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
quasidecomposable 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-decomposable operators satisfying an additional duality
property. Also an asymptotic spectral decomposition with strongly analytic
subspaces implies decomposability. Strongly bi-decomposable operators are also
studied.
