Ragoubi, S. (2018). On linear maps preserving certain pseudospectrum and condition spectrum subsets. Advances in Operator Theory, 3(2), 423-432. doi: 10.15352/aot.1705-1159

Sayda Ragoubi. "On linear maps preserving certain pseudospectrum and condition spectrum subsets". Advances in Operator Theory, 3, 2, 2018, 423-432. doi: 10.15352/aot.1705-1159

Ragoubi, S. (2018). 'On linear maps preserving certain pseudospectrum and condition spectrum subsets', Advances in Operator Theory, 3(2), pp. 423-432. doi: 10.15352/aot.1705-1159

Ragoubi, S. On linear maps preserving certain pseudospectrum and condition spectrum subsets. Advances in Operator Theory, 2018; 3(2): 423-432. doi: 10.15352/aot.1705-1159

On linear maps preserving certain pseudospectrum and condition spectrum subsets

^{}Department of Mathematic, Univercity of Monastir, Preparatory Institute for Engineering Studies of Monastir, Tunisia

Receive Date: 04 May 2017,
Revise Date: 01 October 2017,
Accept Date: 29 October 2017

Abstract

We define two new types of spectrum, called the $\varepsilon$-left (or right) pseudospectrum and the $\varepsilon$-left (or right) condition spectrum, of an element $a$ in a complex unital Banach algebra $A$. We prove some basic properties among them the property that the $\varepsilon$-left (or right) condition spectrum is a particular case of Ransford spectrum. We study also the linear preserver problem for our defined functions and we establish the following: (1) Let $A$ and $B$ be complex unital Banach algebras and $\varepsilon>0$. Let $\phi : A\longrightarrow B $ be an $\varepsilon$-left (or right) pseudospectrum preserving onto linear map. Then $\phi$ preserves certain standart spectral functions. (2) Let $A$ and $B$ be complex unital Banach algebras and $0< \varepsilon<1$. Let $\phi : A\longrightarrow B $ be unital linear map. Then (a) If $\phi $ is $\varepsilon$-almost multiplicative map, then $\sigma^{l}(\phi(a))\subseteq \sigma^{l}_\varepsilon(a)$ and $\sigma^{r}(\phi(a))\subseteq \sigma^{r}_\varepsilon(a)$, for all $a \in A$. (b) If $\phi$ is an $\varepsilon$-left (or right) condition spectrum preserving, then (i) if $A$ is semi-simple, then $\phi$ is injective; (ii) if B is spectrally normed, then $\phi$ is continuous.