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
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.