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CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 09 September 2019, pp. 412-430
- Print publication:
- June 2020
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- Article
- Export citation
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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.
