DUALITY AND ASYMPTOTIC SPECTRAL DECOMPOSITIONS

Asymptotic spectral decomposition for an operator on a Banach space is studied in light of the well-known theory of decomposable operators of Foias type. It is proved that adjoints of strongly quasidecom-posable operators have the single-valued extension property. Duality theorems for strongly decomposable operators are given, for example, an operator has strongly decomposable adjoint iff it has a rich supply of strongly analytic subspaces. For reflexive spaces sharper results are obtained. Decomposable operators are characterized as those quasi-de-composable operators satisfying an additional duality property. Also an asymptotic spectral decomposition with strongly analytic subspaces im-plies decomposability. Strongly bi-decomposable operators are also studied