Decomposition of the $$(n,\epsilon )$$-pseudospectrum of an element of a Banach algebra
| Autor: | Kousik Dhara, S. H. Kulkarni |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Advances in Operator Theory. 5:248-260 |
| ISSN: | 2538-225X 2662-2009 |
| DOI: | 10.1007/s43036-019-00016-x |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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