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HERMITIANS IN MATRIX ALGEBRAS WITH OPERATOR NORM – II

Part of: General theory of linear operators Basic linear algebra

Published online by Cambridge University Press:  10 June 2021

JOHN DUNCAN Open the ORCID record for JOHN DUNCAN [Opens in a new window]  and
COLIN M. McGREGOR
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

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$. We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.

MSC classification

Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A12: Numerical range, numerical radius
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 376 - 396
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Bauer, F. L., Theory of norms, infolab.stanford.edu/pub/cstr/reports/cs/tr/67/75/CS-TR-67-75.pdf (Stanford University, 1967).Google Scholar
Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras , LMS Lecture Note Series, vol. 2 (Cambridge University Press, New York, 1971).Google Scholar
Bonsall, F. F. and Duncan, J., Numerical ranges II , LMS Lecture Note Series, vol. 10 (Cambridge University Press, New York, 1973).Google Scholar
Crabb, M. J., Duncan, J. and McGregor, C. M., Hermitians in matrix algebras with operator norm, Glasgow Math J. 63 (2021) 280–290.CrossRefGoogle Scholar