Abstract
An operator T means a bounded linear transformation on a complex Banach space X. For an operator T and for a closed subset F of the complex plane C, we let {X_T}(F) = \left{ {x \in X{: there exists an analytic function f|C\backslash F} → {X such that (z - T)f(z)} ≡ x} \
ight}, and if E is an arbitrary subset of C, we let {X_T}(E) = \bigcup {\left{ {{X_T}(F):F \subset E{ and }F{ is closed}} \
ight}}. If {X_T}(E) is closed for all closed subsets F of C, we say that T satisfies the closure condition (C). In this paper, we show that an operator T is decomposable if and only if (1) T satisfies the closure condition (C) and (2) {X_T}({G_1} \cup {G_2}) = {X_T}({G_2}) for any pair of open subsets {G_1} and {G_2} of C. This is a generalization of Plafker's result in [5] for strongly decomposable operators. And we show some applications of this result.
