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Continuity of the \(\varvec{(n,\epsilon )}\)-Pseudospectrum in Banach Algebras

  • Kousik DharaEmail author
  • S. H. Kulkarni
  • Markus Seidel
Article

Abstract

Let \(\epsilon >0\), n a non-negative integer, and A a complex unital Banach algebra. Define \(\gamma _n: A\times {\mathbb {C}}\rightarrow [0,\infty ]\) by
$$\begin{aligned} \gamma _n(a,z)={\left\{ \begin{array}{ll} \Vert (z -a)^{-2^n}\Vert ^{-1/2^n}, &{}\quad \text {if } (z-a) \text{ is } \text{ invertible }\\ 0, &{}\quad \text {if } (z-a) \text { is not invertible}. \end{array}\right. } \end{aligned}$$
The \((n,\epsilon )\)-pseudospectrum \(\Lambda _{n,\epsilon }(a)\) of an element \(a\in A\) is defined by \(\Lambda _{n,\epsilon }(a):= \{\lambda \in {\mathbb {C}}:\gamma _n(a,\lambda )\le \epsilon \}\). We show that \(\gamma _0\) is Lipschitz on \(A\times {\mathbb {C}}\), \(\gamma _n\) is uniformly continuous on bounded subsets of \(A\times {\mathbb {C}}\) for \(n\ge 1\), and \(\gamma _n\) is Lipschitz on some particular domains for \(n\ge 1\). Using these properties, we establish that the map \((\epsilon ,a)\mapsto \Lambda _{n,\epsilon }(a)\) is continuous at \((\epsilon _0,a_0)\) if and only if the level set \(\{\lambda \in {\mathbb {C}}: \gamma _n(a_0,\lambda )= \epsilon _0 \}\) does not contain any non-empty open set. In particular, this happens when a is a compact operator on a Banach space or a bounded operator on a Hilbert space or on an \(L^p \) space with \(1\le p\le \infty \). We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous.

Keywords

Banach algebra Spectrum Pseudospectrum \((n, \epsilon )\)-Pseudospectrum 

Mathematics Subject Classification

47A10 47A58 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments that led to a considerable improvement in this paper. The first author would like to thank the Department of Atomic Energy (DAE), India (Ref No: 2/39(2)/2015/NBHM/R&D-II/7440) for financial support.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.University of Applied Sciences ZwickauZwickauGermany

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