Interpolation in de Branges-Rovnyak spaces

Ball, Joseph A.; Bolotnikov, Vladimir; Horst, S. ter

doi:10.1090/s0002-9939-2010-10505-1

Cited by 16 publications

(24 citation statements)

References 6 publications

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“…This operator-argument problem was considered in [4] for scalar-valued functions and for the nondegenerate case where the solution of the Stein equation (1.7) is positive definite (i.e., invertible). The eventual parametrization for the set of all solutions, which we obtain in Theorem 5.1 below, is connected with previously appearing representations for almost invariant subspaces and Toeplitz kernels in terms of an isometric multiplier between two de Branges-Rovnyak spaces.…”

Section: The Operatorsmentioning

confidence: 99%

“…In the scalar-valued case = = ℂ, the latter problem has been considered recently in [4], with the additional assumption that > 0. Similarly to the situation in [4], the operator-valued version contains left-tangential Nevanlinna-Pick and Carathéodory-Fejér interpolation problems as particular cases corresponding to special choices of and .…”

Section: Interpolation With Operator Argumentmentioning

confidence: 99%

“…Similarly to the situation in [4], the operator-valued version contains left-tangential Nevanlinna-Pick and Carathéodory-Fejér interpolation problems as particular cases corresponding to special choices of and . We now show that on the other hand, the problem OAP ℋ( ) can be considered as a particular case of the problem AIP ℋ( ) .…”

Section: Interpolation With Operator Argumentmentioning

confidence: 99%

“…However, very little work exists on interpolation for functions in de Branges-Rovnyak spaces themselves. In this paper we pursue our studies of interpolation problems for functions in de Branges-Rovnyak spaces, which started in [4]. We consider a norm constrained interpolation problem (denoted by AIP ℋ( ) in what follows), which is sufficiently fine so as to include on the one hand interpolation problems with operator argument (considered for the nondegenerate and scalar-valued case in [4]) and, on the other hand, boundary interpolation problems; it is only recent work [10,11,12,22] which has led to a systematic understanding of boundarypoint evaluation on de Branges-Rovnyak spaces from an operator-theoretic point of view.…”

Section: Introductionmentioning

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: The Operatorsmentioning

confidence: 99%

Section: Interpolation With Operator Argumentmentioning

confidence: 99%

Section: Interpolation With Operator Argumentmentioning

confidence: 99%

Section: Introductionmentioning

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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“…This unified approach to these spaces has obvious advantages. For example, we will show how to exploit the results in the papers [8,9], in which the authors formulate and solve interpolation problems in the de Branges-Rovnyak spaces, to explicitly solve interpolation problems in the Fock space. This is a far reaching result, and it is not clear how to obtain it using different techniques.…”

Section: Introductionmentioning

confidence: 99%

The Fock Space as a De Branges–Rovnyak Space

Alpay

Colombo

Sabadini

2019

Integr. Equ. Oper. Theory

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Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

Alpay

Jørgensen

Lewkowicz

et al. 2012

Multiscale Signal Analysis and Modeling

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We introduce connections between the Cuntz relations and the Hardy space H 2 of the open unit disk D. We then use them to solve a new kind of multipoint interpolation problem in H 2 , where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.1991 Mathematics Subject Classification. 42C40, 47A57, 93B28. Key words and phrases. Cuntz relations, Leech's theorem, Schur analysis. D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The work was done in part while the second named author visited Department of Mathematics, Ben Gurion University of the Negev, supported by a BGU distinguished visiting scientist program. Support and hospitality is much appreciated. We acknowledge discussions with colleagues there, and in the US, Dorin Dutkay, Myung-Sin Song, and Erin Pearse.

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Interpolation in de Branges-Rovnyak spaces

Cited by 16 publications

References 6 publications

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

The Fock Space as a De Branges–Rovnyak Space

Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

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