Restrictions and extensions of semibounded operators

TL;DR

This paper investigates the restriction and extension theory of semibounded Hermitian operators in the Hardy space, classifying their selfadjoint extensions and spectral properties based on boundary conditions on measure-zero sets.

Contribution

It provides a comprehensive classification of selfadjoint extensions of boundary-restricted operators in the Hardy space, linking boundary conditions to spectral characteristics.

Findings

Different boundary sets F lead to distinct spectral configurations.

Selfadjoint extensions can be classified explicitly for each boundary restriction.

Reproducing kernel Hilbert spaces with Hurwitz zeta-function kernels are used in the analysis.

Abstract

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.