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Spectral approximations of unbounded nonselfadjoint operators

Spectral approximations of unbounded nonselfadjoint operators We consider the operator $$A=S+B,$$ where $$S$$ is an unbounded normal operator in a separable Hilbert space $$H,$$ having a compact inverse one and $$B$$ is a linear operator in $$H,$$ such that $$BS^{-1} $$ is compact. Let $$\{e_k\}_{k=1}^\infty $$ be the normalized eigenvectors of $$S$$ and $$B$$ be represented in $$\{e_k\}_{k=1}^\infty $$ by a matrix $$(b_{jk})_{j,k=1}^\infty .$$ We approximate the eigenvalues of $$A$$ by a combination of the eigenvalues of $$S$$ and the eigenvalues of the finite matrix $${(b_{jk})}_{j,k=1}^{n}.$$ Applications of to differential operators are also discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Spectral approximations of unbounded nonselfadjoint operators

Analysis and Mathematical Physics , Volume 3 (1) – Sep 14, 2012

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Basel AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-012-0037-2
Publisher site
See Article on Publisher Site

Abstract

We consider the operator $$A=S+B,$$ where $$S$$ is an unbounded normal operator in a separable Hilbert space $$H,$$ having a compact inverse one and $$B$$ is a linear operator in $$H,$$ such that $$BS^{-1} $$ is compact. Let $$\{e_k\}_{k=1}^\infty $$ be the normalized eigenvectors of $$S$$ and $$B$$ be represented in $$\{e_k\}_{k=1}^\infty $$ by a matrix $$(b_{jk})_{j,k=1}^\infty .$$ We approximate the eigenvalues of $$A$$ by a combination of the eigenvalues of $$S$$ and the eigenvalues of the finite matrix $${(b_{jk})}_{j,k=1}^{n}.$$ Applications of to differential operators are also discussed.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Sep 14, 2012

References