# Article

Full entry | PDF   (0.2 MB)
Keywords:
Banach space; spectrum; local spectrum
Summary:
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.
References:
[1] I. Colojoara, C. Foiaş: Quasinilpotent equivalence of not necessarily commuting operators. Journal Math. Mech. 15 (1966), 521–540. MR 0192344
[2] I. Colojoara, C. Foiaş: Theory of Generalized Spectral Operators. Gordon and Breach, New York, 1968. MR 0394282
[3] C. Foiaş: Spectral maximal spaces and decomposable operators in Banach spaces. Arch. Math. 14 (1963), 341–349. DOI 10.1007/BF01234965 | MR 0152893
[4] J.D.Gray: Local analytic extensions of the resolvent. Pacific J. Math. 27(2) (1968), 305–324. DOI 10.2140/pjm.1968.27.305 | MR 0236738 | Zbl 0172.17204
[5] R.C. Sine: Spectral decomposition of a class of operators. Pacific J. Math. 14 (1964), 333–352. DOI 10.2140/pjm.1964.14.333 | MR 0164242 | Zbl 0197.39501
[6] P. Vrbová: On local spectral properties of operators in Banach spaces. Czechoslovak Math. J. 23 (1973), 483–492. MR 0322536

Partner of