Pseudospectra of elements of reduced Banach algebras II

Arundhathi Krishnan, Sudhir H Kulkarni

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. Let $x\in A$ be such that $pxp=xp$, and $\varepsilon>0$. We examine the relationship between the spectrum of $x\in A$, $\sigma(A,x)$, and the spectra of $pxp\in pAp$, $\sigma(pAp,pxp)$ and $qxq\in qAq$, $\sigma(qAq,qxq)$. Similarly, we examine the relationship betweeen the $\varepsilon$-pseudospectrum of $x\in A$, $\Lambda_{\varepsilon}(A,x)$ and $\varepsilon$-pseudospectra of $pxp\in pAp$, $\Lambda_{\varepsilon}(pAp, pxp)$ and of $qxq\in qAq$, $\Lambda_{\varepsilon}(qAq, qxq)$.  

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References


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