Variational principles for eigenvalues of block operator matrices

2002 ◽  
Vol 51 (6) ◽  
pp. 1427-1460 ◽  
Author(s):  
H. Langer ◽  
M. Langer ◽  
Christiane Tretter
Keyword(s):  
Variational Principles ◽  
Operator Matrices ◽  
Block Operator ◽  
Block Operator Matrices

Sharp Inequalities for the Numerical Radii of Block Operator Matrices

2019 ◽  
Vol 45 (4) ◽  
pp. 687-703
Author(s):  
M. Ghaderi Aghideh ◽  
M. S. Moslehian ◽  
J. Rooin
Keyword(s):  
Operator Matrices ◽  
Block Operator ◽  
Block Operator Matrices

scholarly journals Essential numerical ranges for linear operator pencils

2019 ◽  
Vol 40 (4) ◽  
pp. 2256-2308
Author(s):  
Sabine Bögli ◽  
Marco Marletta
Keyword(s):  
Linear Operator ◽  
Numerical Range ◽  
Projection Methods ◽  
Operator Pencil ◽  
Operator Matrices ◽  
New Concepts ◽  
Block Operator ◽  
Block Operator Matrices ◽  
Spectral Pollution ◽  
Operator Pencils

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.

On the group invertibility of operators

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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