Eigenvalues, numerical radius and norms of operators inequalities in Hilbert space

Abstract

Our main target in this research is to refine some well-known numerical radius inequalities of operators on a Hilbert space or to obtain new bounds. In this thesis, we establish some bounds for the numerical radius of one operator and for the numerical radii of 2×2 operator matrices. Also, we provide new upper bounds for the numerical radii of n x n operator matrices. Applying some of our results, we have succeeded to give a new bound for the zeros of polynomials. Furthermore, we present an improvement of the triangle inequality for the operator norm. On the other hand, we establish several upper and lower bounds for the p-numerical radius of one operator and of 2 x 2 operator matrices as well as of n x n operator matrices. An application to 2-nilpotent operators is provided. A p-numerical radius inequality involving power is also given.