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5.—Semi-bounded Dirichlet Integrals and the Invariance of the Essential Spectra of Self-adjoint Operators

Published online by Cambridge University Press:  14 February 2012

W. D. Evans
W. D. Evans
Affiliation:
Department of Pure Mathematics, University College, Cardiff.
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Synopsis

In the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integral

associated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 75 , Issue 1 , 1976 , pp. 41 - 66
Copyright
Copyright © Royal Society of Edinburgh 1976

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