Radical Numerical Range

Operator theory is an important field in the study and discussion of mathematical problems,which reflects the essence of many mathematical problems and has profound research significance.The local spectral radius and numerical range of linear operators are important tools to understand the properties and structures of linear operators.Based on the definition of classes of contractions,local spectral radius and numerical range of linear operators,this paper propose a new concept:Radical numerical range.Definition.A ∈ B(H),the radical numerical range of A is denoted by F(A)={h(A,x),x∈H,x≠0}.where h(A,x)=(?)(|An|x,x>1/n.This paper studies a series of basic properties of radical numerical range for this new concept.The first chapter introduces the idea and background of the new concept of radical numerical range and the structure of this paper.The second chapter gives some basic properties of radical numerical range,including the radical numerical range of multiplicative operators,the range of the radical numerical range of operators,and the radical numerical range of direct sum of finite operators is equal to the union of the radical numerical range of these operators.Finally.It is pointed out that the radical numerical range of unitary equivalent operators is equal.The third chapter introduces the radical numerical range of weighted shift operators,gives the general formula for calculating the radical numerical range of weighted shift operators,the more accurate range of the radical numerical range of weighted shift operators.Gives the relationship between the radical numerical range of injective weighted shift operators and its approximate point spectrum,furthermore,points out that quasi-similar injective unilateral weighted shift operators have equal radical numerical range.Finally.Finally,gives two methods are given to determine whether the bilateral weighted shift operator is strongly irreducible by using the radical numerical range.The fourth chapter introduces the equivalent definition of the radical numerical range of normal operators,and points out that the radical numerical range of quasi-similar normal operators is equal.