Q-functions of Quasi-selfadjoint Contractions

Arlinskii̇̆, Yury; Hassi, Seppo; Snoo, Henk de

doi:10.1007/3-7643-7516-7_2

Cited by 10 publications

(26 citation statements)

References 23 publications

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“…The next theorem is a consequence of Theorem 2.5; see [4]. H is a selfadjoint contraction and K ∈ L(D A , N), X ∈ L(D K * ) are contractions.…”

Section: Quasi-selfadjoint Contractions and Associated Passive Systemsmentioning

confidence: 91%

“…The equivalence (i) ⇔ (ii) was proved in [4]. For the proof of (i) ⇔ (iii), observe that the simplicity of A 0 is equivalent to …”

mentioning

confidence: 87%

“…is a Herglotz-Nevanlinna function of the class N N [−1, 1] with W (∞) = 0, see [4]. It follows from the condition (S2) that the strong limit values W (±1) exist.…”

Section: The Realization Of the Class S Qs (N)mentioning

confidence: 99%

“…[4]. This last condition alone is equivalent to A = A * and C = B * ; for contractivity of T see , then the realization of Θ(λ) as the transfer function of a satisfactory (passive) quasi-selfadjoint system depends essentially on the value Θ(0) = D: a quasi-selfadjoint system (C = B * , A = A * , A ≤ 1) which is in addition passive (i.e., a pqs-system) exists if and only if the value Θ(0) belongs to an operator ball, which is determined by the (bounded) strong limit values of Θ(λ) at the points ±1; see Proposition 5.6.…”

Section: Introductionmentioning

confidence: 99%

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Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Arlinskii̇̆

Hassi

Snoo

2008

Complex Anal. Oper. Theory

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Passive systems τ = {T, M, N, H} with M and N as an input and output space and H as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system τ with M = N is said to be quasi-selfadjoint if ran (T − T * ) ⊂ N. The subclass S qs (N) of the Schur class S(N) is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass S qs (N) is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass S qs (N) and the Q-function of T is given. Mathematics Subject Classification (2000). Primary 47A45, 47A48, 47A56; Secondary 93B15, 93B28.

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“…The next theorem is a consequence of Theorem 2.5; see [4]. H is a selfadjoint contraction and K ∈ L(D A , N), X ∈ L(D K * ) are contractions.…”

Section: Quasi-selfadjoint Contractions and Associated Passive Systemsmentioning

confidence: 91%

“…The equivalence (i) ⇔ (ii) was proved in [4]. For the proof of (i) ⇔ (iii), observe that the simplicity of A 0 is equivalent to …”

mentioning

confidence: 87%

“…is a Herglotz-Nevanlinna function of the class N N [−1, 1] with W (∞) = 0, see [4]. It follows from the condition (S2) that the strong limit values W (±1) exist.…”

Section: The Realization Of the Class S Qs (N)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Arlinskii̇̆

Hassi

Snoo

2008

Complex Anal. Oper. Theory

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“…The present new proof is based on an approximation procedure and is along the lines of the proof in [8] for the parametrization of all quasi-selfadjoint extensions of a symmetric contraction. The operator matrix…”

Section: Contractive Block Operatorsmentioning

confidence: 99%

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Arlinskii̇̆¹,

Hassi

Snoo

2007

Complex anal.oper.theory

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Passive linear systems τ = {A, B, C, D; H, M, N} have their transfer function Θτ (λ) = D + λC(I − λA) −1 B in the Schur class S(M, N). Using a parametrization of contractive block operators the transfer function Θτ (λ) is connected to the Sz.-Nagy-Foiaş characteristic function ΦA(λ) of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions Θτ (λ) and ΦA(λ). The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Mathematics Subject Classification (2000). Primary 47A45, 47A48, 47A56, 47N70; Secondary 93B15.

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