Decomposition of the $(n,\epsilon )$-pseudospectrum of an element of a Banach algebra

Abstract

Let A be a complex Banach algebra with unit. For an integer $n\ge 0$ and $\epsilon >0$, the $(n,\epsilon )$-pseudospectrum of $a\in A$ is defined by

$$\begin{aligned} \varLambda _{n,\epsilon } (A,a):=\left\{ \lambda \in \mathbb {C}: (\lambda -a) \text { is not invertible in } A \text { or } \Vert (\lambda -a)^{-2^{n}}\Vert ^{1/2^n} \ge \frac{1}{\epsilon }\right\} . \end{aligned}$$

Let $p\in A$ be a nontrivial idempotent. Then $pAp=\{pbp:b\in A\}$ is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose $ap=pa$. We study the relationship of $\varLambda _{n,\epsilon }(A,a)$ and $\varLambda _{n,\epsilon }(pAp,pa)$. We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the $(n,\epsilon )$-pseudospectrum of a can be decomposed into the union of the $(n,\epsilon )$-pseudospectra of some elements in reduced Banach algebras.