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7 - Boundary triplets and Weyl functions. Recent developments

Published online by Cambridge University Press:  05 November 2012

By
V.A. Derkach ,
S. Hassi ,
M.M. Malamud  and
H.S.V. de Snoo
Edited by
Seppo Hassi ,
Hendrik S. V. de Snoo  and
Franciszek Hugon Szafraniec
V.A. Derkach
Affiliation:
Donetsk National University
S. Hassi
Affiliation:
University of Vaasa
M.M. Malamud
Affiliation:
Donetsk National University
H.S.V. de Snoo
Affiliation:
University of Groningen
Seppo Hassi
Affiliation:
University of Vaasa, Finland
Hendrik S. V. de Snoo
Affiliation:
Rijksuniversiteit Groningen, The Netherlands
Franciszek Hugon Szafraniec
Affiliation:
Jagiellonian University, Krakow
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Summary

Abstract Selfadjoint extensions of a closed symmetric operator in a Hilbert space with equal deficiency indices are described by means of ordinary boundary triplets. In certain problems the more general notion of a boundary triplet of bounded type is needed. It will be shown that such triplets correspond in a certain way with the, in general infinite dimensional, graph perturbations of selfadjoint operators or relations. However, when considering selfadjoint exit spaces extensions for a symmetric operator by means of the so-called Kreĭn's formula one meets the notion of boundary relation (even when the deficiency indices are finite). Whereas ordinary boundary triplets and boundary triplets of bounded type correspond to bounded unitary or unitary operators in a Kreĭn space, respectively, boundary relations correspond to general unitary relations in a Kreĭn space, which are not necessarily single-valued. It is shown that the study of isometric relations in a Kreĭn space has useful applications. This present overview of recent developments includes illustrations, for instance, by means of elliptic differential operators and Schrödinger operators with local point interactions.

Introduction

The extension theory of densely defined symmetric operators was developed in the 1930s by J. von Neumann. A complete description was given for all selfadjoint extensions in terms of the defect subspaces; see [von Neumann, 1932; Stone, 1932]. Then M.H. Stone suggested to J.W. Calkin to develop another approach based on the notion of “abstract boundary conditions”, which reduces the extension problem to a description of the hyper-maximal symmetric subspaces of some auxiliary Hilbert space.

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Operator Methods for Boundary Value Problems , pp. 161 - 220
Publisher: Cambridge University Press
Print publication year: 2012

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