Some Properties Of C - Valued Domain

In this paper,we first give the definition of C-numerical range and C-numerical radius,and then study the properties of the C-numerical range.According to the content of this paper is divided into the following three chapters:In chapter 1,we give some fundamental knowledge of C-numerical range and the existing theorietical results about C-numerical range.In chapter 2,we have proved the inequality about C-numerical range,If A∈Mn C(A)=diag(a11,a22,…,ann),then wc(C(A))≤wc(A) andIf A1,A2,…,Am∈Mn,A=diag(A1,A2,…,Am)is a quasi-diagonal matrix,thenIf A,B,C,D∈Mn are positive definite matrices,and AB=BA,Am…+Bm…=Dm(m is positive integer),thenIn chapter 3,we first define wc{0}={A∈Mn,0∈wC(A),where C∈Mn},Then we prove the following theorems and results.Let η be an arbitrary non-empty subset of Mn,if U is unitary,then U∈lη if and only if U∈Rη.We also prove that if η is an arbitrary non-empty subset of invertible matrix Mn,then lRlη=lη and RlRη=Rη.Let A∈lWC{0} and U be unitary, then U*AU∈lWCc{0}.At the end of this chapter we prove that lWC{0} is a semigroup which contains the identity matrix E.