Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space |
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Authors: | R T W Martin |
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Institution: | 1. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA
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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
L2 (\mathbbR, dn){L^2 (\mathbb{R}, d\nu)} onto a de Branges space of entire functions which acts as multiplication by a measurable function. |
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Keywords: | |
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