A system coupling and Donoghue classes of Herglotz-Nevanlinna functions

Transcription

1 A system coupling and Donoghue classes of Herglotz-Nevanlinna functions S. Belyi, K. A. Makarov and E. Tsekanovskiĭ Abstract. We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class M κ of Herglotz-Nevanlinna functions considered by the authors earlier, we introduce inverse generalized Donoghue classes of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function V Θz of an L-system Θ to belong the class M κ presented. In addition, we establish a connection between geometrical properties of two L-systems whose impedance functions belong to the classes M κ and M κ, respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes M κ M κ and M κ M κ, then the impedance function of the coupling falls into the class M κ κ. Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class M = M 0 is coupled with any other L-system, the impedance function of the coupling belongs to M the absorbtion property. Observing the result of coupling of n L-systems as n goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which in the position and momentum representations are presented. All major results are illustrated by various examples. M κ Mathematics Subject Classification 00. Primary: 8Q0, Secondary: 35P0, 47N50. Keywords. L-system, transfer function, impedance function, Herglotz- Nevanlinna function, Weyl-Titchmarsh function, Livšic function, characteristic function, Donoghue class, symmetric operator, dissipative extension, von Neumann parameter.

2 Belyi, Makarov and Tsekanovskii. Introduction This article is a part of an ongoing project studying the connections between various subclasses of Herglotz-Nevanlinna functions and conservative realizations of L-systems see [], [3], [6]. Let T be a densely defined closed operator in a Hilbert space H such that its resolvent set ρt is not empty. We also assume that DomT DomT is dense and that the restriction A = T DomT DomT is a closed symmetric operator with finite and equal deficiency indices. Let H + H H be the rigged Hilbert space associated with A see Section. One of the main objectives of the current paper is the study of the conservative L-system Θ = A K J H + H H E,. where the state-space operator A is a bounded linear operator from H + into H such that A T A, A T A, E is a finite-dimensional Hilbert space, K is a bounded linear operator from the space E into H, and J = J = J is a self-adjoint isometry on E such that the imaginary part of A has a representation Im A = KJK. Due to the facts that H ± is dual to H and that A is a bounded linear operator from H + into H, Im A = A A /i is a well defined bounded operator from H + into H. Note that the main operator T associated with the system Θ is uniquely determined by the statespace operator A as its restriction on the domain DomT = {f H + Af H. Recall that the operator-valued function given by W Θ z = I ik A zi KJ, z ρt, is called the transfer function of the L-system Θ and V Θ z = i[w Θ z+i] [W Θ z I] = K Re A zi K, R z ρt C\R, is called the impedance function of Θ. The main goal of the paper is to study L-systems the impedance functions of which belong to the generalized Donoghue class M κ consisting of all analytic mappings M from C + into itself that admits the representation Mz = λ z λ + λ dµλ, where µ is an infinite Borel measure such that dµλ R + λ = + κ 0 κ <. κ Note that L-systems with the impedance functions from the generalized Donoghue class M κ have been studied earlier in [6] see eq. 5. below for the definition of the classes M κ. A new twist in our exposition is the definition of a coupling of two L-systems associated with various generalized Donoghue classes, with the study of analytic properties of the impedance functions of the coupling being of primarily interest.

3 A system coupling and the Donoghue classes 3 The paper is organized as follows. In Section we recall the definition of an L-system and provide necessary background. Section 3 is of auxiliary nature and it contains some basic facts about the Livšic function associated with a pair A, A of a symmetric operator with deficiency indices, and its self-adjoint extension A as presented in [5], [6], [8]. In Section 4 we present an explicit construction of two L-systems whose impedance functions belong to generalized Donoghue classes considered in Section 5. In Section 5, following our development in [6], we introduce yet another generalized Donoghue class of functions M κ and establish a connection between geometrical properties of two L-systems whose impedance functions belong to the classes M κ and M κ, respectively. Then see Theorem 5.4 we present a criterion for the impedance function V Θ z of an L-system Θ to be a member of the class M κ. In Section 6 we introduce a concept of a coupling of two L-system see Definition 6.3 and show Theorem 6.4 that the procedure of coupling serves as a serial connection the transfer function of the coupling is a simple product of the transfer functions of L-systems coupled. Moreover, it is proved Theorem 6.5 that if the impedance functions of two L-systems belong to the generalized Donoghue classes M κ and M κ, then the impedance function of the constructed coupling falls into the class M κκ. An immediate consequence of this is the absorbtion property of the regular Donoghue class M = M 0 : if an L-system whose impedance function belongs to the class M is coupled with any other L-system, the impedance function of the coupling belongs to M see Corollary 6.6. We also show that if the impedance functions of two L-systems belong to the generalized Donoghue classes M κ and M κ, then the impedance function of the coupling still falls into the class M κκ see Corollary 6.8. A similar result takes place if the impedance function of one L-system belongs to the generalized Donoghue class M κ while the impedance function of the second L-system is from the class M κ. Then the coupling belongs to the class M κ κ see Corollary 6.9. The classes M, M k and M k can be considered symbolically as classes of mass preserving, mass decreasing and mass increasing classes respectively see 3.5, 5., 5., 5.8. In Section 7 we put forward the concept of a limit coupling see Definition 7. and define the system attractor, two models of which in the position and momentum representations are discussed in the end of the section. We conclude the paper by providing several examples that illustrate all the main results and concepts. For the sake of completeness, a functional model for a prime dissipative triple used in our considerations is presented in Appendix A.

4 4 Belyi, Makarov and Tsekanovskii. Preliminaries For a pair of Hilbert spaces H, H we denote by [H, H ] the set of all bounded linear operators from H to H. Let A be a closed, densely defined, symmetric operator with finite equal deficiency indices acting on a Hilbert space H with inner product f, g, f, g H. Any operator T in H such that A T A is called a quasi-self-adjoint extension of A. Consider the rigged Hilbert space see [7], [4] H + H H, where H + = Dom A and f, g + = f, g + A f, A g, f, g DomA.. Let R be the Riesz-Berezansky operator R see [7], [4] which maps H onto H + such that f, g = f, Rg + f H +, g H and Rg + = g. Note that identifying the space conjugate to H ± with H, we get that if A [H +, H ], then A [H +, H ]. An operator A [H +, H ] is called a self-adjoint bi-extension of a symmetric operator A if A = A and A A. Let A be a self-adjoint bi-extension of A and let the operator  in H be defined as follows: Dom = {f H + : Âf H,  = A DomÂ. The operator  is called a quasi-kernel of a self-adjoint bi-extension A see [], [3, Section.]. According to the von Neumann Theorem see [3, Theorem.3.] if  is a self-adjoint extension of A, then Dom = Dom A I UN i,. where U is a and +-isometric operator from N i into N i and N ±i = Ker A ii are the deficiency subspaces of A. A self-adjoint bi-extension A of a symmetric operator A is called t-self-adjoint see [3, Definition 3.3.5] if its quasi-kernel  is self-adjoint operator in H. An operator A [H +, H ] is called a quasiself-adjoint bi-extension of an operator T if A T A and A T A. In what follows we will be mostly interested in the following type of quasiself-adjoint bi-extensions. Definition. [3]. Let T be a quasi-self-adjoint extension of A with nonempty resolvent set ρt. A quasi-self-adjoint bi-extension A of an operator T is called a -extension of T if Re A is a t-self-adjoint bi-extension of A. Under the assumption that A has equal finite deficiency indices we say that a quasi-self-adjoint extension T of A belongs to the class Λ A if ρt, Dom A = DomT DomT, and hence T admits -extensions. The description of all -extensions via Riesz-Berezansky operator R can be found in [3, Section 4.3].

5 A system coupling and the Donoghue classes 5 Definition.. A system of equations { A zix = KJϕ ϕ + = ϕ ik, x or an array A K J Θ =.3 H + H H E is called an L-system if: A is a -extension of an operator T of the class Λ A; J = J = J [E, E], dim E < ; 3 Im A = KJK, where K [E, H ], K [H +, E], and RanK = RanIm A. In the definition above ϕ E stands for an input vector, ϕ + E is an output vector, and x is a state space vector in H. The operator A is called the state-space operator of the system Θ, T is the main operator, J is the direction operator, and K is the channel operator. A system Θ in.3 is called minimal if the operator A is a prime operator in H, i.e., there exists no non-trivial reducing invariant subspace of H on which it induces a self-adjoint operator. An L-system Θ defined above is conservative in the sense explained in [3, Section 6.3]. Systems of this type with bounded operators were studied in [8], [9], [5], [6], [7]. We associate with an L-system Θ the operator-valued function W Θ z = I ik A zi KJ, z ρt,.4 which is called the transfer function of the L-system Θ. We also consider the operator-valued function V Θ z = K Re A zi K, z ρâ..5 It was shown in [4], [3, Section 6.3] that both.4 and.5 are well defined. The transfer operator-function W Θ z of the system Θ and an operatorfunction V Θ z of the form.5 are connected by the following relations valid for Im z 0, z ρt, V Θ z = i[w Θ z + I] [W Θ z I]J, W Θ z = I + iv Θ zj I iv Θ zj..6 The function V Θ z defined by.5 is called the impedance function of an L-system Θ of the form.3. The class of all Herglotz-Nevanlinna functions in a finite-dimensional Hilbert space E, that can be realized as impedance functions of an L-system, was described in [4], [3, Definition 6.4.]. Two minimal L-systems Θ j = A j K j J H +j H j H j E, j =,, are called bi-unitarily equivalent [3, Section 6.6] if there exists a triplet of operators U +, U, U that isometrically maps the triplet H + H H

6 6 Belyi, Makarov and Tsekanovskii onto the triplet H + H H such that U + = U H+ is an isometry from H + onto H +, U = U+ is an isometry from H onto H, and UT = T U, U A = A U +, U K = K..7 It is shown in [3, Theorem 6.6.0] that if the transfer functions W Θ z and W Θ z of two minimal L-systems Θ and Θ are the same on z ρt ρt C ±, then Θ and Θ are bi-unitarily equivalent. 3. The Livšic function Another important object of interest and a tool that we use in this paper is the characteristic function of a symmetric operator introduced by Livšic in [4]. In [8] two of the authors K.A.M. and E.T. suggested to define a characteristic function of a symmetric operator as well of its dissipative extension as the one associated with the pairs A, A and T, A, rather than with the single operators A and T, respectively. Following [8], [9] we call the characteristic function associated with the pair A, A the Livšic function. For a detailed treatment of the Livšic function and characteristic function we refer to [8] and references therein. Suppose that A is a self-adjoint extension of a symmetric operator A with deficiency indices,. Let g ± be deficiency elements g ± Ker A ii, g ± =. Assume, in addition, that g + g DomA. 3. Following [8], the Livšic function s A, A associated with the pair A, A is sz = z i z + i g z, g g z, g +, z C +, 3. Here g ± Ker A ii are normalized appropriately chosen deficiency elements and g z 0 are arbitrary deficiency elements of the symmetric operators A. The Livšic function sz is a complete unitary invariant of a pair A, A with a prime, densely defined symmetric operator with deficiency indices, symmetric operator A and its self-adjoint extension A. Suppose also that T T is a maximal dissipative extension of A, ImT f, f 0, f DomT. Since A is symmetric, its dissipative extension T is automatically quasi-selfadjoint [3], that is, A T A, and hence, according to 4. with K = κ, g + κg DomT for some κ <. 3.3 A symmetric operator Ȧ is prime if there does not exist a subspace invariant under Ȧ such that the restriction of Ȧ to this subspace is self-adjoint.

7 A system coupling and the Donoghue classes 7 Based on the parametrization 3.3 of the domain of T and following [4], [8], the Möbius transformation Sz = sz κ κ sz, z C +, 3.4 where s is given by 3., is called the characteristic function of the dissipative extension T [4]. The characteristic function Sz of a dissipative maximal extension T of a densely defined prime symmetric operator A is a complete unitary invariant of the triple A, A, T see [8]. Recall that Donoghue [0] introduced a concept of the Weyl-Titchmarsh function M A, A associated with a pair A, A by M A, Az = Az + IA zi g +, g +, z C +, g + Ker A ii, g + =, where A is a symmetric operator with deficiency indices,, def A =,, and A is its self-adjoint extension. Denote by M the Donoghue class of all analytic mappings M from C + into itself that admits the representation Mz = λ z R λ + λ dµ, 3.5 where µ is an infinite Borel measure and dµλ =, equivalently, Mi = i. + λ R It is known [0], [], [], [8] that M M if and only if M can be realized as the Weyl-Titchmarsh function M A, A associated with a pair A, A. A standard relationship between the Weyl-Titchmarsh and the Livšic functions associated with the pair A, A was described in [8]. In particular, if we denote by M = M A, A and by s = s A, A the Weyl-Titchmarsh function and the Livšic function associated with the pair A, A, respectively, then sz = Mz i Mz + i, z C Hypothesis 3.. Suppose that T T is a maximal dissipative extension of a symmetric operator A with deficiency indices,. Assume, in addition, that A is a self-adjoint extension of A. Suppose, that the deficiency elements g ± Ker A ii are normalized, g ± =, and chosen in such a way that g + g DomA and g + κg DomT for some κ <. 3.7 Under Hypothesis 3., consider the characteristic function S = S A, T, A associated with the triple of operators A, T, A Sz = sz κ κ sz, z C +, 3.8 where s = s A, A is the Livšic function associated with the pair A, A.

8 8 Belyi, Makarov and Tsekanovskii We remark that given a triple A, T, A, one can always find a basis g ± in the deficiency subspace Ker A ii +Ker A + ii, such that and then, in this case, g ± =, g ± A ii, g + g DomA and g + κg DomT, κ = S A, T, Ai. 3.9 It was pointed out in [8], that if κ = 0, the quasi-self-adjoint extension T coincides with the restriction of the adjoint operator A on DomT = Dom A +Ker A ii. and that the prime triples A, T, A with κ = 0 are in one-to-one correspondence with the set of prime symmetric operators. In this case, the characteristic function S and the Livšic function s coincide up to a sign, Sz = sz, z C +. Similar constructions can be obtained if Hypothesis 3. is replaced with an anti-hypothesis as follows. Hypothesis 3.. Suppose that T T is a maximal dissipative extension of a symmetric operator A with deficiency indices,. Assume, in addition, that A is a self-adjoint extension of A. Suppose, that the deficiency elements g ± Ker A ii are normalized, g ± =, and chosen in such a way that g + + g DomA and g + κg DomT for some κ <. 3.0 Remark 3.3. Without loss of generality, in what follows we assume that κ is real and 0 κ < : if κ = κ e iθ, change the basis g to e iθ g in the deficiency subspace Ker A + ii. Thus, for the remainder of this paper we assume that κ is real and 0 κ <. Both functions sz and Sz can also be defined via 3. and 3.8 under the condition of Hypothesis 3.. It follows directly from the definition of the function sz and 3. that s A, A + = s A, A, where A + and A are two self-adjoint extensions of A satisfying the first part of Hypotheses 3. and 3., respectively. Similar relation takes place between the characteristic functions of the triplets A, T, A + and A, T, A, namely S A, T, A + = S A, T, A. 3. Here the triplets A, T, A + and A, T, A satisfy Hypotheses 3. and 3., respectively. To prove 3. one substitutes sz and κ for sz and κ in the formula 3.8 under the assumption that κ is real and 0 < κ <.

9 A system coupling and the Donoghue classes extensions as state-space operators of L-systems In this section we provide an explicit construction of two L-systems based upon given -extensions that become state-space operators of these systems. We will show that the corresponding operators of these L-systems satisfy the conditions of Hypotheses 3. and 3., respectively. Let A be a densely defined closed symmetric operator with finite deficiency indices n, n. In this case all operators T from the class Λ A with i ρt are of the form see [3, Theorem 4..9], [] DomT = Dom A I KN i, T = A DomT, DomT = Dom A I K N i, T = A DomT, 4. where K [N i, N i ]. Let M = N i N i and P + be a +-orthogonal projection operator onto a corresponding subspace shown in its subscript. Then see [], [6] all quasi-self-adjoint bi-extensions of T can be parameterized via an operator H [N i, N i ] as follows A = A + R S A P + M, A = A + R S A P + M, 4. where the block-operator matrices S A and S A are of the form HK H S A =, KHK + ii ii + KH K S A = H ii K H iik H H K. 4.3 Let T Λ A, i ρt and A be a self-adjoint extension of A such that U defines DomA via. and K defines T via 4.. It was shown in [6, Proposition 3] that A is a -extension of T whose real part Re A has the quasi-kernel A if and only if UK I is invertible and the operator H in 4.3 takes the form H = ii K K [I K UI U K K U]U. 4.4 For the remainder of this paper we will be interested in the case when the deficiency indices of A are,. Then S A and S A in 4.3 become matrices with scalar entries and As it was announced in [0] any quasi-selfadjoint bi-extension A of T takes a form A = A + [p, ϕ + q, ψ] ϕ + [v, ϕ + w, ψ] ψ, 4.5 p q where S A = is a matrix with scalar entries such that v w p = HK, q = H, v = KHK + i, and w = i + KH. Also, ϕ and ψ are For the convenience of the reader and to ease some further calculations we choose to have parameterizing operators U and K to carry the opposite signs in von Neumann s formulas. and 4.. As a result, formulas 4.3, 4.4, and 4.7 slightly deviate from their versions that appear in [6].

10 0 Belyi, Makarov and Tsekanovskii -normalized elements in R N i and R N i, respectively. Both parameters H and K are complex numbers in this case and K <. Similarly we write A = A + [ p, ϕ + q, ψ ] ϕ + [ v, ϕ + w, ψ ] ψ, 4.6 p where S A = q v w is such that p = K H i, q = K H i K, v = H, and w = H K. Following [6] let us assume 3 first that K = K = K = κ and U =. We are going to use 4.5 and 4.6 to describe a -extension A parameterized by these values of K and U. The formula 4.4 then becomes see [6] and S A = H = iκ +κ i +κ iκ +κ i κ [ κ κ κ] = i + κ + iκ i iκ +κ Performing direct calculations gives, S A = i S A S A = κ + κ S A + S A = i iκ +κ i iκ i iκ +κ +κ. Using 4.9 with 4.5 one obtains see [6] Im A = κ, ϕ ψϕ ψ + κ where Also, χ = +κ iκ , 4.9 =, χ χ, 4.0 κ κ ϕ ψ = ϕ ψ κ + κ Re A = A + i, ϕ + ψϕ ψ. 4. As one can see from 4., the domain Dom of the quasi-kernel  of Re A consists of such vectors f H + that are orthogonal to ϕ + ψ. Consider a special case when κ = 0. Then the corresponding -extension A,0 is such that Im A,0 =, ϕ ψϕ ψ =, χ,0χ,0, 4.3 where χ,0 = ϕ ψ Throughout this paper κ will be called the von Neumann extension parameter.

11 A system coupling and the Donoghue classes The -extension A or A,0 that we have just described for the case of K = κ and U = can be included in an L-system A Θ = K 4.5 H + H H C with K c = c χ, c C. Now let us assume that K = K = K = κ but U = and describe a -extension A parameterized by these values of K and U. Then formula 4.4 yields i H = κ [ + κ + κ + κ] = i κ. 4.6 Similarly to the above, we substitute this value of H into 4.3 and obtain S A = S A = iκ κ iκ κ + iκ iκ i κ κ i Performing direct calculations gives Furthermore, where Also, i κ i + iκ κ iκ i S A S A = + κ κ S A + S A = i κ iκ iκ κ, 4.7. Im A = + κ [, ϕ +, ψ]ϕ + [, ϕ +, ψ]ψ κ = + κ, ϕ + ψϕ + ψ κ =, χ χ, χ =., κ + κ ϕ + ψ = ϕ + ψ. 4.0 κ κ Re A = A i, ϕ ψϕ + ψ. 4. As one can see from 4., the domain Dom of the quasi-kernel  of Re A consists of such vectors f H + that are orthogonal to ϕ ψ. Similarly to the above when κ = 0, then the corresponding -extension A,0 is such that Im A,0 =, ϕ + ψϕ + ψ =, χ,0χ,0, 4. where χ,0 = ϕ + ψ. 4.3

12 Belyi, Makarov and Tsekanovskii Again we include A or A,0 into an L-system A Θ = K H + H H C 4.4 with K c = c χ, c C. Note that two L-systems Θ and Θ in 4.5 and 4.4 are constructed in a way that the quasi-kernels  of Re A and  of Re A satisfy the conditions of Hypotheses 3. and 3., respectively, as it follows from 4. and Impedance functions and Generalized Donoghue classes We say see [6] that an analytic function M from C + into itself belongs to the generalized Donoghue class M κ, 0 κ < if it admits the representation 3.5 where µ is an infinite Borel measure such that dµλ + λ = κ κ, equivalently, Mi = i + κ + κ, 5. R and to the generalized Donoghue class 4 M κ, 0 κ < if it admits the representation 3.5 and dµλ + λ = + κ + κ, equivalently, Mi = i κ κ. 5. R Clearly, M 0 = M 0 = M, the standard Donoghue class introduced above. Let A K Θ = 5.3 H + H H C be a minimal L-system with one-dimensional input-output space C whose main operator T and the quasi-kernel  of Re A satisfy the conditions of Hypothesis 3.. It is shown in [6] that the transfer function of W Θ z and the characteristic function Sz of the triplet A, T,  are reciprocals of each other, i.e., W Θ z = Sz, z C + ρt ρâ. 5.4 It is also shown in [6] that the impedance function V Θ z of Θ can be represented as κ V Θ z = V Θ0 z, κ where V Θ0 z is the impedance function of an L-system Θ 0 with the same set of conditions but with κ 0 = 0, where κ 0 is the von Neumann parameter of the main operator T 0 of Θ 0. 4 Introducing classes M κ and M κ we followed the notation of [3].

13 A system coupling and the Donoghue classes 3 Moreover, if Θ is an arbitrary minimal L-system of the form 5.3, then the transfer function of W Θ z and the characteristic function Sz of the triplet A, T,  are related as follows W Θ z = η Sz, z C + ρt ρâ, 5.6 where η C and η = see [6, Corollary ]. Lemma 5.. Let Θ and Θ be two minimal L-system of the form 5.3 whose components satisfy the conditions of Hypothesis 3. and Hypothesis 3., respectively. Then the impedance functions V Θ z and V Θ z admit the integral representation V Θk z = t z t + t dµ k t, k =,. 5.7 R Proof. We know see [3] that both V Θ z and V Θ z have the integral representation V Θk z = Q k + t z t + t dµ k t, k =,, 5.8 R and hence our goal is to show that Q = Q = 0. It was shown in [6, Theorem ] that under the conditions of Hypothesis 3. the function V Θ z belongs to the class M κ and thus Q = 0. We are going to show that the same property takes place for the function V Θ z. Consider the system Θ satisfying Hypothesis 3.. In Section 4 we showed that if an L-system satisfies Hypothesis 3. and hence the parameter U = in von Neumann s representation. of the quasi-kernel  of Re A, then Im A =, χχ, where the vector χ is given by 4.0. We know see [3], [6] that V Θ z = Re A zi χ, χ =  zi χ, χ, 5.9 χ = + κ κ ϕ + ψ, where  is the quasi-kernel of Re A of Θ and ϕ, ψ are vectors in H described in Section 4. According to [6, Theorem ], the impedance function of Θ belongs to the Donoghue class M if and only if κ = 0. Thus, if we set κ = 0 in 5.9, then V Θ z M and has Q = 0 in 5.7. But since the quasi-kernel  does not depend on κ, the expression for V Θ z when κ 0 is only different from the case κ = 0 by the constant factor +κ κ. Therefore Q = 0 in 5.7 no matter what value of κ is used. Now let us consider a minimal L-system Θ of the form 5.3 that satisfies Hypothesis 3.. Let also A Θ α = α K α, α [0, π, 5.0 H + H H C be a one parametric family of L-systems such that W Θα z = W Θ z e iα, α [0, π. 5.

14 4 Belyi, Makarov and Tsekanovskii The existence and structure of Θ α were described in details in [3, Section 8.3]. In particular, it was shown that Θ and Θ α share the same main operator T and that V Θα z = cos α + sin αv Θz sin α cos αv Θ z. 5. The following theorem takes place. Theorem 5.. Let Θ be a minimal L-system of the form 5.3 that satisfies Hypothesis 3.. Let also Θ α be a one parametric family of L-systems given by Then the impedance function V Θα z has an integral representation V Θα z = if and only if α = 0 or α = π/. R t z t + t dµ α t Proof. It was proved in [6, Theorem ] that V Θ z belongs to the generalized Donoghue class M κ holds if and only if the quasi-kernel  of Re A satisfies the conditions of Hypothesis 3. which is true in our case. Thus, Let us set V Θ i = i κ + κ. = κ + κ. Clearly, > 0 since we operate under assumption that 0 < κ <. It follows from 5. that V Θα i = cos α + sin αv Θi cos α + isin α = sin α cos αv Θ i sin α icos α cos α + isin α sin α + icos α = sin α icos α sin α + icos α = cos α sin α + i sin α + cos α = cos α sin α sin α + cos α + i sin α + cos α. On the other hand, V Θα z admits the integral representation of the form 5.7 and hence V Θα i = Q α + R t i t + t dµ α t = Q α + R t i t dµ α t t it + i = Q α + t dµ α t = Q α + i t i t + i + t dµ αt. R Comparing the two representations of V Θα i we realize that Q α = cos α sin α sin α + cos α. 5.3 R

15 A system coupling and the Donoghue classes 5 Analyzing 5.3 yields that Q α = 0 for κ 0 only if either α = 0 or α = π/. Consequently, the only two options for Q α to be zero are either W Θπ/ z = W Θ z or W Θ0 z = W Θ z. In the former case Θ π/ = Θ and hence Θ π/ satisfies Hypothesis 3.. Taking into account Lemma 5. we conclude that Θ 0 must comply with the conditions of Hypothesis 3. as the only available option. The next result describes the relationship between two L-systems of the form 5.3 that comply with Hypotheses 3. and 3.. Theorem 5.3. Let A Θ = K H + H H C 5.4 be a minimal L-system whose main operator T and the quasi-kernel  of Re A satisfy the conditions of Hypothesis 3. and let A Θ = K 5.5 H + H H C be another minimal L-system with the same operators A and T as Θ but with the quasi-kernel  of Re A that satisfies the conditions of Hypothesis 3.. Then W Θ z = W Θ z, z C + ρt, 5.6 and V Θ z = V Θ z, z C + ρt. 5.7 Proof. Using Lemma 5. and the reasoning above we conclude that since V Θ z and V Θ z do not have constant terms in their integral representation and Θ is different from Θ, equation 5.6 must take place. Then we use relations.6 to obtain 5.7. Next, we present a criterion for the impedance function of a system to belong to the generalized Donoghue class M κ. cf. [6, Theorem ] for the M κ -membership criterion. Theorem 5.4. Let Θ of the form 5.3 be a minimal L-system with the main operator T that has the von Neumann parameter κ, 0 < κ < and impedance function V Θ z which is not an identical constant in C +. Then V Θ z belongs to the generalized Donoghue class M κ if and only if the quasi-kernel  of Re A satisfies the conditions of Hypothesis 3.. Proof. Suppose Θ is such that the quasi-kernel  of Re A satisfies the conditions of Hypothesis 3.. Then according to Lemma 5. it has an integral representation V Θ z = R t z t + t dµt. 5.8

16 6 Belyi, Makarov and Tsekanovskii Also, if Θ is another system of the form 5.3 with the same main operator that satisfies the conditions of Hypothesis 3., then V Θ z M κ. Then we can utilize 5.7 to get V Θ i = R + t dµt = V Θ i = κ +κ i = + κ κ i. 5.9 Therefore, V Θ z M κ and we have shown necessity. Now let us assume V Θ z M κ and hence satisfies 5.8 and 5.9. Then, according to Theorem 5., equation 5.8 implies that V Θ z should either match the case α = 0 or α = π/. But α = π/ would cause Θ to comply with Hypothesis 3. which would violate 5.9 since see [6] then V Θ z M κ. Thus, α = 0 and Θ complies with Hypothesis 3.. Remark 5.5. Let us consider the case when the condition of V Θ z not being an identical constant in C + is omitted in the statement of Theorem 5.4. Suppose V Θ z is an identical constant and Θ satisfies the conditions of Hypothesis 3.. Let Θ be the L-system described in the proof of Theorem 5.4 that satisfies the conditions of Hypothesis 3.. Then, because of 5.7, its impedance function V Θ z is also an identical constant in C + and hence, as it was shown in [6, Remark 3] V Θ z = i κ + κ, z C Applying 5.7 combined with 5.0 we obtain V Θ z = i + κ κ, z C +, 5. and hence V Θ z M κ. Now let us assume that V Θ z M κ. We will show that in this case the L-system Θ from the statement of Theorem 5.4 is bi-unitarily equivalent to an L-system Θ a that satisfies the conditions of Hypothesis 3.. Let V Θ z from Theorem 5.4 takes a form 5.. Let also µλ be a Borel measure on R given by the simple formula µλ = λ, λ R, 5. π and let V 0 z be a function with integral representation 3.5 with the measure µ, i.e., V 0 z = R λ z λ + λ dµ. Then by direct calculations one immediately finds that V 0 i = i and that V 0 z V 0 z = 0 for any z z in C +. Therefore, V 0 z i in C + and hence using 5. we obtain V Θ z = i + κ κ = + κ κ V 0z, z C Let us construct a model triple B, T B, B defined by A. A.3 in the Hilbert space L R; dµ using the measure µ from 5. and our value of

17 A system coupling and the Donoghue classes 7-0 Im Re Figure. Parametric region 0 κ <, 0 α < π κ see Apendix A for details. Using the formula A.6 for the deficiency elements g z λ of Ḃ and the definition of sḃ, Bz in 3. we evaluate that sḃ, Bz 0 in C +. Then, 3.8 yields S B, T B, Bz κ in C +. Moreover, applying formulas A.4 and A.5 to the operator T B in our triple we obtain κ T B zi = B zi + i, g z g z. 5.4 κ Consider the operator B a which another self-adjoint extension of B and whose domain is given by the formula DomB a = Dom B +lin span { The model triple B, T B, B a is now consistent with Hypothesis 3.. Also, 3. yields S B, T B, B a z = S B, T B, Bz κ. Let us now follow Step of the proof of [6, Theorem 7] to construct a model L-system Θ a corresponding to our model triple B, T B, B a see Appendix A for details. Note, that this L-system Θ a is minimal by construction, its main operator T B has regular points in C + due to 5.4, and, according to 5.4 and 5.6, W Θa z /κ. But formulas.6 yield that in the case under consideration W Θ z /κ. Therefore W Θ z = W Θa z and we can taking into account the properties of Θ a we mentioned apply the Theorem on bi-unitary equivalence [3, Theorem 6.6.0] for L-systems Θ and Θ a. Thus we have successfully constructed an L-system Θ a that is bi-unitarily equivalent to the L-system Θ and satisfies the conditions of Hypothesis 3..

18 8 Belyi, Makarov and Tsekanovskii The results of Theorem 5.4 can be illustrated with the help of Figure describing the parametric region for the family of L-systems Θ α in 5.0. When κ = 0 and α changes from 0 to π, every point on the unit circle with cylindrical coordinates, α, 0, α [0, π describes an L-system Θ α with κ = 0. Then in this case according to [6, Theorem ] V Θα z belongs to the class M for any α [0, π. On the other hand, for any κ 0 such that 0 < κ 0 < we apply [6, Theorem ] to conclude that only the point, 0, κ 0 on the wall of the cylinder is responsible for an L-system Θ 0 such that V Θ0 z belongs to the class M κ0. Similarly, Theorem 5.4 yields that only the point, 0, κ 0 corresponds to an L-system Θ π/ such that V Θπ/ z belongs to the class M κ 0. We should also note that Theorem 5. and the reasoning above imply that for the family of L-systems Θ α with a fixed value of κ, 0 < κ < the only two family members Θ 0 and Θ π/ are such that the impedance functions V Θ0 z and V Θπ/ z belong to the classes M κ and M κ, respectively. Theorem 5.6. Let V z belong to the generalized Donoghue class M κ, 0 κ <. Then V z can be realized as the impedance function V Θκ z of an L- system Θ κ of the form 5.3 with the triple A, T,  that satisfies Hypothesis 3. with A = Â, the quasi-kernel of Re A. Moreover, V z = V Θκ z = + κ κ M A, Âz, z C +, 5.6 where M A, Âz is the Weyl-Titchmarsh function associated with the pair A, Â. Proof. It is easy to see that since V z Mκ, then V z = /V z M κ. Indeed, by direct check V z is a Herglotz-Nevanlinna function and condition 5. for V z immediately implies condition 5. written for V z, i.e., V i = i. Hence, V z M κ. Consequently, according to [6, Theorem 3], V z can be realized as the impedance function of an L-system Θ κ of the form 5.3 with the triple A, T,  that satisfies Hypothesis 3. with A = Â, the quasi-kernel of Re A. Moreover, V z = κ + κ M A, Âz, z C +, 5.7 where M A, Âz is the Weyl-Titchmarsh function associated with the pair A, Â. Let Θ κ be another L-system with the same operators A and T as Θ κ but with the quasi-kernel  of Re A that satisfies the conditions of Hypothesis 3.. Then Theorem 5.3 and 5.7 yield V Θκ z = V Θ κ z = V z = V z. Therefore, V z is realized by Θ κ as required. In order to show 5.6 we use 5.7 together with the fact that M A, Âz = /M A, Âz. The latter

19 A system coupling and the Donoghue classes 9 follows from the formula M A, Â cos α M A, Â sin α = cos α + sin α M A, α [0, π, Â, see [9, Subsection.] applied for α = π/. As a closing remark to this section we will make one important observation. Let Θ and Θ be two minimal L-systems given by 5.4 and 5.5 described in the statement of Theorem 5.3. Both L-systems share the same main operator T with the von Neumann parameter κ while Θ satisfies the conditions of Hypothesis 3. and Θ satisfies the conditions of Hypothesis 3.. As we have shown above in this case the impedance functions V Θ z and V Θ z belong to the classes M κ and M κ, respectively. Consequently, V Θ z and V Θ z admit integral representations 5.7 with the measures µ λ and µ λ, respectively. Thus, R dµ λ + λ = κ <, and + κ for the same fixed value of κ. Setting, dµ λ L = + λ and L = R and solving both parts of 5.8 for κ we obtain R κ = L + L = L L +. It also follows from 5.8 that L L = or dµ λ + λ = / dµ λ + λ. R R dµ λ + λ = + κ >, 5.8 κ R dµ λ + λ, 6. The coupling of two L-systems In this Section we will heavily rely on the concept of coupling of two quasiself-adjoint dissipative unbounded operators introduced in [9]. Let us denote the set of all such dissipative unbounded operators satisfying Hypothesis 3. by D. Definition 6.. Suppose that T DH and T DH are maximal dissipative unbounded operators acting in the Hilbert spaces H and H, respectively. We say that the maximal dissipative operator T DH H is an operator coupling of T and T, in writing, T = T T, if the Hilbert space H is invariant for T, i.e., DomT H = DomT, T DomT = T,

20 0 Belyi, Makarov and Tsekanovskii and the symmetric operator A = T DomT DomT has the property A T T. The following procedure provides an explicit algorithm for constructing an operator coupling of two dissipative operators T and T. Introduce the notation, A k = T k DomTk DomTk, k =,, and fix a basis g k ± Ker A k ii, g ± =, k =,, in the corresponding deficiency subspaces such that g k + κ k g k DomT k, with 0 κ k <. It was shown in [9, Lemma 5.3] that there exists a one parameter family [0, π θ A θ of symmetric restrictions with deficiency indices, of the operator A A such that A A A θ T T, θ [0, π. Without loss of generality we can pick the case when θ = 0 and set A = A 0. Then by [9, Lemma 5.3] the domain of A admits the representation Dom A = Dom A A +L 0, where { lin span sin α g+ κ g cos α g+ κ L 0 = g, κ 0 { lin span g+ κ g κ g, κ = 0 6. and tan α = κ κ κ, κ In accordance with [9, Theorem 5.4], we introduce the maximal dissipative extension T of A defined as the restriction of A A on DomT = Dom A +lin span {G + κ κ G, 6.3 where the deficiency elements G ± of A are given by Here and By construction, G + = cos α g + sin α g +, 6.4 G = cos βg sin β g. sin β = κ sin α and cos β = κ cos α, if κ 0, sin β = κ and cos β = κ, if κ = 0. A = T DomT DomT, 6.5 and the operator T is a unique coupling of T and T.

21 A system coupling and the Donoghue classes We should also mention that the self-adjoint extension A of A is also uniquely defined by the requirements G + G DomA, The following theorem is proved in [9]. A = A DomA. 6.6 Theorem 6.. Suppose that T = T T is an operator coupling of maximal dissipative operators T k DH k, k =,. Denote by A, A and A the corresponding underlying symmetric operators with deficiency indices,, respectively. That is, A = T DomT DomT and A k = T k DomTk DomTk k =,. Then there exist self-adjoint reference operators A, A,and A, extensions of A, A and A, respectively, such that ST T, A = ST, A ST, A. 6.7 One important corollary to Theorem 6. was presented in [9] and can be summarized as the following multiplication formula for the von Neumann parameters κt T = κt κt. 6.8 In this formula each κ represents the corresponding von Neumann parameter for the operator in parentheses. Our goal is to give a definition of a coupling of two L-systems of the form 5.3. Let us explain how all elements of this coupling are constructed. The main operator T of the system Θ is defined to be the coupling of main operators T and T of L-systems Θ and Θ. That is T = T T, 6.9 where operator coupling is defined by Definition 6. and its construction is described earlier in this section. The symmetric operator A of the system Θ is naturally given by A = T DomT DomT. The rigged Hilbert space H + H H H is constructed as the operator based rigging see [3, Section.] such that H + = Dom A. A self-adjoint extension  of A is the quasi-kernel of Re A and is defined by 6.6. The -extension A of the system is defined uniquely based on operators T and  and is described by formula 4.5. Then using formulas 4.0 and 4. one gets κ Im A =, χ χ, χ = Φ Ψ κ Here, κ = κ κ see 6.8 is the von Neumann parameter of the operator T in 6.9 while κ and κ are corresponding von Neumann parameters of T and T of Θ and Θ, respectively. Also, Φ = R G + and Ψ = R g, where R is the Riesz-Berezansky operator for the rigged Hilbert state space

22 Belyi, Makarov and Tsekanovskii of Θ. The channel operator K of the system is then given by Kc = c χ, c C and K f = f, χ, f H +. Now we are ready to give a definition of a coupling of two L-systems of the form 5.3. Definition 6.3. Let A Θ i = i K i, i =,, 6. H +i H i H i C be two minimal L-systems of the form 5.3 that satisfy the conditions of Hypothesis 3.. An L-system A K Θ = H + H H H C is called a coupling of two L-systems if the operators A, K and the rigged space H + H H H are defined as described above. In this case we will also write Θ = Θ Θ. Clearly, by construction, the elements of L-system Θ satisfy the conditions of Hypothesis 3. for κ = κ κ. Theorem 6.4. Let an L-system Θ be the coupling of two L-systems Θ and Θ of the form 6. that satisfy the conditions of Hypothesis 3.. Then if z ρt ρt, we have W Θ z = W Θ z W Θ z. 6. Proof. The proof of the Theorem immediately follows from Theorems 6. and formula 5.4. A different type of L-system coupling with property 6. was constructed in [5] see also [3, Section 7.3]. Remark 6 Definition 6.3 for the coupling of two L-systems can be meaningfully extended to the case when participating L-systems Θ and Θ do not satisfy the requirements of Hypothesis 3.. We use the following procedure to explain the system Θ = Θ Θ. We begin by introducing L-systems Θ and Θ which are only different from Θ and Θ by the fact that their quasi-kernels  of Re A and  of Re A satisfy the conditions of Hypothesis 3. for the same values of κ and κ. Then Definition 6.3 applies to them and we can obtain the coupling Θ = Θ Θ. Note that Θ is an L-system that satisfies the conditions of Hypothesis 3.. Theorem 6.4 also applies and we have W Θ z = W Θ z W Θ z = ν W Θ z ν W Θ z, where ν = ν =. Equivalently, We set ν ν W Θ z = W Θ z W Θ z. W Θ z = ν ν W Θ z, 6.3

23 A system coupling and the Donoghue classes 3 and recover the L-system Θ whose transfer function is W Θ z and that is different from the system Θ by changing the quasi-kernel  to  in a way that produces the unimodular factor ν ν in 6.3. This construction of Θ is well defined and unique due to the theorems about a constant J-unitary factor [3, Theorem 8..] and [6, Proposition 3]. Theorem 6.5. Let an L-system Θ be the coupling of two L-systems Θ and Θ of the form 6. that satisfy the conditions of Hypothesis 3.. Suppose also that V Θ z M κ and V Θ z M κ. Then V Θ z M κκ. Proof. As we mentioned earlier, the elements of L-system Θ satisfy the conditions of Hypothesis 3. for κ = κ κ. Thus, according to [6, Theorem ], V Θ z satisfies 5.5 and consequently V Θ i = i κ + κ, and hence V Θ z M κκ. The following statement immediately follows from Theorem 6.5. Corollary 6.6. Let an L-system Θ be the coupling of two L-systems Θ 0 and Θ of the form 6. that satisfy the conditions of Hypothesis 3.. Suppose also that V Θ0 z M and V Θ z M κ. Then V Θ z M. The coupling absorbtion property described by Corollary 6.6 can be enhanced as follows. Corollary 6.7. Let an L-system Θ be the coupling of two L-systems Θ 0 and Θ of the form 6. and let V Θ0 z M. Then V Θ z M. Proof. The procedure of forming the coupling of two L-systems that don t necessarily obey the conditions of Hypothesis 3. is described in Remark 6. It is clear that the main operator T of such coupling is also found via 6.9 and its von Neumann parameter κ is still a product given by 6.8. Hence in our case κ = 0 regardless of whether Θ 0 and Θ satisfy the conditions of Hypothesis 3.. Therefore applying [6, Theorem ] yields V Θ z M. Corollary 6.8. Let an L-system Θ be the coupling of two L-systems Θ and Θ of the form 6. that satisfy the conditions of Hypothesis 3.. Suppose also that V Θ z M κ and V Θ z M κ. Then Θ satisfies the conditions of Hypothesis 3. and V Θ z M κκ. Proof. In order to create the coupling of two systems Θ and Θ that do not obey the conditions of Hypothesis 3. we follow the procedure described in Remark 6. We know, however, that according to Theorem 5.3 and 5.6 both unimodular factors ν and ν in 6.3 are such that ν = ν =. Then 6.3 yields W Θ z = W Θ z = W Θ z = W Θ z W Θ z, where W Θ z is the transfer function of the L-system Θ that obeys Hypothesis 3..

24 V 4 Belyi, Makarov and Tsekanovskii V V Figure. Coupling Θ = Θ Θ and its impedance Let us show that in fact the L-systems Θ and Θ coincide. Recall see Remark 6 that Θ and Θ can only be different by the quasi-kernels  and  and thus share the same operator T. Then according to the theorem about a constant J-unitary factor [3, Theorem 8..], W Θ z = νw Θ z, where ν = is a unimodular complex number. In our case, however, we have just shown that W Θ z = W Θ z and hence ν =. As it can be seen in the proof of [3, Theorem 8..] then only possibility for ν = occurs when von Neumann s parameters for the quasi-kernels  and  match see also [3, Theorem 4.4.6]. But this implies that  =  and thus Θ and Θ coincide. Consequently, Θ satisfies the conditions of Hypothesis 3.. Also, both Θ and Θ share the same main operators T and T with Θ and Θ, respectively. Therefore, according to Theorem 6.5, V Θ z = V Θ z M κκ. Corollary 6.9. Let an L-system Θ be the coupling of two L-systems Θ and Θ of the form 6. that both satisfy the conditions of Hypotheses 3. and 3., respectively. Suppose also that V Θ z M κ and V Θ z M κ. Then Θ satisfies the conditions of Hypothesis 3. and V Θ z M κ κ. Proof. In order to prove the statement of the corollary we replicate all the steps of the proof of Corollary 6.8 above with ν = and ν = implying W Θ z = W Θ z = W Θ z = W Θ z W Θ z = W Θ z, where W Θ z is the transfer function of the L-system Θ that has the same operators A and T as Θ but with the quasi-kernel  of Re A that satisfies the conditions of Hypothesis 3.. Then similar reasoning is used to show that the L-system Θ satisfies the conditions of Hypothesis 3. and that V Θ z M κ κ. Note that the conclusion of Corollary 6.9 remains valid if V Θ z M κ and V Θ z M κ. Figure above is provided to illustrate Theorems as well as Corollaries including the absorbtion property.

25 7. The Limit coupling A system coupling and the Donoghue classes 5 Let Θ, Θ,..., Θ n be L-systems of the form 5.3. Relying on Definition 6.3 and Remark 6 from the previous section we can inductively define the coupling of n L-systems by n Θ n = Θ k = Θ Θ... Θ n, 7. k= with all the elements of the L-system Θ n constructed according to the repeated use of the coupling procedure described in Section 6. In particular, if each of the L-systems Θ k, k =,..., n satisfies Hypothesis 3., then so does the L-system Θ n. Definition 7.. Let Θ, Θ, Θ 3,... be an infinite sequence of L-systems with von Neumann s parameters κ, κ, κ 3,..., respectively. We say that a minimal L-system Θ is a limit coupling of the sequence {Θ n and write n Θ = lim n Θn = lim Θ k = Θ Θ... Θ n..., 7. n k= if the von Neumann parameter κ of Θ is such that n κ = lim κ n, κ <, and W Θ z = lim n k= n k= n W Θn z, z C. Formula 7. and Definition 7. allow us to introduce the concepts of the power of an L-system and of the system attractor. Indeed, let Θ be an L-system of the form 5.3 satisfying the conditions of Hypothesis 3.. We refer to the repeated self-coupling Θ n = Θ Θ... Θ {{ n times 7.3 as an n-th power of an L-system Θ. Furthermore, the L-system Ξ such that Ξ = lim n Θn, 7.4 will be called a system attractor. Let κ and κ Ξ be the von Neumann parameters of the L-systems Θ and Ξ, respectively. It follows from Definition 7. and the fact that κ < that κ Ξ = lim n κn = 0. Moreover, it is known see [3, Section 6.3] that W Θ z < for all z C. Hence W Ξ z = lim WΘ z n = 0, z C. 7.5 n

26 6 Belyi, Makarov and Tsekanovskii Figure 3. System attractor Ξ Consequently, for the impedance function V Ξ z relation.6 yields V Ξ z = i W Ξz W Ξ z + i, z C. Since V Ξ z is a Herglotz-Nevanlinna function, then V Ξ z i for all z C +. To show that a system attractor exists we consider the model triple B, T B0, B of the form A. A.3 with S B, T B0, Bi = 0 see Apendix A for details and a model system B Ξ = 0 K 0, 7.6 H + H H C constructed in [6, Section 5] upon this triple. Recall, that this system Ξ the state-space operator B 0 [H +, H ] is a -extension of T B0 such that Re B 0 B = B, K0 f = f, g 0, f H +, and κ = 0. It is shown in [6, Theorem 7] that for the system Ξ we have V Ξ z = M B, Bz, z C +. Taking into account that κ = 0, we note that resolvent formula A.4 A.5 takes the form T B0 zi = B zi M B, Bz i, gz g z, 7.7 z ρt B0 ρb. In the case when M B, Bz = i for all z C +, formula 3.6 would imply that s B, Bz 0 in the upper half-plane. Then, 7.7 see also [8, Lemma 5.] yields that all the points z C + are eigenvalues for T B0. Consequently, the point spectrum of the dissipative operator T B0 fills in the whole open upper half-plane C +. The following lemma will be a useful tool in understanding the geometric structure of system attractors. Lemma 7.. Let Θ and Θ be two minimal L-systems of the form 6. such that the corresponding main operators T and T of these systems have the von Neumann parameters κ = κ = 0. Let also V Θ z = V Θ z = i for all z C +. Then Θ and Θ are bi-unitary equivalent to each other. Proof. Let A k and Âk, k =, be the corresponding symmetric operators and quasi-kernels of Re A k, k =, in L-systems Θ and Θ. Then, as it was explained in Section 3 and Remark 5.5, s A k, Âk = S A k, T k, Âk 0, k =, in C +. Hence, see [8] the triplets A, T, Â and A, T, Â are

27 A system coupling and the Donoghue classes 7 unitarily equivalent to each other, that is there exists an isometric operator U from H onto H such that UT = T U, U A = A U, UÂ = ÂU. 7.8 Let g ±,k be deficiency elements g ±,k Ker A k ii, g ±,k =, k =,. Then g +, e iα g, DomÂ, g +, e iβ g, DomÂ, for some α, β [0, π. Also, g +, DomT and g +, DomT. Moreover, 7.8 implies Ug +, = g +, DomT and Ue iα g, = e iβ g,. Following [3, Theorem 6.6.0] we introduce U + = U H+ to be an isometry from H + onto H +, and U = U+ is an isometry from H onto H. Performing similar to Section 4 steps we obtain Re A = A + i, ϕ + e iα ψ ϕ e iα ψ, Re A = A + i, ϕ + e iβ ψ ϕ e iβ ψ, and Im A =, ϕ e iα ψ ϕ e iα ψ, Im A =, ϕ e iβ ψ ϕ e iβ ψ, where ϕ k and ψ k, k =, are -normalized elements in R Ker A k ii and R Ker A k + ii, k =,, respectively. It is not hard to show that U ϕ e iα ψ = ϕ e iβ ψ. Indeed, for any f H + we have f, ϕ e iβ ψ = f, g +, e iβ g, + = f, U + g +, e iα g, + = U + U + f, U + g +, e iα g, + = U + f, g +, e iα g, + = U + f, ϕ e iα ψ = f, U + ϕ e iα ψ = f, U ϕ e iα ψ. Combining this with the above formulas for Im A and Im A we obtain that Im A = U Im A U +. Taking into account that 7.8 implies A = UA U = U A U+ and using similar steps we also get Re A = U Re A U+. Thus, A = U A U+. Note that the L-system Ξ in 7.6 is minimal since the symmetric operator B is prime. Moreover, it follows from Definition 7. and Lemma 7. that a system attractor Ξ is not unique but all system attractors are bi-unitary equivalent to each other. The L-system Ξ in 7.6 will be called the position system attractor.