On operators with the same local spectra
Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 77-83
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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 $\Vert (T_1 - T_2)^n\Vert ^{1/n} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.
@article{CMJ_1998__48_1_a6,
author = {Torga\v{s}ev, Aleksandar},
title = {On operators with the same local spectra},
journal = {Czechoslovak Mathematical Journal},
pages = {77--83},
publisher = {mathdoc},
volume = {48},
number = {1},
year = {1998},
mrnumber = {1614080},
zbl = {0926.47002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_1998__48_1_a6/}
}
Torgašev, Aleksandar. On operators with the same local spectra. Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 77-83. http://geodesic.mathdoc.fr/item/CMJ_1998__48_1_a6/