On a class of model Hilbert spaces

Fritz Gesztesy; Rudi Weikard; Maxim Zinchenko

doi:10.3934/dcds.2013.33.5067

Article Contents

2013, Volume 33, Issue 11&12: 5067-5088. Doi: 10.3934/dcds.2013.33.5067

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On a class of model Hilbert spaces

1.
Department of Mathematics, University of Missouri, Columbia, MO 65211
2.
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294
3.
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131

Received: October 31, 2011

Published: October 2013

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Abstract

A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.

Keywords:

Direct integrals of Hilbert spaces,
model Hilbert spaces.

Mathematics Subject Classification: Primary: 28B05, 46E40, 47A10; Secondary: 46G10, 47E05.

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