On the Numerical Range of some Bounded Operators

In this paper, we investigate conditions under which the numerical range of a composition operator, acting on a Hilbert space, contains zero as an interior point and we investigate extreme points of the numerical range of an operator acting on an arbitrary Banach space. Also, we give necessary and sufficient conditions under which the numerical range of an operator on some Banach spaces, to be closed. Finally, we characterize the structure of the numerical range of an operator acting on Banach weighted Hardy spaces.