| Title:
|
On operators with the same local spectra (English) |
| Author:
|
Torgašev, Aleksandar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
77-83 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Banach space |
| Keyword:
|
spectrum |
| Keyword:
|
local spectrum |
| MSC:
|
47A10 |
| MSC:
|
47A11 |
| idZBL:
|
Zbl 0926.47002 |
| idMR:
|
MR1614080 |
| . |
| Date available:
|
2009-09-24T10:11:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127400 |
| . |
| Reference:
|
[1] I. Colojoara, C. Foiaş: Quasinilpotent equivalence of not necessarily commuting operators.Journal Math. Mech. 15 (1966), 521–540. MR 0192344 |
| Reference:
|
[2] I. Colojoara, C. Foiaş: Theory of Generalized Spectral Operators.Gordon and Breach, New York, 1968. MR 0394282 |
| Reference:
|
[3] C. Foiaş: Spectral maximal spaces and decomposable operators in Banach spaces.Arch. Math. 14 (1963), 341–349. MR 0152893, 10.1007/BF01234965 |
| Reference:
|
[4] J.D.Gray: Local analytic extensions of the resolvent.Pacific J. Math. 27(2) (1968), 305–324. Zbl 0172.17204, MR 0236738, 10.2140/pjm.1968.27.305 |
| Reference:
|
[5] R.C. Sine: Spectral decomposition of a class of operators.Pacific J. Math. 14 (1964), 333–352. Zbl 0197.39501, MR 0164242, 10.2140/pjm.1964.14.333 |
| Reference:
|
[6] P. Vrbová: On local spectral properties of operators in Banach spaces.Czechoslovak Math. J. 23 (1973), 483–492. MR 0322536 |
| . |
