Representation of simple symmetric operators with deficiency indices (1,1) in de Branges space

TL;DR

This paper demonstrates that simple symmetric operators with deficiency indices (1,1) can be represented as multiplication operators in de Branges spaces, providing new insights into their spectral properties and functional models.

Contribution

It proves that such symmetric operators are unitarily equivalent to multiplication in de Branges spaces, extending previous results and offering new spectral and functional analysis tools.

Findings

Operators are unitarily equivalent to multiplication in de Branges spaces.

New criteria for the density of multiplication by z in these spaces.

Conditions for isometries from subspaces of L^2(R) onto de Branges spaces.

Abstract

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1,1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges-Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges-Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of $L^2 (R, dv) onto a de Branges space of entire functions which acts as multiplication by a measurable function.