Geodesic
On operators with the same local spectra
Czechoslovak Mathematical Journal, Tome 48 (1998) no. 1, pp. 77-83 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 47A10, 47A11
Keywords: Banach space; spectrum; local spectrum 
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     author = {Torga\v{s}ev, Aleksandar},
     title = {On operators with the same local spectra},
     journal = {Czechoslovak Mathematical Journal},
     pages = {77--83},
     year = {1998},
     volume = {48},
     number = {1},
     mrnumber = {1614080},
     zbl = {0926.47002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_1998_48_1_a6/}
}
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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/

[1] I. Colojoara, C. Foiaş: Quasinilpotent equivalence of not necessarily commuting operators. Journal Math. Mech. 15 (1966), 521–540. | MR

[2] I. Colojoara, C. Foiaş: Theory of Generalized Spectral Operators. Gordon and Breach, New York, 1968. | MR

[3] C. Foiaş: Spectral maximal spaces and decomposable operators in Banach spaces. Arch. Math. 14 (1963), 341–349. | DOI | MR

[4] J.D.Gray: Local analytic extensions of the resolvent. Pacific J. Math. 27(2) (1968), 305–324. | DOI | MR | Zbl

[5] R.C. Sine: Spectral decomposition of a class of operators. Pacific J. Math. 14 (1964), 333–352. | DOI | MR | Zbl

[6] P. Vrbová: On local spectral properties of operators in Banach spaces. Czechoslovak Math. J. 23 (1973), 483–492. | MR