Journal article
Restrictions and Extensions of Semibounded Operators
Complex Analysis and Operator Theory, Vol.8(3), pp.591-663
03/2014
DOI: 10.1007/s11785-012-0241-y
Abstract
We study restriction and extension theory for semibounded Hermitian operators in the Hardy space $$\fancyscript{H}_{2}$$ of analytic functions on the disk $$\mathbb D $$ . Starting with the operator $${z\frac{d}{dz}}$$ , we show that, for every choice of a closed subset $$F\subset \mathbb T =\partial \mathbb D $$ of measure zero, there is a densely defined Hermitian restriction of $$z\frac{d}{dz}$$ 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 $$F\times F$$ , as reproducing kernel.
Details
- Title: Subtitle
- Restrictions and Extensions of Semibounded Operators
- Creators
- Palle Jorgensen - Department of Mathematics The University of Iowa Iowa IA 52242-1419 USASteen Pedersen - Department of Mathematics Wright State University Dayton OH 45435 USAFeng Tian - Department of Mathematics Wright State University Dayton OH 45435 USA
- Resource Type
- Journal article
- Publication Details
- Complex Analysis and Operator Theory, Vol.8(3), pp.591-663
- DOI
- 10.1007/s11785-012-0241-y
- ISSN
- 1661-8254
- eISSN
- 1661-8262
- Publisher
- Springer Basel; Basel
- Language
- English
- Date published
- 03/2014
- Academic Unit
- Mathematics
- Record Identifier
- 9983986093202771
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