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An operator satisfying Dunford's condition (C) but without bishop's property (β)

Published online by Cambridge University Press:  18 May 2009

T. L. Miller  and
V. G. Miller
T. L. Miller
Affiliation:
Department of Mathematics, Mississipi State University, Drawer MA, Mississippi State 39762, USA
V. G. Miller
Affiliation:
Department of Mathematics, Mississipi State University, Drawer MA, Mississippi State 39762, USA
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For X a complex Banach space and U an open subset of the complex plane С, let O (U, X) denote the space of analytic X- valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping TU on O(U, X) given by (Tuf)(λ) = (λ – T)f(λ) for every f e O(U, X) and λ e U. Corresponding to each closed F ⊂ С there is also an associated analytic subspace XT(F) = X ∩ ran(7c//F). For an arbitrary T e L(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 3 , September 1998 , pp. 427 - 430
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

References

REFERENCES

1.Albrecht, E., On decomposable operators, Integral Equations Operator Theory, 2 (1979) 1–10.CrossRefGoogle Scholar
2.Albrecht, E. and Eschmeier, J., Analytic functional models and local spectral theory, preprint.Google Scholar
3.Albrecht, E., Eschmeier, J. and Neumann, M., Some topics in the theory of decomposable operators, in Operator Theory: Advances and Applications, Vol. 17 (Birkhäuser, Basel, 1986), 15–34.Google Scholar
4.Dunford, N. and Schwartz, J. T., Linears operators, Vol.III (Wiley-Interscience, New York, 1971).Google Scholar
5.Laursen, K. B., Miller, V. G. and Neumann, M. M., Local spectral properties of commutators, Proc. Edinburgh Math. Soc. (2) 38 (1995), 313–329.Google Scholar
6.Laursen, K. and Neumann, M., Asymptotic intertwining and spectral inclusions on Banach spaces, Czechoslovak Math. J. 43, (1993), 483–497.CrossRefGoogle Scholar
7.Miller, T. and Miller, V., Local spectral theory and orbits of operators, preprint.Google Scholar
8.Miller, T., Miller, V. and Smith, R., The Cesàro operator and Bishop's property, J. London Math. Soc. (to appear).Google Scholar
9.Putinar, M., Hyponormal operators are subscalar, J. Operator Theory 12 (1984), 385–395.Google Scholar
10.Radjibalipour, M. and Radjavi, H., On decomposability of compact perturbations of normal operators, Canada, J. Math. 27 (1975), 725–735.Google Scholar
11.Shields, A. L., Weighted shift operators and analytic function theory, Topics in Operator Theory (Pearcy, C., ed.), Mathematical Surveys 13 (AMS, Providence, RI, 1974).Google Scholar