Bishop's Condition Beta and Decomposable Operators

102 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982.This thesis considers the notions of decomposable operators in the sense of C. Foias, and a certain analytic condition--called condition ((beta))--due to Errett Bishop. It is shown that the concepts are related, and that each can be used to study the other. In particular, it is shown how Bishop's condition can be exploited in the study of decomposable operators.The main part of the thesis is divided into three parts. In the first part, condition ((beta)) itself is studied intensively. The stability of condition ((beta)) under restriction, similarity transformations, the adjoint operation, compact perturbations and other changes of operator is explored. Some necessary conditions for an operator to have condition ((beta)) are established, and its relationship to the related notion of the single-valued extension property is studied.In the second part, results from the first are used to study strongly analytic subspaces in the sense of Lange, and to obtain a characterization of strongly decomposable operators on a reflexive Banach space.In the last part, a subclass of the decomposable operators, called the T-strongly decomposable operators, is studied. Results from the first two parts are used to obtain sufficient conditions for an operator to be T-strongly decomposable.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD