Pseudospectra of elements of reduced Banach algebras

Abstract

Let $A$ be a Banach algebra with identity $1$ and $p \in A$ be a non-trivial idempotent. Then $q=1-p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $a \in A$ and $\varepsilon > 0$, we examine the relationship between the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A,a)$ of $a \in A$, and $\varepsilon$-pseudospectra of $pap \in pAp$ and $qaq \in qAq$. We also extend this study by considering a finite number of idempotents $p_{1},\cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.