Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X3220180401On linear maps preserving certain pseudospectrum and condition spectrum subsets4234325146010.15352/aot.1705-1159ENSayda RagoubiDepartment of Mathematic, Univercity of Monastir, Preparatory Institute for Engineering Studies of Monastir, TunisiaJournal Article20170504 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:<br /> (1) Let $A$ and $B$ be complex unital Banach algebras and $varepsilon>0$. Let $phi : Alongrightarrow B $ be an $varepsilon$-left (or right) pseudospectrum preserving onto linear map. Then $phi$ preserves certain standart spectral functions.<br />(2) Let $A$ and $B$ be complex unital Banach algebras and $0< varepsilon<1$. Let $phi : Alongrightarrow B $ be unital linear map. Then<br />(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 $a in A$.<br />(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.<br /><br />http://www.aot-math.org/article_51460_43332c93297a57ce21f16a69e9f0e63e.pdf