Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 43-62.

Geometry of higher order relative spectra and projection methods

Authors: Eugene Shargorodsky
Author institution: School of Mathematical Sciences, University of Sussex, Falmer, East Sussex, BN1 9QH, UK

Summary: Let

$H$ be a densely defined linear operator acting on a Hilbert space

$\cH$ , let

$P$ be the orthogonal projection onto a closed linear subspace

$\cL$ and let

$n \in \bn$ . The

$n$ -th order spectrum

${\rm Spec}_n(H,\cL)$ of

$H$ relative to

$\cL$ is the set of

$z\in\bC$ such that the restriction to

$\cL$ of the operator

$P(H-zI)^nP$ is not invertible within the subspace

$\cL$ . We study restrictions which may be placed on this set under given assumptions on

${\rm Spec}(H)$ and the behaviour of

${\rm Spec}_n(H,\cL)$ as

$\cL$ increases towards

$\cH$ .

Keywords: Higher order relative spectra, orthogonal projections, projection methods

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