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Perturbation Theorems for Relative Spectral Problems

Published online by Cambridge University Press:  20 November 2018

Edward Hughes
Edward Hughes*
Affiliation:
The University of British Columbia, Vancouver, British Columbia; Carleton University, Ottawa, Ontario
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Eigenvalue problems of the form Af = λBf, where λ is a complex parameter and A and B are operators on a Hilbert Space, have been considered by a number of authors (e.g., [1; 3; 5; 7; 10]). In this paper, we shall be concerned with the existence and nature of eigenfunction expansions associated with such problems, with no assumptions of self-adjointness. The form of the theorems to be given here is: if the system (A, B) is spectral and complete (definitions below), and F and G are operators satisfying certain “smallness” conditions, then (A + F, B + G) is also spectral and complete. The hypotheses for these theorems are chosen with an eye to applying the results to boundary-value problems on a compact interval. Such applications, together with an examination of circumstances under which the system (Dn, Dm) (D denoting differentiation) is spectral and complete under a broad class of boundary conditions, will be made in a later paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 72 - 81
Copyright
Copyright © Canadian Mathematical Society 1972

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