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Pseudospectra of elements of reduced Banach algebras. (English) Zbl 06804223

Summary: 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.

MSC:

47A10 Spectrum, resolvent
46H05 General theory of topological algebras
47A12 Numerical range, numerical radius
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