On decomposable operators

Local spectral theory and orbits of operators

“…That decomposable operators have property (β) is due to Albrecht [1]. It is a theorem of Bishop [7] that an operator T ∈ L(X) with (β) has the property that the dual space decomposes as X * = X *…”

“…An operator T on a complex Banach space X is decomposable in the sense of Foiaş [1] provided that whenever…”

Power Regular Operators

“…We recall that according to [1] an operator T in £f(X) is decomposable if for every cover of the complex plane by a pair of open sets U and V, there exist subspaces M and K in Lat(T) such that M + K = X,a(TM) c U, and o-(TK)cV.…”

Bishop's Condition (β)

“…By [1] and [20], it suffices to show that if , T is decomposable if and only if T* is. Thus for a decomposable operator, we have that cr(T) = <x a (T).…”