Restrictions and extensions of semibounded operators

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.