Perturbations of L-systems Sergey Belyi Troy University (USA) - PowerPoint PPT Presentation

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct and Inverse Realization Problems Direct Problem Given an L-system Θ we need to derive transfer function W Θ ( z ) and classify the impedance function V Θ ( z ) Inverse Problem Given a function V ( z ) of a certain class we need to construct an L-system Θ such that V ( z ) = i [ W Θ ( z ) + I ] − 1 [ W Θ ( z ) − I ] J

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct and Inverse Realization Problems Direct Problem Given an L-system Θ we need to derive transfer function W Θ ( z ) and classify the impedance function V Θ ( z ) Inverse Problem Given a function V ( z ) of a certain class we need to construct an L-system Θ such that V ( z ) = i [ W Θ ( z ) + I ] − 1 [ W Θ ( z ) − I ] J

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems An L-system w/1-D input-output One-dimensional L-system � � A K 1 Θ = (3) H + ⊂ H ⊂ H − C

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems An L-system w/1-D input-output One-dimensional L-system � � A K 1 Θ = (3) H + ⊂ H ⊂ H − C

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Main operator T In L-system (3) T � = T ∗ is a maximal dissipative extension of a symmetric operator ˙ A with deficiency indices ( 1 , 1 ) , Im ( Tf , f ) ≥ 0 , f ∈ Dom ( T ) . A ∗ and Operator T is quasi-self-adjoint that is, ˙ A ⊂ T ⊂ ˙ g + − κ g − ∈ Dom ( T ) for some | κ | < 1 . (4) Operator T is the main operator of L-system (3).

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Quasi-kernel ˆ A of Re A Let ˆ A be a self-adjoint extension of ˙ A such that A ∗ ⊃ ˙ Re A ⊃ ˆ A = ˆ A . By von Neumann’s formula A ∗ − iI ) , Dom (ˆ A ) = Dom ( ˙ A ) ⊕ ( 1 + U ) ker ( ˙ where U is a unimodular parameter, | U | = 1. Operator ˆ A is the quasi-kernel of the real part Re A of the state-space operator A .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems A unique L-system w/1-D input-output A triple ( ˙ A , T , ˆ A ) of a symmetric operator, main operator, and a quasi-kernel in a Hilbert space H defines an L-system � � K 1 A Θ = (5) H + ⊂ H ⊂ H − C uniquely. This L-system Θ has a one-dimensional input-output space E = C .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Hypothesis 1, U = − 1 Hypothesis (1) Suppose that T � = T ∗ is a maximal dissipative extension of a symmetric operator ˙ A with deficiency indices ( 1 , 1 ) and ˆ A is a self-adjoint (reference) extension of ˙ A. Let deficiency elements A ∗ ∓ iI ) be normalized, � g ± � = 1 , and such that g ± ∈ ker ( ˙ g + − g − ∈ Dom ( A ) and g + − κ g − ∈ Dom ( T ) for some | κ | < 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Hypothesis 1, U = − 1 Hypothesis (1) Suppose that T � = T ∗ is a maximal dissipative extension of a symmetric operator ˙ A with deficiency indices ( 1 , 1 ) and ˆ A is a self-adjoint (reference) extension of ˙ A. Let deficiency elements A ∗ ∓ iI ) be normalized, � g ± � = 1 , and such that g ± ∈ ker ( ˙ g + − g − ∈ Dom ( A ) and g + − κ g − ∈ Dom ( T ) for some | κ | < 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Hypothesis 2 (“Anti-hypothesis”), U = 1 Hypothesis (2) Suppose that T � = T ∗ is a maximal dissipative extension of a symmetric operator ˙ A with deficiency indices ( 1 , 1 ) and ˆ A is a self-adjoint (reference) extension of ˙ A. Let deficiency elements A ∗ ∓ iI ) be normalized, � g ± � = 1 , and such that g ± ∈ ker ( ˙ g + + g − ∈ Dom ( A ) and g + − κ g − ∈ Dom ( T ) for some | κ | < 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Hypothesis 2 (“Anti-hypothesis”), U = 1 Hypothesis (2) Suppose that T � = T ∗ is a maximal dissipative extension of a symmetric operator ˙ A with deficiency indices ( 1 , 1 ) and ˆ A is a self-adjoint (reference) extension of ˙ A. Let deficiency elements A ∗ ∓ iI ) be normalized, � g ± � = 1 , and such that g ± ∈ ker ( ˙ g + + g − ∈ Dom ( A ) and g + − κ g − ∈ Dom ( T ) for some | κ | < 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Donoghue class M Denote by M the Donoghue class of all analytic mappings M from C + into itself that admits the representation � � � 1 λ M ( z ) = λ − z − d µ, (6) 1 + λ 2 R where µ is an infinite Borel measure and � d µ ( λ ) 1 + λ 2 = 1 , equivalently, M ( i ) = i . (7) R

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes M κ and M − 1 κ An analytic function M from C + into itself belongs to the generalized Donoghue class M κ , (0 ≤ κ < 1) if it admits the representation (6) and � 1 + λ 2 = a = 1 − κ d µ ( λ ) M ( i ) = i 1 − κ 1 + κ < 1 ⇔ (8) 1 + κ R and to the generalized Donoghue class M − 1 κ , (0 ≤ κ < 1) if it admits the representation (6) and 1 + λ 2 = a = 1 + κ d µ ( λ ) M ( i ) = i 1 + κ � 1 − κ > 1 ⇔ 1 − κ. (9) R Clearly, M 0 = M − 1 = M . 0

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes M κ and M − 1 κ An analytic function M from C + into itself belongs to the generalized Donoghue class M κ , (0 ≤ κ < 1) if it admits the representation (6) and � 1 + λ 2 = a = 1 − κ d µ ( λ ) M ( i ) = i 1 − κ 1 + κ < 1 ⇔ (8) 1 + κ R and to the generalized Donoghue class M − 1 κ , (0 ≤ κ < 1) if it admits the representation (6) and 1 + λ 2 = a = 1 + κ d µ ( λ ) M ( i ) = i 1 + κ � 1 − κ > 1 ⇔ 1 − κ. (9) R Clearly, M 0 = M − 1 = M . 0

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes M κ and M − 1 κ An analytic function M from C + into itself belongs to the generalized Donoghue class M κ , (0 ≤ κ < 1) if it admits the representation (6) and � 1 + λ 2 = a = 1 − κ d µ ( λ ) M ( i ) = i 1 − κ 1 + κ < 1 ⇔ (8) 1 + κ R and to the generalized Donoghue class M − 1 κ , (0 ≤ κ < 1) if it admits the representation (6) and 1 + λ 2 = a = 1 + κ d µ ( λ ) M ( i ) = i 1 + κ � 1 − κ > 1 ⇔ 1 − κ. (9) R Clearly, M 0 = M − 1 = M . 0

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes M κ and M − 1 κ An analytic function M from C + into itself belongs to the generalized Donoghue class M κ , (0 ≤ κ < 1) if it admits the representation (6) and � 1 + λ 2 = a = 1 − κ d µ ( λ ) M ( i ) = i 1 − κ 1 + κ < 1 ⇔ (8) 1 + κ R and to the generalized Donoghue class M − 1 κ , (0 ≤ κ < 1) if it admits the representation (6) and 1 + λ 2 = a = 1 + κ d µ ( λ ) M ( i ) = i 1 + κ � 1 − κ > 1 ⇔ 1 − κ. (9) R Clearly, M 0 = M − 1 = M . 0

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbed Donoghue classes M Q , M Q κ and M − 1 , Q κ A scalar Herglotz-Nevanlinna function V ( z ) belongs to the class M Q if it admits the following integral representation � � 1 � λ Q = ¯ V ( z ) = Q + λ − z − d µ, Q , (10) 1 + λ 2 R and has condition (7) on the measure µ . Similarly, we introduce κ and M − 1 , Q perturbed classes M Q if normalization conditions κ (8) and (9), respectively, hold on measure µ in (10).

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbed Donoghue classes M Q , M Q κ and M − 1 , Q κ A scalar Herglotz-Nevanlinna function V ( z ) belongs to the class M Q if it admits the following integral representation � � 1 � λ Q = ¯ V ( z ) = Q + λ − z − d µ, Q , (10) 1 + λ 2 R and has condition (7) on the measure µ . Similarly, we introduce κ and M − 1 , Q perturbed classes M Q if normalization conditions κ (8) and (9), respectively, hold on measure µ in (10).

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Donoghue class impedance functions Theorem (B., Makarov, Tsekanovski˘ i, ’15) Let Θ of the form (5) be an L-system whose main operator T has the von Neumann parameter κ , ( 0 ≤ κ < 1 ) . Then its impedance function V Θ ( z ) belongs to the Donoghue class M if and only if κ = 0 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Donoghue class impedance functions Theorem (B., Makarov, Tsekanovski˘ i, ’15) Let Θ of the form (5) be an L-system whose main operator T has the von Neumann parameter κ , ( 0 ≤ κ < 1 ) . Then its impedance function V Θ ( z ) belongs to the Donoghue class M if and only if κ = 0 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes impedance functions Theorem (B., Makarov, Tsekanovski˘ i, ’15) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M κ if and only if the triple ( ˙ A , T , ˆ A ) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M − 1 if and only if the triple κ ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes impedance functions Theorem (B., Makarov, Tsekanovski˘ i, ’15) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M κ if and only if the triple ( ˙ A , T , ˆ A ) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M − 1 if and only if the triple κ ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Generalized Donoghue classes impedance functions Theorem (B., Makarov, Tsekanovski˘ i, ’15) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M κ if and only if the triple ( ˙ A , T , ˆ A ) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θ κ , 0 ≤ κ < 1 , of the form (5) be an L-system with the main operator T. Then its impedance function V Θ κ ( z ) belongs to the generalized Donoghue class M − 1 if and only if the triple κ ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � K 1 A Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M T κ = 0 has U is an arbitrary unimodular number.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ T has von Neumann parameter κ ˆ A is parameterized with U = − 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 1.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M − 1 κ T has von Neumann parameter κ ˆ A is parameterized with U = 1 ( ˙ A , T , ˆ A ) satisfies Hypothesis 2.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M Function class V ( z ) ∈ M κ 0 Function class T has von Neumann parameter κ =? . V ( z ) ∈ M − 1 κ 0 ˆ A is parameterized with U =? .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M Function class V ( z ) ∈ M κ 0 Function class T has von Neumann parameter κ =? . V ( z ) ∈ M − 1 κ 0 ˆ A is parameterized with U =? .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M Function class V ( z ) ∈ M κ 0 Function class T has von Neumann parameter κ =? . V ( z ) ∈ M − 1 κ 0 ˆ A is parameterized with U =? .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes � � A K 1 Θ = H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M Function class V ( z ) ∈ M κ 0 Function class T has von Neumann parameter κ =? . V ( z ) ∈ M − 1 κ 0 ˆ A is parameterized with U =? .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes Function class � � K 1 A Θ = V ( z ) ∈ M H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ 0 Function class V ( z ) ∈ M − 1 κ 0 T has von Neumann parameter κ =? . Perturbed function ˆ A is parameterized with U =? . Q + V ( z )

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes Function class � � K 1 A Θ = V ( z ) ∈ M H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ 0 Function class V ( z ) ∈ M − 1 κ 0 T has von Neumann parameter κ =? . Perturbed function ˆ A is parameterized with U =? . Q + V ( z )

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes Function class � � K 1 A Θ = V ( z ) ∈ M H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ 0 Function class V ( z ) ∈ M − 1 κ 0 T has von Neumann parameter κ =? . Perturbed function ˆ A is parameterized with U =? . Q + V ( z )

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes Function class � � K 1 A Θ = V ( z ) ∈ M H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ 0 Function class V ( z ) ∈ M − 1 κ 0 T has von Neumann parameter κ =? . Perturbed function ˆ A is parameterized with U =? . Q + V ( z )

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of perturbed Donoghue classes Function class � � K 1 A Θ = V ( z ) ∈ M H + ⊂ H ⊂ H − C Function class V ( z ) ∈ M κ 0 Function class V ( z ) ∈ M − 1 κ 0 T has von Neumann parameter κ =? . Perturbed function ˆ A is parameterized with U =? . Q + V ( z )

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M Q . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is determined as a function of Q. by the formula | Q | √ κ = , Q � = 0 . (11) Q 2 + 4 Moreover, the unimodular parameter U of the quasi-kernel ˆ A of Θ is also uniquely defined by Q. U = Q | Q | · − Q + 2 i √ Q � = 0 . (12) , Q 2 + 4

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M Q . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is determined as a function of Q. by the formula | Q | √ κ = , Q � = 0 . (11) Q 2 + 4 Moreover, the unimodular parameter U of the quasi-kernel ˆ A of Θ is also uniquely defined by Q. U = Q | Q | · − Q + 2 i √ Q � = 0 . (12) , Q 2 + 4

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q κ Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M Q κ 0 and have a normalization parameter 0 < a < 1 . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is uniquely determined as a function of Q and a. √ √ � 2 � 2 � b − 2 Q 2 − � b 2 + 4 Q 2 b 2 + 4 Q 2 + 4 Q 2 a ( a − 1 ) − a b − κ = √ √ � 2 � 2 � � b − 2 Q 2 − b 2 + 4 Q 2 b 2 + 4 Q 2 + a b − + 4 Q 2 a ( a + 1 ) (13) where Q � = 0 and b = Q 2 + a 2 − 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q κ Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M Q κ 0 and have a normalization parameter 0 < a < 1 . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is uniquely determined as a function of Q and a. √ √ � 2 � 2 � b − 2 Q 2 − � b 2 + 4 Q 2 b 2 + 4 Q 2 + 4 Q 2 a ( a − 1 ) − a b − κ = √ √ � 2 � 2 � � b − 2 Q 2 − b 2 + 4 Q 2 b 2 + 4 Q 2 + a b − + 4 Q 2 a ( a + 1 ) (13) where Q � = 0 and b = Q 2 + a 2 − 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q κ Theorem (B., Tsekanovski˘ i, ’19) Moreover, the quasi-kernel ˆ A of Re A of the realizing L-system Θ is uniquely defined with U = ( a + Qi )( 1 − κ 2 ) − 1 − κ 2 Q � = 0 . (14) , 2 κ

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M Q κ Theorem (B., Tsekanovski˘ i, ’19) Moreover, the quasi-kernel ˆ A of Re A of the realizing L-system Θ is uniquely defined with U = ( a + Qi )( 1 − κ 2 ) − 1 − κ 2 Q � = 0 . (14) , 2 κ

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M − 1 , Q κ Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M − 1 , Q and have a normalization κ 0 parameter a > 1 . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is uniquely determined as a function of Q and a. √ √ � 2 � 2 � � b 2 + 4 Q 2 b − 2 Q 2 + b 2 + 4 Q 2 − 4 Q 2 a ( a − 1 ) a b + − κ = √ √ � 2 � 2 � � b − 2 Q 2 + b 2 + 4 Q 2 b 2 + 4 Q 2 + a b + + 4 Q 2 a ( a + 1 ) (15) where Q � = 0 and b = Q 2 + a 2 − 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M − 1 , Q κ Theorem (B., Tsekanovski˘ i, ’19) Let V ( z ) belong to the class M − 1 , Q and have a normalization κ 0 parameter a > 1 . Then V ( z ) can be realized by a minimal L-system Θ with the main operator T whose von Neumann’s parameter κ is uniquely determined as a function of Q and a. √ √ � 2 � 2 � � b 2 + 4 Q 2 b − 2 Q 2 + b 2 + 4 Q 2 − 4 Q 2 a ( a − 1 ) a b + − κ = √ √ � 2 � 2 � � b − 2 Q 2 + b 2 + 4 Q 2 b 2 + 4 Q 2 + a b + + 4 Q 2 a ( a + 1 ) (15) where Q � = 0 and b = Q 2 + a 2 − 1 .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M − 1 , Q κ Theorem (B., Tsekanovski˘ i, ’19) Moreover, the quasi-kernel ˆ A of Re A of the realizing L-system Θ is uniquely defined with U = ( a + Qi )( 1 − κ 2 ) − 1 − κ 2 Q � = 0 . (16) , 2 κ

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Realization of class M − 1 , Q κ Theorem (B., Tsekanovski˘ i, ’19) Moreover, the quasi-kernel ˆ A of Re A of the realizing L-system Θ is uniquely defined with U = ( a + Qi )( 1 − κ 2 ) − 1 − κ 2 Q � = 0 . (16) , 2 κ

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Direct theorem for L-systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ be a minimal L-system of the form (5) with the main operator T and its von Neumann’s parameter κ , ( 0 ≤ κ < 1 ) . Then only one of the following takes place: V Θ ( z ) belongs to class M Q and κ is determined by (11) for 1 some Q; V Θ ( z ) belongs to class M Q κ 0 and κ is determined by (13) 2 for some Q and a = 1 − κ 0 1 + κ 0 ; V Θ ( z ) belongs to class M − 1 , Q and κ is determined by (15) 3 κ 0 for some Q and a = 1 + κ 0 1 − κ 0 . The values of Q and κ 0 are determined from integral representation (10) of V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Suppose we are given an L-system Θ whose impedance function V Θ ( z ) belongs to one of the Donoghue classes M , M κ 0 , or M − 1 κ 0 . Let also Q � = 0 be any real number. Perturbation of an L-system An L-system Θ Q whose construction is based on the elements of a given L-system Θ (subject to either of Hypotheses 1 or 2) is called the perturbation of an L-system Θ if V Θ Q ( z ) = Q + V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Suppose we are given an L-system Θ whose impedance function V Θ ( z ) belongs to one of the Donoghue classes M , M κ 0 , or M − 1 κ 0 . Let also Q � = 0 be any real number. Perturbation of an L-system An L-system Θ Q whose construction is based on the elements of a given L-system Θ (subject to either of Hypotheses 1 or 2) is called the perturbation of an L-system Θ if V Θ Q ( z ) = Q + V Θ ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Unperturbed L-system � � A K 1 Θ = H + ⊂ H ⊂ H − C Given unperturbed L-system.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Unperturbed L-system � � A K 1 Θ = H + ⊂ H ⊂ H − C Given unperturbed L-system.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Unperturbed L-system � � 1 Θ = H + ⊂ H ⊂ H − C Keep the symmetric operator ˙ A and state space H + ⊂ H ⊂ H − .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Unperturbed L-system � � 1 Θ = H + ⊂ H ⊂ H − C Construct state-space operator A Q and channel operator K Q .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbing an L-system Perturbed L-system � A Q K Q � 1 Θ Q = H + ⊂ H ⊂ H − C Obtain perturbed L-system Θ Q such that V Θ Q ( z ) = Q + V Θ 0 ( z ) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ 0 be an L-system satisfying the conditions of Hypothesis 1 and such that V Θ 0 ( z ) ∈ M . Then for any real number Q � = 0 there exists another L-system Θ Q with the same symmetric operator ˙ A as in Θ 0 and such that V Θ Q ( z ) = Q + V Θ ( z ) . Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (11) while the quasi-kernel ˆ A Q is defined by U from (12) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ 0 be an L-system satisfying the conditions of Hypothesis 1 and such that V Θ 0 ( z ) ∈ M . Then for any real number Q � = 0 there exists another L-system Θ Q with the same symmetric operator ˙ A as in Θ 0 and such that V Θ Q ( z ) = Q + V Θ ( z ) . Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (11) while the quasi-kernel ˆ A Q is defined by U from (12) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M κ systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ κ 0 be an L-system and such that V Θ 0 ( z ) ∈ M κ 0 . Then for any real number Q � = 0 there exists another L-system Θ Q κ with the same symmetric operator ˙ A as in Θ κ 0 and such that κ ( z ) = Q + V Θ κ 0 ( z ) . V Θ Q Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (13) while the quasi-kernel ˆ A Q is defined by U from (14) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M κ systems Theorem (B., Tsekanovski˘ i, ’19) Let Θ κ 0 be an L-system and such that V Θ 0 ( z ) ∈ M κ 0 . Then for any real number Q � = 0 there exists another L-system Θ Q κ with the same symmetric operator ˙ A as in Θ κ 0 and such that κ ( z ) = Q + V Θ κ 0 ( z ) . V Θ Q Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (13) while the quasi-kernel ˆ A Q is defined by U from (14) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M − 1 systems κ Theorem (B., Tsekanovski˘ i, ’19) Let Θ κ 0 be an L-system such that V Θ 0 ( z ) ∈ M − 1 κ 0 . Then for any real number Q � = 0 there exists another L-system Θ Q κ with the same symmetric operator ˙ A as in Θ κ 0 and such that V Θ Q κ ( z ) = Q + V Θ κ 0 ( z ) . Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (15) while the quasi-kernel ˆ A Q is defined by U from (16) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems Perturbation of class M − 1 systems κ Theorem (B., Tsekanovski˘ i, ’19) Let Θ κ 0 be an L-system such that V Θ 0 ( z ) ∈ M − 1 κ 0 . Then for any real number Q � = 0 there exists another L-system Θ Q κ with the same symmetric operator ˙ A as in Θ κ 0 and such that V Θ Q κ ( z ) = Q + V Θ κ 0 ( z ) . Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (15) while the quasi-kernel ˆ A Q is defined by U from (16) .

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems References I Yu. Arlinskii, S. Belyi, E. Tsekanovskii, Conservative Realizations of Herglotz-Nevanlinna functions, Oper. Theory Adv. Appl., Vol. 217, Birkhauser Verlag, (2011). S. Belyi, E. Tsekanovski˘ ı, Perturbations of Donoghue classes and inverse problems for L-systems , Complex Analysis and Operator Theory, vol. 13 (3), (2019), pp. 1227-1311.