0, 446 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators we obtain the equality Ψ∗σ̃2(S( 1 z )−K)− z(1− zT1)−1Φ = T ∗2 ΦN − T ∗2 (1− zT1)−1Φ(N + Γz), Ψ∗σ̃2S( 1 z )− z(1− zT1)−1Φ = −T ∗2 (1− zT1)−1Φ(N + Γz), T ∗2 (1− zT1)−1Φ = z(1− zT1)−1Φ(N + zΓ)−1 −Ψ∗σ̃2S( 1 z )(N + zΓ)−1, T ∗2 (1−zT1)−1Φ(1, 1) = z(1−zT1)−1Φ(N+zΓ)−1(1, 1)−Ψ∗σ̃2S( 1 z )(N+zΓ)−1(1, 1). Let us introduce the function m(z) satisfying the equation (N + zΓ)−1(1, 1) = 1 m(z) (1, 1), i.e., suppose that (1,1) is a latent vector of (N + zΓ). Then the statement T ∗2 Lx(z) = Lx(z) m(z) + Ψ∗σ̃2S(1 z )(1, 1) m(z) can be presented in the form Ψ∗σ̃2S( 1 z )(1, 1) = µ̃Lx(z) + ν̃L̃x(z), or by using the operator Ψ∗, (1, 1) √ 2S ( 1 z ) σ̃2Φx = µ̃Lx(z) + ν̃L̃x(z). (3.15) Multiplying (3.15) by Lx(z) and L̃x(z), we obtain two statements l∫ 0 (1, 1) √ 2S ( 1 z ) σ̃2ΦtdFtL ∗ t (z) = µ̃(z) l∫ 0 Lt(z)dFtL ∗ t (z) + ν̃ l∫ 0 L̃t(z)dFtL ∗ t (z), l∫ 0 (1, 1) √ 2S ( 1 z ) σ̃2ΦtdFtL̃ ∗ t (z) = µ̃(z) l∫ 0 Lt(z)dFtL̃ ∗ t (z) + ν̃ l∫ 0 L̃t(z)dFtL̃ ∗ t (z). By using previously obtained expressions for integrals (2.26)–(2.29), we introduce the following coefficients: d1(z) = l∫ 0 Lt(z)dFtL ∗ t (z) = E0(z)E0(z)− Ẽ0(z)Ẽ0(z) 1− \|z\|2 , Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 447 V.N. Syrovatskyi d2(z) = l∫ 0 L̃t(z)dFtL ∗ t (z) = G′ l(z) + G̃′ l(z) 2 , d3(z) = l∫ 0 Lt(z)dFtL̃ ∗ t (z) = E′ 0(z)− Ẽ′ 0(z) 2 , d4(z) = l∫ 0 L̃t(z)dFtL̃ ∗ t (z) = Gl(z)Gl(z)− G̃l(z)G̃l(z) 1− \|z\|2 . Now, using the calculations from Lemma 5, we obtain l∫ 0 ΦtdFtL ∗ t (z) = 1− z 2z ( ΦlJJ ( 1 1 ) 1 1− z − IJ 1 1− z (E0(z), Ẽ0(z)) ) , l∫ 0 ΦxaxL̃∗x(z) = 1− z 2z ( ΦlJ 1 1− z ( Gl(z) G̃l(z) ) − IJ 1 1− z I ( 1 −1 )) . By I1(z) and I2(z), we denote the following expressions: I1(z) = 1 2z (1, 1) √ 2S ( 1 z ) σ̃2 ( Φl ( 1 1 ) − J(E0(z), Ẽ0(z)) ) , I2(z) = 1 2z (1, 1) √ 2S ( 1 z ) σ̃2 ( ΦlJ ( Gl(z) G̃l(z) ) + ( 1 1 )) , then I1(z) d1(z) = µ̃(z) + ν̃(z) d2(z) d1(z) , I2(z) d3(z) = µ̃(z) + ν̃(z) d4(z) d3(z) , I1(z)d3(z)− I2(z)d1(z) d1(z)d3(z) = ν(z) d2(z)d3(z)− d1(z)d4(z) d1(z)d3(z) . Hence we have ν(z) = I1(z)d3(z)− I2(z)d1(z) d2(z)d3(z)− d1(z)d4(z) , ν̃(z) = I1(z)d4(z)− I2(z)d2(z) d1(z)d4(z)− d2(z)d3(z) and obtain the expression BL(T2f(z)) = F1(z) m(z) + µ̃(z) m(z) F1(z) + ν̃(z) m(z) F2(z), which proves the lemma. 448 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 Functional Models in De Branges Spaces of One Class Commutative Operators From Lemmas 6–11 we have the following theorem. Theorem. Let a commutative knot 4 (1.21) be such that E = Ẽ, dimE = 2, σ1 = σ̃1 = JN (1.13), the spectrum of the operator T1 be located at the point {1} and the vector (1, 1) be latent for (N + zΓ), i.e., let the function m(z) be such that (N +zΓ)(1, 1)T = m(z)(1, 1)T , and the vector (1,−1) be latent for Ñ∗+zΓ̃∗, i.e., let the function n(z) be such that (Ñ∗ + zΓ̃∗)(1,−1)T = n(z)(1,−1)T . Then the main system of the commutative operators {T1, T2} of the knot 4 (1.21) is unitarily equivalent to the system of operators that operates in the de Branges space B(E,G) in the following way: (T1F )1(z) = (z + µ(z))F1(z) + ν(z)F2(z) + E0(z)− Ẽ0(z) 2 F2(0), (T1F )2(z) = F2(z)− F2(0) z , (T2F )1(z) = F1(z) m(z) + µ̃(z) m(z) F1(z) + ν̃(z) m(z) F2(z) , (T2F )2(z) = F2(z)n(z)− F (0)n(0) z , where (F1(z), F2(z)) ∈ B(E, G). The coefficients µ(z), ν(z) and µ̃(z), ν̃(z) have the forms of (2.17), (2.18) and (3.7), (3.8), respectively, N, Ñ,Γ, Γ̃ are defined by (1.22). The correctness of this definition follows from the reversibility of σ and σ̃. Note. Let us consider the conditions (N + zΓ)(1, 1)T = m(z)(1, 1)T and (Ñ∗ + zΓ̃∗)(1,−1)T = n(z)(1,−1)T from the theorem. If we use (1.22), then the condition of intertwining [7] S(z)N1 −1(N2 + zΓ1) = Ñ−1 1 (Ñ2 + zΓ̃1)S(z) will have the form S(z)(N + zΓ) = (Ñ + zΓ̃)S(z). After multiplying this equation from left by (1,−1) and from right by (1, 1)T and using m(z)(1, 1)T = (N+zΓ)(1, 1)T and (1,−1)(Ñ+zΓ̃) = (1,−1)n(z), we obtain m(z)(1,−1)S(z)(1, 1)T = n(z)(1,−1)S(z)(1, 1)T . Hence the conditions imply that either m(z) = n(z) or (1, 1)S(z)(1, 1)T = 0 for ∀z ∈ C. Thus the functional model is built for the commutative system of the op- erators T1, T2, which is the main for the commutative knot 4(1.21) satisfying the conditions of the theorem. However, T1 and T2 affect one of the compo- nents [F1(z), F2(z)] as a shift and the other one as a multiplication by special holomorphic functions. Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 449 V.N. Syrovatskyi References [1] M.S. Livshits and A.A. Yantsevich, Theory of Operator Knots in Hilbert Spaces. Izdat. Kharkiv Univ., Kharkov, 1971. (Russian) [2] V.A. Zolotarev, Analytical Methods of Spectral Representation of Non-unitary and Non-selfadjoint Operators. Kharkiv National University, Kharkov, 2003. (Russian) [3] Lui De Branges, Hilbert Spaces of Entire Functions. Prentice-Hall, London, 1968. [4] M.S. Livshits, About one Class of Linear Operators in Hilbert Space. — Math. Collection 19 (1946), No. 2, 236–260. [5] B.S. Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space. Mir, Moscow, 1970. (Russian) [6] V.A. Zolotarev and V.N. Syrovatsky, De Branges Transform on the Cicle. Vestnik KNU V.N. Karazin. Number 711, Series ”Mathematics and Applied Mechanics”, Kharkiv National University, Kharkiv, 2005. (Russian) [7] V.A. Zolotarev, Model Representations of Commutative Systems of Linear Opera- tors. — Funct. Anal. and Appl. 22 (1988), No. 1, 66–68. [8] V.A. Zolotarev, Functional Model of Commutative Operator Systems. — J. Math. Phys., Anal., Geom. 4 (2008), No. 3, 420–440. [9] V.N. Syrovatsky, Functional Models of Commutative Systems of Operators Close to the Unitary. Vestnik KNU V.N. Karazin. Number 1018, Series ”Mathematics and Applied Mechanics”, Kharkiv National University, Kharkiv, 2012. (Russian) 450 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4

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