On operators with the same local spectra

Torgašev, Aleksandar

Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 77-83.

Résumé

Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation ${\sigma }_{T_1}(x)={\sigma }_{T_2}(x)$ for any vector $x\in X$, where ${\sigma }_T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1-T_2$ is a quasinilpotent operator, that is $\Vert (T_1-T_2{)}^n{\Vert }^{1/n}\to 0$ as $n\to \mathrm{\infty }$. Without these conditions, we describe the operators with the same local spectra only in some particular cases.

Détail

Classification : 47A10, 47A11
Mots-clés : Banach space; spectrum; local spectrum