A-numerical Radius Of Operator Matrices On Semi-Hilbertian Space

The theory of the numerical radius is often used in operator trigonometry,optimization algorithms,the estimation for zeros of polynomials and other fields.This thesis mainly studies the A-numerical radius on semi-Hilbertian space.Because semi-Hilbertian space is induced by semi-inner product,many theories in Hilbert space are not applicable to it.One of the notable points is that operators are not necessarily able to perform conjugate operation on semi-Hilbertian space.But through Douglas theorem,we can find the operator which can carry on conjugate operation.The main object of this thesis is the operator which satisfies the condition.The thesis is divided into four chapters.In Chapter One,we summarize some research results on the-numerical radius in recent years and describe the main work of this thesis.In Chapter Two,we introduce the basic knowledge of semi-Hilbertian space and the-numerical radius.In Chapter Three,we study the A-numerical radius of 2×2 operator matrices on semi-Hilbertian space.Here A=(?)and its diagonal elementis a positive operator.We obtain some inequalities by generalized Cauchy inequality,adjoint operation,matrix decomposition and other techniques for the A-numerical radius.These inequalities give upper bounds of the A-numerical radius for different types of 2×2 operator matrices.Applying these inequalities,we can obtain two inequalities about(·).In particular,we improve an earlier inequality by using a result of this thesis.In addition to the theoretical proof,we also give a numerical example to see it.In Chapter Four,we study the classical numerical radius.In this chapter,we mainly focus on 3×3 and n×n operator matrices.We obtain some estimations for the upper and lower bounds of the numerical radius by polar decomposition,functional calculus and other methods.