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THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

Published online by Cambridge University Press:  01 January 2008

THOMAS L. MILLER  and
VLADIMIR MÜLLER
THOMAS L. MILLER
Affiliation:
Dept. of Mathematics and Statistics, Mississippi State University, Drawer MA, Mississippi State, MS 39762 e-mail: [email protected]
VLADIMIR MÜLLER
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic e-mail: [email protected]
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Abstract

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Let T be a bounded operator on a complex Banach space X. Let V be an open subset of the complex plane. We give a condition sufficient for the mapping f(z)↦ (T−z)f(z) to have closed range in the Fréchet space H(V, X) of analytic X-valued functions on V. Moreover, we show that there is a largest open set U for which the map f(z)↦ (T−z)f(z) has closed range in H(V, X) for all V⊆U. Finally, we establish analogous results in the setting of the weak–* topology on H(V, X*).

Keywords

47A11
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 17 - 26
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Aiena, P., Fredholm and local spectral theory, with applications to multipliers (Kluwer Academic Publ., Dordrecht, 2004).Google Scholar
2.Aiena, P. and Villafane, F., 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, 1–14. Zbl 1057.47006CrossRefGoogle Scholar
3.Albrecht, E. and Eschmeier, J., Analytic functional models and local spectral theory, Proc. London Math. Soc. (3) 75 (1997), 323–348.CrossRefGoogle Scholar
4.Bishop, E., A duality theory for an arbitrary operator, Pacific J. Math. 9 (1959), 379–397.CrossRefGoogle Scholar
5.Eschmeier, J., Analytische Dualität und Tensorprodukte in der mehrdimensionalen Spektraltheorie, Habilitationsschrift, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie, Heft 42 (Münster, 1987).Google Scholar
6.Eschmeier, J., On the essential spectrum of Banach space operators, Proc. Edinburgh Math. Soc. (2) 43 (2000), 511–528.CrossRefGoogle Scholar
7.Herrero, D., On the essential spectra of %quasisimilar operators, Can. J. Math. 40 (1988), 1436–1457.CrossRefGoogle Scholar
8.Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math. 6 (1958), 261–322.CrossRefGoogle Scholar
9.Labrousse, J.-P., Les opérateurs quasi Fredholm: une généralisation des opérateurs semi Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), 161–258.CrossRefGoogle Scholar
10.Laursen, K. B. and Neumann, M. M., An introduction to local spectral theory (Clarendon Press, Oxford, 2000).CrossRefGoogle Scholar
11.Miller, T. L. and Miller, V. G., An operator satisfying Dunford's condition (C) but without Bishop's property(β), Glasgow Math. J. 40 (1998), 427–430.CrossRefGoogle Scholar
12.Miller, T. L., Miller, V. G. and Neumann, M. M., Localization in the spectral theory of operators on Banach spaces, in Proceedings of the Fourth Conference on Function Spaces at Edwardsville, Contemp. Math. 328 (Amer. Math. Soc., Providence, RI, 2003), 247–262.Google Scholar
13.Miller, T. L., Miller, V. G. and Neumann, M. M., The Kato-type spectrum and local spectral theory, Czech. Math. J. 57 (2007), 831–842.CrossRefGoogle Scholar
14.Müller, V., Spectral theory of linear operators and spectral systems in banach algebras (Birkhäuser Verlag, Basel, 2003).CrossRefGoogle Scholar
15.Nagy, B., On S-decomposable operators, J. Operator Theory 2 (1979), 277–286.Google Scholar
16.Putinar, M., Quasi-similarity of tuples with Bishop's property (β), Int. Eq. and Oper. Theory 15 (1992), 1047–1052.CrossRefGoogle Scholar
17.Vasilescu, F.-H., Analytic functional calculus and spectral decompositions (Editura Academiei and D. Reidel Publishing Company, Bucharest and Dordrecht, 1982).Google Scholar
18.Shaefer, H. H., Topological Vector Spaces, Graduate Texts in Mathematics, Springer-Verlag, New York, 1971.CrossRefGoogle Scholar