### Linear maps preserving pseudospectrum and condition spectrum

G. Krishna Kumar and S. H. Kulkarni
Source: Banach J. Math. Anal. Volume 6, Number 1 (2012), 45-60.

#### Abstract

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

First Page:
Primary Subjects: 47B49
Secondary Subjects: 46H05, 46J05, 47S48
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Permanent link to this document: http://projecteuclid.org/euclid.bjma/1337014664
Mathematical Reviews number (MathSciNet): MR2862542
Zentralblatt MATH identifier: 06058275

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