On invariance of the numerical range and some classes of operators in Hilbert spaces

Abstract

In this thesis the invariance of the numerical range under isometric similarity is examined. It is a well known fact that unitarily similar operators have the same numerical range. Chapter one is devoted to basic definitions and some well known results on the numerical range. In chapter two, some conditions are examined for operators that are isometrically equivalent to have the same numerical range In chapter three the norm properties of an operator with a norm attaining vector are examined.It is found that such an operator satisfies the generalized Daugavet equation. Chapter four draws some conditions for two 0perator's to commute. up to a scalar factor. Also in this chapter the numerical range of a Hyponormal operator in finite dimensional spaces is shown to be polygonal in the complex plane.The vertices of the polygon are the eigenvalues of the operator. The thesis was supervised by Prof J.M.Khalaghai and Prof. G.P.Pokhariyal.