| Title:
|
The Kato-type spectrum and local spectral theory (English) |
| Author:
|
Miller, T. L. |
| Author:
|
Miller, V. G. |
| Author:
|
Neumann, M. M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
831-842 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $T\in {\mathcal{L}}(X)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f(z)\mapsto (z-T)f(z)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves. (English) |
| Keyword:
|
decomposable operator |
| Keyword:
|
semi-Fredholm operator |
| Keyword:
|
semi-regular operator |
| Keyword:
|
Kato decomposition |
| Keyword:
|
Bishop’s property ($\beta $) |
| Keyword:
|
property ($\delta $) |
| MSC:
|
47A11 |
| MSC:
|
47A53 |
| idZBL:
|
Zbl 1174.47001 |
| idMR:
|
MR2356283 |
| . |
| Date available:
|
2009-09-24T11:49:46Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128209 |
| . |
| Reference:
|
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| Reference:
|
[2] P. Aiena and F. Villafañe: Components of resolvent sets and local spectral theory.Proceedings of the Fourth Conference on Function Spaces at Edwardsville, Contemp. Math. 328, Amer. Math. Soc., Providence, RI, 2003, pp. 1–14. MR 1990385 |
| Reference:
|
[3] E. Albrecht and J. Eschmeier: Analytic functional models and local spectral theory.Proc. London Math. Soc. 75 (1997), 323–348. MR 1455859 |
| Reference:
|
[4] J. Eschmeier: Analytische Dualität und Tensorprodukte in der mehrdimensionalen Spektraltheorie.Habilitationsschrift, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie, Heft 42, Münster, 1987. Zbl 0619.47030, MR 0876484 |
| Reference:
|
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| Reference:
|
[6] D. Herrero: On the essential spectra of quasisimilar operators.Can. J. Math. 40 (1988), 1436–1457. Zbl 0723.47015, MR 0990108, 10.4153/CJM-1988-066-x |
| Reference:
|
[7] T. Kato: Perturbation theory for nullity, deficiency and other quantities of linear operators.J. Anal. Math. 6 (1958), 261–322. Zbl 0090.09003, MR 0107819, 10.1007/BF02790238 |
| Reference:
|
[8] J.-P. Labrousse: Les opérateurs quasi Fredholm: une généralisation des opérateurs semi Fredholm.Rend. Circ. Mat. Palermo 29 (1980), 161–258. Zbl 0474.47008, MR 0636072, 10.1007/BF02849344 |
| Reference:
|
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| Reference:
|
[10] T. L. Miller and V. G. Miller: Equality of essential spectra of quasisimilar operators with property $(\delta )$.Glasgow Math. J. 38 (1996), 21–28. MR 1373954, 10.1017/S0017089500031219 |
| Reference:
|
[11] T. L. Miller, V. G. Miller and M. M. Neumann: Localization in the spectral theory of operators on Banach spaces.Proceedings of the Fourth Conference on Function Spaces at Edwardsville, Contemp. Math. 328, Amer. Math. Soc., Providence, RI, 2003, pp. 247–262. MR 1990406 |
| Reference:
|
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| Reference:
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| Reference:
|
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| . |
