Elementary Operators And Centralizers On Certain Reflexive Algebras

In this thesis, we mainly study some important maps of isolated point algebras and nest algebras, which are elementary operators, centralizers and Jordan centralizers. In the following text, reflexive algebras refer to isolated point algebras. The thesis consists of six chapters.In Chapter I, we introduce some terminology and notation, and summarize the background and the main contents of this paper.In Chapter II, we characterize the surjective elementary operators on standard subalgebras of reflexive algebras on Banach spaces.In Chapter III, we show that an additive mappingφ:A→A is a left centralizer, ifφ( A~2 ) =φ(A ) A,holds for allA∈A. In particular, we characterize the centralizers and prove that every linear local centralize is a centralizer on reflexive algebras.In Chapter IV, we show that an additive mappingφ:A→A is a centralizer, if 2φ( A~2 ) =φ( A ) A + Aφ(A ), holds for allA∈A.In Chapter V, it is proved that an additive mappingφ:A→A of nest algebras is a left centralizer, ifφ( A~2 ) =φ(A ) A, holds for allA∈A.In the last Chapter, we conclude and give some questions.