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

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.