On the Study of Eigenvalues and Operator Matrices Inequalities.

Abstract

Our main goal in this research is to refine some well-known numerical radius in-equalities of operators on a Hilbert space or to discover new bounds for the numerical radius. In this thesis, after expressing concepts and prerequisites, we give some new upper bounds for the numerical radius of operators as well as for the numerical radii of 2 × 2 operator matrices. Also, we improve the triangle inequality of the operator norm. We refine some earlier existing bounds of the numerical radius. Furthermore, we derive some new Hilbert-Schmidt numerical radius inequalities for operators as well as for 2 × 2 operator matrices. Some of these inequalities refine some existing ones. Then we define a new norm and we study the basic properties of this norm. Finally, we provide new upper and lower bounds for the p-numerical radius of operators as well as for 2 × 2 operator matrices.