Class of operators with superiorly closed numerical ranges

Abstract

‎The aim of this paper is to introduce a class of operators acting on a complex Hilbert space‎. ‎This class contains‎, ‎among others‎, ‎nonzero compact operators‎. ‎We give a characterization of this class in term of generalized numerical ranges and deduce that if $A$ is a compact operator‎, ‎then $ w(A)=\vert \lambda \vert $ with $ \lambda \in\mathit W(A) $‎, ‎where $ \mathit W(A)$ and $ w(A) $ are the numerical range and the numerical radius of $ A $‎, ‎respectively‎. ‎We will give some new necessary conditions for an operator to be compact‎. ‎We also show some light on the generalized numerical ranges of the elementary operators $\delta_{2,A,B}$ and $\mathcal{M}_{2,A,B}$‎.