# 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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