Restrictions and Extensions of Semibounded Operators (original) (raw)
2014, Complex Analysis and Operator Theory
We study restriction and extension theory for semibounded Hermitian operators in the Hardy space H 2 of analytic functions on the disk D. Starting with the operator z d dz , we show that, for every choice of a closed subset F ⊂ T = ∂D of measure zero, there is a densely defined Hermitian restriction of z d 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 F × F , as reproducing kernel.
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