Power Regularity Of Adjoint Abelian Operators On Banach Space

Local spectra,orbit and invariant subspaces of operators have always been important research contents of operator theory.Power regularity of operators has attracted much attention as an important tool to study local spectral,orbit and invariant subspaces of operators.If {wn}n=1? is a sequence of nonnegative numbers which is submultiplicative,it's equivalent to wm+n?wmwn,(?)m,n,then(?)exists.Obviously,if T ? L(X),where X is Banach space,then the sequence {?Tn?}n=1? is submultiplicative,so sequence {?Tn?1/n}n=1? converges.But for all x in X,the sequence {?Tnx?1/n}n=1? is in general not submultiplicative.In view of this,the definition of power regular operators is given as follows:if the sequence{?Tnx?1/n}n=1? is convergent for all x in X,we shall call operators power-regular.As we all know,the inner-product space has rich geometric properties,so it has always been subject to mathematical researchers.In order to apply this to a normal normed linear space,we defined a semi-inner-product on a normed linear space,which makes it have properties similar to inner-product space,is called semi-inner-product space.It is of great significance to study the properties of adjoint operators,normal operators and numberical ranges and the relationship between the spectrum of operators and numberical ranges.By analogy with the definition of the self-adjoint operator on Hilbert space,the definition of adjoint Abelian operators can be obtained.Let's think about whether adjoint Abelian operators are power regular operators?This is the main research content of this paper.This paper first describes the research history and present situation of the power regular operators,semi-inner-product Spaces and the adjoint abelian operators on Banach space;The second chapter gives some elementary knowledge and the definition of special operators and basic properties which are mentioned in this paper;The third chapter introduces the definition of power regular operators and its general judgment,summarizes some special operators which are power regular operators;In chapter four,the first section mainly introduces the definition of semi-inner-product space and the properties of the numberical ranges in the sense of semi-inner-product and the boundary operator norm formula of semi-inner product space are given.The second section describes the definition and properties of adjoint abelian operators on Banach space.Finally in the third section,we prove the adjoint abelian operators are power regular operators.