Showing papers in "Kyungpook Mathematical Journal in 1993"

(e)-super-decomposable operators

Abstract: The decomposable operator theory was introduced by C. Foias in the 1960’s as an extension of the spectral operator theory which was developed for a period of over thirty years since 1940’s, and since then it has been studied as one of the key research topics. In 1979, E. Albrecht discovered that the definition of decomposable operator can be replaced by a more weakened condition. Weaker conditions then appeared in the decomposable operator are weakly decomposable operator and analytically decomposable operator, and stronger conditions then appeared in the decomposable operator are strongly decomposable operator. Many research results were produced on these theories. At the same time, theories on classification of invariant subspace problem were developed. In this paper, motivated by [7] we introduces the notion of an (E)-superdecomposable operator which is a type of a super-decomposable operator and solves the problems on dual operator and restriction and quotient operators which has not been solved under strongly decomposable operator theory. At the same time, some interesting research results are disclosed. Throughout this paper we shall use the standard notions and some basic results of the theory of decomposable operators as presented in [8] and [10]. Let .X/ be the space of all continuous linear operators on a complex Banach space X. For an operator T 2 .X/, Lat.T/ stands for the collection of all closed T -invariant linear subspaces of X. Also, .C/ stands for the family of closed subsets ofC: T denotes the dual operator of T2 .X/ .I f Y is a closed T -invariant subspace, we write TjY for the restriction and T for the operator induced by T on X=Y: For Y X ,l et Y ? be its annihilator in X. We use. T/for spectrum of T and.T/ for its resolvent set. We put Y for the closure of Y in appropriate topology. An operator T has the single-valued