The Local Radical Numerical Radius And The Local Spectral Radius Of Operators

The numerical range and local spectrum of operators are important contents in operator theory,which play an important role in the study of spectral structure of operators and invariant subspaces,so they have been widely concerned for a long time.In addition,the concept of Cαβ classification of contractions proposed by Sz.-Nagy and Foias also further promotes the research on the problem of invariant subspaces.Furthermore,the proposal of the concept of power set provides a new idea and method for the study of invariant subspaces of quasi-nilpotent operators.Based on the definition of numerical range and power set,and drawing on the guiding idea of limit differentiation for the Cαβ classification of contractions,professor Ji Youqing proposes the concept of radical numerical range as follows.Definition Let T ∈B(H),x∈H{0}.We define h(T,x)=(?)<|Tn|x,x>1/n,and call h(T,x)the local radical numerical radius of T at x.Moreover,let F(T)={h(T,x):x ≠0},and we call F(T)the radical numerrical range of T.Referring to the research path and method of numerical range,we study the properties of radical numerical range.Besides,the inequality h(T,x)≤r(T,x)is true for arbitrary T ∈B(H).In particular,if T is quasi-normal then the equal sign is established,where r(T,x)is the local spectral radius of T at x.This also makes us consider whether the local radical numerical radius has similar properties to local spectral radius,which may provide a new perspective for the study of local spectrum.On the other hand,according to the Cαβclassification of contractions,we expect to supplement the classification with this new concept.This paper is divided into four parts.The first part is the introduction of the numerical range,power set,the Cαβ classification of contractions and local spectrum,we introduce their research background and some existing research results.The second part studies the properties of the local radical numerical radius by analogy with local spectral radius and studies their relationship of size.The third part calculates the local radical numerical radius of 2 × 2 matrices,and we obtain the characterization of the radical numerical range for 2 × 2 matrices with spectrum of a single point or of real values.The fourth part calculates the local radical numerical radius of injective unilateral weighted shift operators,and studies the condition that the local radical numerical radius is equal to the local spectral radius.