The Functional Calculus For Numerical Ranges And Numerical Ranges Of Idempotents

Numerical ranges of operators and operator matrices are actively discussed top-ics in operator theory. The research of these subjects has related to algebra, matrix analysis and quantum computation,.ect. In this paper we study some applications of operator theory in numerical ranges. The research methods mainly focus on techniques of block operators. The dissertation studies the functional calculus for numerical ranges and the numerical ranges of idempotents, which is divided into three chapters:In Chapter 1, we introduce some notations, definitions and some conceptions and conclusions in the later chapters.In Chapter 2. by giving the definition of an analytic funtion,it is characterized that the functional calculus for numerical ranges of bounded linear operators on a complex Hilbert space. Tt is proved that for any bounded linear operators A and B on a complex Hilbert space H and a non-linear analytic function g on the numerical range W(B) of B. if (Ax. x)= g((Bx.x)) for all unit vectors x∈H. then both A and B are multiples of identity.In Chapter 3. a relationship between the numerical range of an idempotent and its adjoint operator is given by the technique of block operators and the spectral theorey of operators. And the geomtrical structure of the numerical range of an idempotent is also given.