### 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.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 475-493.

Dates
Accepted: 12 July 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431723

Digital Object Identifier
doi:10.22034/aot.1702-1112

Mathematical Reviews number (MathSciNet)
MR3730042

Zentralblatt MATH identifier
06804223

#### Citation

Krishnan, Arundhathi; Kulkarni, S. H. Pseudospectra of elements of reduced Banach algebras. Adv. Oper. Theory 2 (2017), no. 4, 475--493. doi:10.22034/aot.1702-1112. https://projecteuclid.org/euclid.aot/1512431723

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