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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

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