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CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS

Part of: General theory of linear operators Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  09 September 2019

D. SUKUMAR  and
S. VEERAMANI
D. SUKUMAR
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hyderabad, India email suku@iith.ac.in
S. VEERAMANI*
Affiliation:
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India email ma13p1005@iith.ac.in
*
✉ma13p1005@iith.ac.in
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Abstract

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\unicode[STIX]{x1D716}<1$. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$-level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

Keywords

condition spectrumupper and lower hemicontinuous correspondence

MSC classification

Primary: 46H05: General theory of topological algebras
Secondary: 47A10: Spectrum, resolvent
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 3 , June 2020 , pp. 412 - 430
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author thanks the University Grants Commission (UGC), India for the financial support (Ref. no. 23/12/2012(ii)EU-V) provided as a form of Research Fellowship to carry out this research work at IIT Hyderabad.

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