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HERMITIANS IN MATRIX ALGEBRAS WITH OPERATOR NORM

Part of:General theory of linear operatorsBasic linear algebra

Published online by Cambridge University Press:  27 April 2020

MICHAEL J. CRABB,
JOHN DUNCAN and
COLIN M. McGREGOR
MICHAEL J. CRABB
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, Scotland, UK
JOHN DUNCAN
Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA, e-mail: jduncan@uark.edu
COLIN M. McGREGOR
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, Scotland, UK, e-mail: Colin.McGregor@glasgow.ac.uk
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Abstract

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We investigate the real space H of Hermitian matrices in $M_n(\mathbb{C})$ with respect to norms on $\mathbb{C}^n$. For absolute norms, the general form of Hermitian matrices was essentially established by Schneider and Turner [Schneider and Turner, Linear and Multilinear Algebra (1973), 9–31]. Here, we offer a much shorter proof. For non-absolute norms, we begin an investigation of H by means of a series of examples, with particular reference to dimension and commutativity.

MSC classification

Primary:15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Secondary:47A12: Numerical range, numerical radius

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021, pp. 280 - 290
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Footnotes

To the memory of Michael J. Crabb

References

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