A new concept for block operator matrices:The quadratic numerical range

H. Langer; A. Markus; V. Matsaev; C. Tretter

doi:10.1016/S0024-3795(01)00230-0

A new concept for block operator matrices:The quadratic numerical range

H. Langer, A. Markus, V. Matsaev, C. Tretter

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

58 Scopus citations

Abstract

In this paper a new concept for 2×2-block operator matrices - the quadratic numerical range - is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.

Original language	English
Pages (from-to)	89-112
Number of pages	24
Journal	Linear Algebra and Its Applications
Volume	330
Issue number	1-3
DOIs	https://doi.org/10.1016/S0024-3795(01)00230-0
State	Published - 15 Jun 2001

Keywords

Angular operator
Block operator matrix
Quadratic numerical range
Riccati equation
Schur complement

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/S0024-3795(01)00230-0

Cite this

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title = "A new concept for block operator matrices:The quadratic numerical range",

abstract = "In this paper a new concept for 2×2-block operator matrices - the quadratic numerical range - is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.",

keywords = "Angular operator, Block operator matrix, Quadratic numerical range, Riccati equation, Schur complement",

author = "H. Langer and A. Markus and V. Matsaev and C. Tretter",

note = "Funding Information: A. Markus and V. Matsaev have been supported by grants from the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. C. Tret-ter gratefully acknowledges the support of the German Research Foundation, DFG, Grant Tr 368/4-1. All authors acknowledge the support of the Research Training Network HPRN-CT-2000-00116 of the European Union.",

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T1 - A new concept for block operator matrices:The quadratic numerical range

AU - Langer, H.

AU - Markus, A.

AU - Matsaev, V.

AU - Tretter, C.

N1 - Funding Information: A. Markus and V. Matsaev have been supported by grants from the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. C. Tret-ter gratefully acknowledges the support of the German Research Foundation, DFG, Grant Tr 368/4-1. All authors acknowledge the support of the Research Training Network HPRN-CT-2000-00116 of the European Union.

PY - 2001/6/15

Y1 - 2001/6/15

N2 - In this paper a new concept for 2×2-block operator matrices - the quadratic numerical range - is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.

AB - In this paper a new concept for 2×2-block operator matrices - the quadratic numerical range - is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.

KW - Angular operator

KW - Block operator matrix

KW - Quadratic numerical range

KW - Riccati equation

KW - Schur complement

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JO - Linear Algebra and Its Applications

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ER -

A new concept for block operator matrices:The quadratic numerical range

Abstract

Keywords

ASJC Scopus subject areas

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