Functional Models of Operators and Their Multivalued Extensions in Hilbert Space

Abstract

This paper presents the first part of a study of functional models of selfadjoint and nonselfadjoint extensions \(\widetilde{A}\) of symmetric and nonsymmetric operators A in a Hilbert space \(\mathfrak {H}\). The extensions will be considered in the framework of linear relations (which may also be interpreted as the graphs of multivalued operators) that are required to have a nonempty set of regular points \(\rho (\widetilde{A})\). In these models \(\mathfrak {H}\) is modelled by a reproducing kernel Hilbert space \(\mathcal {H}\) of vector valued holomorphic functions that are defined on some nonempty open set \(\Omega \subseteq \rho (\widetilde{A})\) and \(\mathcal {H}\) is invariant under the action of the (generalized) backward shift operator \(R_\alpha \) for every \(\alpha \in \Omega \); A is modelled by the operator \(\mathfrak {A}\) of multiplication by the independent variable [i.e., \((\mathfrak {A}f)(\lambda )=\lambda f(\lambda )\) for \(f\in \mathcal {H}\) for which \(\mathfrak {A}f\in \mathcal {H}\)]; and \(\widetilde{A}\) is modelled by a linear relation \(\widetilde{\mathfrak {A}}\) with the property that \((\widetilde{\mathfrak {A}}-\alpha I)^{-1}=R_\alpha \) for all points \(\alpha \in \Omega \).

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Correspondence to Harry Dym.

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D. Arov acknowledges with thanks the support of a Belkin visiting Professorship from the Weizmann Institute of Science.

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Arov, D.Z., Dym, H. Functional Models of Operators and Their Multivalued Extensions in Hilbert Space. Integr. Equ. Oper. Theory 92, 39 (2020). https://doi.org/10.1007/s00020-020-02595-4

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Keywords

  • Linear relations
  • Extensions of operators in Hilbert space
  • Functional models
  • Reproducing kernel Hilbert spaces
  • Vector valued de Branges spaces
  • Multiplication operators

Mathematics Subject Classification

  • 46E22
  • 47A20
  • 47A06
  • 47B25
  • 47B32