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

Perturbations of L-systems

Sergey Belyi

Troy University (USA)

Operator Theory and Krein Spaces Vienna, Austria 2019

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

Let E and H be Hilbert spaces and let T be an unbounded

• perator in H.

L-system (T − zI)x = KJϕ−, ϕ+ = ϕ− − 2iK ∗x, Im T = KJK ∗. Here ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x ∈ H is a vector of the state space. J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

Let E and H be Hilbert spaces and let T be an unbounded

• perator in H.

L-system (T − zI)x = KJϕ−, ϕ+ = ϕ− − 2iK ∗x, Im T = KJK ∗. Here ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x ∈ H is a vector of the state space. J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

Let E and H be Hilbert spaces and let T be an unbounded

• perator in H.

L-system (T − zI)x = KJϕ−, ϕ+ = ϕ− − 2iK ∗x, Im T = KJK ∗. Here ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x ∈ H is a vector of the state space. J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

L-system

L-system Θ =

• A

K J H+ ⊂ H ⊂ H− E

• (1)

A is a bounded linear operator from H+ to H− ((∗)-extension of T ∈ Ω( ˙ A), i.e., A ⊃ T ⊃ ˙ A, A∗ ⊃ T ∗ ⊃ ˙ A); H+ ⊂ H ⊂ H− is a rigged Hilbert space, dim E < ∞; Im A = KJK ∗; K is a linear bounded operator from E into H−; J = J∗ = J−1 ∈ [E, E].

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Uniqueness of an L-system

T is a main operator of the L-system. ˙ A is a symmetric operator, the largest common Hermitian part of T and T ∗. A is a (∗)-extensions of T, i.e., A ⊃ T, A∗ ⊃ T ∗, ˆ A is a quasi-kernel of Re A, a self-adjoint extension of ˙ A such that Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. The triple of operators ˙ A, T, and ˆ A define an L-system uniquely.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Uniqueness of an L-system

T is a main operator of the L-system. ˙ A is a symmetric operator, the largest common Hermitian part of T and T ∗. A is a (∗)-extensions of T, i.e., A ⊃ T, A∗ ⊃ T ∗, ˆ A is a quasi-kernel of Re A, a self-adjoint extension of ˙ A such that Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. The triple of operators ˙ A, T, and ˆ A define an L-system uniquely.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Uniqueness of an L-system

T is a main operator of the L-system. ˙ A is a symmetric operator, the largest common Hermitian part of T and T ∗. A is a (∗)-extensions of T, i.e., A ⊃ T, A∗ ⊃ T ∗, ˆ A is a quasi-kernel of Re A, a self-adjoint extension of ˙ A such that Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. The triple of operators ˙ A, T, and ˆ A define an L-system uniquely.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Uniqueness of an L-system

T is a main operator of the L-system. ˙ A is a symmetric operator, the largest common Hermitian part of T and T ∗. A is a (∗)-extensions of T, i.e., A ⊃ T, A∗ ⊃ T ∗, ˆ A is a quasi-kernel of Re A, a self-adjoint extension of ˙ A such that Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. The triple of operators ˙ A, T, and ˆ A define an L-system uniquely.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Uniqueness of an L-system

T is a main operator of the L-system. ˙ A is a symmetric operator, the largest common Hermitian part of T and T ∗. A is a (∗)-extensions of T, i.e., A ⊃ T, A∗ ⊃ T ∗, ˆ A is a quasi-kernel of Re A, a self-adjoint extension of ˙ A such that Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. The triple of operators ˙ A, T, and ˆ A define an L-system uniquely.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Transfer and Impedance function of an L-system

Transfer function WΘ(z) = I − 2iK ∗(A − zI)−1KJ Impedance function of Θ = LFT of WΘ(z) VΘ(z) = i[WΘ(z) + I]−1[WΘ(z) − I]J = K ∗(Re A − zI)−1K (2)

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Transfer and Impedance function of an L-system

Transfer function WΘ(z) = I − 2iK ∗(A − zI)−1KJ Impedance function of Θ = LFT of WΘ(z) VΘ(z) = i[WΘ(z) + I]−1[WΘ(z) − I]J = K ∗(Re A − zI)−1K (2)

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Transfer and Impedance function of an L-system

Transfer function WΘ(z) = I − 2iK ∗(A − zI)−1KJ Impedance function of Θ = LFT of WΘ(z) VΘ(z) = i[WΘ(z) + I]−1[WΘ(z) − I]J = K ∗(Re A − zI)−1K (2)

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 WΘ(z) and classify the impedance function VΘ(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[WΘ(z) + I]−1[WΘ(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 WΘ(z) and classify the impedance function VΘ(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[WΘ(z) + I]−1[WΘ(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 WΘ(z) and classify the impedance function VΘ(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[WΘ(z) + I]−1[WΘ(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 Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• (3)

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 Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• (3)

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). Operator T is quasi-self-adjoint that is, ˙ A ⊂ T ⊂ ˙ A∗ and g+ − κg− ∈ 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 Re A ⊃ ˆ A = ˆ A∗ ⊃ ˙ A. By von Neumann’s formula Dom(ˆ A) = Dom( ˙ A) ⊕ (1 + U) ker( ˙ A∗ − iI), 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 Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• (5)
• 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

g± ∈ ker( ˙ A∗ ∓ iI) be normalized, g± = 1, and such that g+ −g− ∈ Dom(A) and g+ −κg− ∈ 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

g± ∈ ker( ˙ A∗ ∓ iI) be normalized, g± = 1, and such that g+ −g− ∈ Dom(A) and g+ −κg− ∈ 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

g± ∈ ker( ˙ A∗ ∓ iI) be normalized, g± = 1, and such that g+ +g− ∈ Dom(A) and g+ −κg− ∈ 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

g± ∈ ker( ˙ A∗ ∓ iI) be normalized, g± = 1, and such that g+ +g− ∈ Dom(A) and g+ −κg− ∈ 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 M(z) =

• R
• 1

λ − z − λ 1 + λ2

• dµ,

(6) where µ is an infinite Borel measure and

• R

dµ(λ) 1 + λ2 = 1 , equivalently, M(i) = i. (7)

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 Mκ and M−1

κ

An analytic function M from C+ into itself belongs to the generalized Donoghue class Mκ, (0 ≤ κ < 1) if it admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 − κ 1 + κ < 1 ⇔ M(i) = i 1 − κ 1 + κ (8) and to the generalized Donoghue class M−1

κ , (0 ≤ κ < 1) if it

admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 + κ 1 − κ > 1 ⇔ M(i) = i 1 + κ 1 − κ. (9) Clearly, M0 = M−1 = M.

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 Mκ and M−1

κ

An analytic function M from C+ into itself belongs to the generalized Donoghue class Mκ, (0 ≤ κ < 1) if it admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 − κ 1 + κ < 1 ⇔ M(i) = i 1 − κ 1 + κ (8) and to the generalized Donoghue class M−1

κ , (0 ≤ κ < 1) if it

admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 + κ 1 − κ > 1 ⇔ M(i) = i 1 + κ 1 − κ. (9) Clearly, M0 = M−1 = M.

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 Mκ and M−1

κ

An analytic function M from C+ into itself belongs to the generalized Donoghue class Mκ, (0 ≤ κ < 1) if it admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 − κ 1 + κ < 1 ⇔ M(i) = i 1 − κ 1 + κ (8) and to the generalized Donoghue class M−1

κ , (0 ≤ κ < 1) if it

admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 + κ 1 − κ > 1 ⇔ M(i) = i 1 + κ 1 − κ. (9) Clearly, M0 = M−1 = M.

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 Mκ and M−1

κ

An analytic function M from C+ into itself belongs to the generalized Donoghue class Mκ, (0 ≤ κ < 1) if it admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 − κ 1 + κ < 1 ⇔ M(i) = i 1 − κ 1 + κ (8) and to the generalized Donoghue class M−1

κ , (0 ≤ κ < 1) if it

admits the representation (6) and

• R

dµ(λ) 1 + λ2 = a = 1 + κ 1 − κ > 1 ⇔ M(i) = i 1 + κ 1 − κ. (9) Clearly, M0 = M−1 = M.

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 MQ, MQ

κ and M−1,Q κ

A scalar Herglotz-Nevanlinna function V(z) belongs to the class MQ if it admits the following integral representation V(z) = Q +

• R
• 1

λ − z − λ 1 + λ2

• dµ,

Q = ¯ Q, (10) and has condition (7) on the measure µ. Similarly, we introduce perturbed classes MQ

κ and M−1,Q κ

if normalization conditions (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 MQ, MQ

κ and M−1,Q κ

A scalar Herglotz-Nevanlinna function V(z) belongs to the class MQ if it admits the following integral representation V(z) = Q +

• R
• 1

λ − z − λ 1 + λ2

• dµ,

Q = ¯ Q, (10) and has condition (7) on the measure µ. Similarly, we introduce perturbed classes MQ

κ and M−1,Q κ

if normalization conditions (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 VΘ(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 VΘ(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 Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class Mκ if and only if the triple ( ˙ A, T, ˆ A) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class M−1

κ

if and only if the triple ( ˙ 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 Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class Mκ if and only if the triple ( ˙ A, T, ˆ A) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class M−1

κ

if and only if the triple ( ˙ 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 Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class Mκ if and only if the triple ( ˙ A, T, ˆ A) satisfies Hypothesis 1. Theorem (B., Makarov, Tsekanovski˘ i, ’16) Let Θκ, 0 ≤ κ < 1, of the form (5) be an L-system with the main

• perator T. Then its impedance function VΘκ(z) belongs to the

generalized Donoghue class M−1

κ

if and only if the triple ( ˙ 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ M Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T

has κ = 0 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ Mκ Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M−1

κ

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• 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

Function class V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Perturbed function Q + V(z) Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Perturbed function Q + V(z) Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Perturbed function Q + V(z) Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Perturbed function Q + V(z) Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 V(z) ∈ M Function class V(z) ∈ Mκ0 Function class V(z) ∈ M−1

κ0

Perturbed function Q + V(z) Θ =

• A

K 1 H+ ⊂ H ⊂ H− C

• T has von Neumann parameter κ =?.

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 class MQ

Theorem (B., Tsekanovski˘ i, ’19) Let V(z) belong to the class MQ. 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 κ = |Q| √ Q2 + 4 , Q = 0. (11) Moreover, the unimodular parameter U of the quasi-kernel ˆ A of Θ is also uniquely defined by Q. U = Q |Q| · −Q + 2i √ Q2 + 4 , Q = 0. (12)

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 MQ

Theorem (B., Tsekanovski˘ i, ’19) Let V(z) belong to the class MQ. 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 κ = |Q| √ Q2 + 4 , Q = 0. (11) Moreover, the unimodular parameter U of the quasi-kernel ˆ A of Θ is also uniquely defined by Q. U = Q |Q| · −Q + 2i √ Q2 + 4 , Q = 0. (12)

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 MQ

κ

Theorem (B., Tsekanovski˘ i, ’19) Let V(z) belong to the class MQ

κ0 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.

κ =

• b − 2Q2 −

√ b2 + 4Q2 2 − a

• b −

√ b2 + 4Q2 2 + 4Q2a(a − 1)

• b − 2Q2 −

√ b2 + 4Q2 2 + a

• b −

√ b2 + 4Q2 2 + 4Q2a(a + 1) (13)

where Q = 0 and b = Q2 + a2 − 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 MQ

κ

Theorem (B., Tsekanovski˘ i, ’19) Let V(z) belong to the class MQ

κ0 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.

κ =

• b − 2Q2 −

√ b2 + 4Q2 2 − a

• b −

√ b2 + 4Q2 2 + 4Q2a(a − 1)

• b − 2Q2 −

√ b2 + 4Q2 2 + a

• b −

√ b2 + 4Q2 2 + 4Q2a(a + 1) (13)

where Q = 0 and b = Q2 + a2 − 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 MQ

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 − κ2) − 1 − κ2 2κ , Q = 0. (14)

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 MQ

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 − κ2) − 1 − κ2 2κ , Q = 0. (14)

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

κ0

and have a normalization 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.

κ = a

• b +

√ b2 + 4Q2 2 −

• b − 2Q2 +

√ b2 + 4Q2 2 − 4Q2a(a − 1)

• b − 2Q2 +

√ b2 + 4Q2 2 + a

• b +

√ b2 + 4Q2 2 + 4Q2a(a + 1) (15)

where Q = 0 and b = Q2 + a2 − 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

κ0

and have a normalization 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.

κ = a

• b +

√ b2 + 4Q2 2 −

• b − 2Q2 +

√ b2 + 4Q2 2 − 4Q2a(a − 1)

• b − 2Q2 +

√ b2 + 4Q2 2 + a

• b +

√ b2 + 4Q2 2 + 4Q2a(a + 1) (15)

where Q = 0 and b = Q2 + a2 − 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 − κ2) − 1 − κ2 2κ , Q = 0. (16)

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 − κ2) − 1 − κ2 2κ , Q = 0. (16)

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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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

• perator T and its von Neumann’s parameter κ, (0 ≤ κ < 1).

Then only one of the following takes place:

1

VΘ(z) belongs to class MQ and κ is determined by (11) for some Q;

2

VΘ(z) belongs to class MQ

κ0 and κ is determined by (13)

for some Q and a = 1−κ0

1+κ0 ;

3

VΘ(z) belongs to class M−1,Q

κ0

and κ is determined by (15) for some Q and a = 1+κ0

1−κ0 .

The values of Q and κ0 are determined from integral representation (10) of VΘ(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 VΘ(z) belongs to one of the Donoghue classes M, Mκ0, or M−1

κ0 . Let also Q = 0 be any real number.

Perturbation of an L-system An L-system ΘQ whose construction is based on the elements

• f a given L-system Θ (subject to either of Hypotheses 1 or 2) is

called the perturbation of an L-system Θ if VΘQ(z) = Q + VΘ(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 VΘ(z) belongs to one of the Donoghue classes M, Mκ0, or M−1

κ0 . Let also Q = 0 be any real number.

Perturbation of an L-system An L-system ΘQ whose construction is based on the elements

• f a given L-system Θ (subject to either of Hypotheses 1 or 2) is

called the perturbation of an L-system Θ if VΘQ(z) = Q + VΘ(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 Θ =

• A

K 1 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 Θ =

• A

K 1 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 Θ =

• 1

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 Θ =

• 1

H+ ⊂ H ⊂ H− C

• Construct state-space operator AQ 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 ΘQ =

• AQ

K Q 1 H+ ⊂ H ⊂ H− C

• Obtain perturbed L-system ΘQ such that VΘQ(z) = Q + VΘ0(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 Θ0 be an L-system satisfying the conditions of Hypothesis 1 and such that VΘ0(z) ∈ M. Then for any real number Q = 0 there exists another L-system ΘQ with the same symmetric

• perator ˙

A as in Θ0 and such that VΘQ(z) = Q + VΘ(z). Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (11) while the quasi-kernel ˆ AQ 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 Θ0 be an L-system satisfying the conditions of Hypothesis 1 and such that VΘ0(z) ∈ M. Then for any real number Q = 0 there exists another L-system ΘQ with the same symmetric

• perator ˙

A as in Θ0 and such that VΘQ(z) = Q + VΘ(z). Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (11) while the quasi-kernel ˆ AQ 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 Mκ systems

Theorem (B., Tsekanovski˘ i, ’19) Let Θκ0 be an L-system and such that VΘ0(z) ∈ Mκ0. Then for any real number Q = 0 there exists another L-system ΘQ

κ with

the same symmetric operator ˙ A as in Θκ0 and such that VΘQ

κ(z) = Q + VΘκ0(z).

Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (13) while the quasi-kernel ˆ AQ 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 Mκ systems

Theorem (B., Tsekanovski˘ i, ’19) Let Θκ0 be an L-system and such that VΘ0(z) ∈ Mκ0. Then for any real number Q = 0 there exists another L-system ΘQ

κ with

the same symmetric operator ˙ A as in Θκ0 and such that VΘQ

κ(z) = Q + VΘκ0(z).

Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (13) while the quasi-kernel ˆ AQ 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

κ

systems

Theorem (B., Tsekanovski˘ i, ’19) Let Θκ0 be an L-system such that VΘ0(z) ∈ M−1

κ0 . Then for any

real number Q = 0 there exists another L-system ΘQ

κ with the

same symmetric operator ˙ A as in Θκ0 and such that VΘQ

κ(z) = Q + VΘκ0(z).

Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (15) while the quasi-kernel ˆ AQ 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

κ

systems

Theorem (B., Tsekanovski˘ i, ’19) Let Θκ0 be an L-system such that VΘ0(z) ∈ M−1

κ0 . Then for any

real number Q = 0 there exists another L-system ΘQ

κ with the

same symmetric operator ˙ A as in Θκ0 and such that VΘQ

κ(z) = Q + VΘκ0(z).

Moreover, the von Neumann parameter κ of its main operator T Q is determined by the formula (15) while the quasi-kernel ˆ AQ 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.

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Thank you!

L-systems L-systems with 1-D input-output Donoghue classes and L-systems Realization of perturbed classes Perturbation of L-systems

Thank you!