An abstract interpolation problem and the extension theory of isometric operators

Katsnelson, Victor; Kheifets, Alexander; Yuditskii, Peter

doi:10.1007/978-3-0348-8944-5_13

Cited by 42 publications

(57 citation statements)

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“…To obtain a more explicit parametrization of the solution set to the AIP ℋ( ) -problem, we need some facts concerning the Abstract Interpolation Problem for functions in the Schur Class ( , ) (denoted as the AIP ( , ) -problem) from [26] (see also [30,28]) which we now recall.…”

Section: Redheffer Transform Related To the Aip-problem On ( )mentioning

confidence: 99%

“…Following ideas from the Abstract Interpolation Problem of [26] for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de BrangesRovnyak space ℋ( ) associated with an operator-valued Schur class function . A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm.…”

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confidence: 99%

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Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Ball

Bolotnikov

Horst

2010

Integr. Equ. Oper. Theory

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Abstract. Following ideas from the Abstract Interpolation Problem of [26] for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de BrangesRovnyak space ℋ( ) associated with an operator-valued Schur class function . A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in [4] (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in ℋ( ). In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.Mathematics Subject Classification (2010). 46E22, 47A57, 30E05.

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Section: Redheffer Transform Related To the Aip-problem On ( )mentioning

confidence: 99%

mentioning

confidence: 99%

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Ball

Bolotnikov

Horst

2010

Integr. Equ. Oper. Theory

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“…Recall (see [25]), that a contractive [L]-valued function ω(ζ) is said to be a solution of the problem…”

Section: Abstract Interpolation Problemmentioning

confidence: 99%

“…In [25], [28] the inclusion of this (and more general bitangential) problem into the scheme of AIP was demonstrated. For the case of Nevanlinna class let us set B 1 = I n ,…”

Section: Tangential Interpolation Problem Letmentioning

confidence: 99%

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Abstract Interpolation Problem in Nevanlinna Classes

Derkach

2009

Modern Analysis and Applications

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Abstract. The abstract interpolation problem (AIP) in the Schur class was posed V. Katznelson, A. Kheifets and P. Yuditskii in 1987 as an extension of the V.P. Potapov's approach to interpolation problems. In the present paper an analog of the AIP for Nevanlinna classes is considered. The description of solutions of the AIP is reduced to the description of L-resolvents of some model symmetric operator associated with the AIP. The latter description is obtained by using the M.G. Krein's theory of L-resolvent matrices. Both regular and singular cases of the AIP are treated. The results are illustrated by the following examples: bitangential interpolation problem, full and truncated moment problems. It is shown that each of these problems can be included into the general scheme of the AIP.

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