Residually decomposable operators in Banach spaces

TLDR: In this paper, a spectral theory for closed linear operators on a Banach space is presented, which is based on the theory of decomposable operators with functional calculus on their spectrum.

Abstract: In this paper we shall construct a certain spectral theoryfor closed linear operators on a Banach space.These operators have a suitable spectral behaviour on subsets of theirspectra but we must eliminate some residual part which do not offer informationabout the intimate structure of the considered objects, at least from our pointof view.It will be easy to see that this theory contains many examples of operators,bounded or not, having a functional calculus on their spectrum [1], [2], [3], [6],[8], [9].A permanent model for our construction will be the theory of decomposableoperators on a Banach space [7], [2].Throughout this paper the sets of points will be taken in CΌo^Cuf