Duality and asymptotic spectral decompositions (Q1819345)

Decomposable operators in the sense of C. Foiaş are operators in a Banach space, for which a reasonably rich supply of invariant subspaces with reasonable spectral localization properties exist. A variety of this notion has been considered in the literature, and the introduction of this paper gives a good review of the main classes of these operators: e.g. strongly, weakly, analytically decomposable operators, and operators having an asymptotic spectral decomposition (ASD). In the paper several other similar classes are also considered, e.g. quasi-decomposable, strongly quasi-decomposable, strongly bi-decomposable and boundedly decomposable operators. 11 theorems are proved on the connections among these classes, using in many cases the duality theory, and assuming in several cases that the underlying space is reflexive. A sample (Theorem 8): Let the space be reflexive. If the operator T has an (ASD) such that each invariant subspace can be chosen strongly analytic, then T is decomposable.