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 $$\alpha \in \Omega $$ .