Spectral Theory of Block Operator Matrices and Applications

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

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Spectral Estimates and Basis Properties for Self-Adjoint Block Operator Matrices

Integral Equations and Operator Theory ◽

10.1007/s00020-010-1780-0 ◽

2010 ◽

Vol 67 (2) ◽

pp. 257-277 ◽

Cited By ~ 1

Author(s):

Michael Strauss

Keyword(s):

Operator Matrices ◽

Spectral Estimates ◽

Block Operator ◽

Block Operator Matrices

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On the group invertibility of operators

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.1967 ◽

2016 ◽

Vol 31 ◽

pp. 492-510

Author(s):

Chunyuan Deng

Keyword(s):

Hilbert Spaces ◽

Main Topic ◽

Reverse Order ◽

Operator Matrices ◽

Reverse Order Law ◽

Block Operator ◽

Block Operator Matrices ◽

Group Inverses ◽

Triangular Block ◽

Equivalent Conditions

The main topic of this paper is the group invertibility of operators in Hilbert spaces. Conditions for the existence of the group inverses of products of two operators and the group invertibility of anti-triangular block operator matrices are studied. The equivalent conditions related to the reverse order law for the group inverses of operators are obtained.

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