Pseudospectra of elements of reduced Banach algebras

Document Type: Original Article


Indian Institute of Technology Madras


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.