Class of operators with superiorly closed numerical ranges

Document Type: Original Article

Authors

‎University Cadi Ayyad‎, ‎Morocco

Abstract

‎The aim of this paper is to introduce a class of operators acting on a complex Hilbert space‎. ‎This class will contain‎, ‎among others‎, ‎non zero compact operators‎. ‎We will give a characterization of this class in term of generalized numerical ranges‎. ‎We will deduce that if $A$ is a compact operator‎, ‎then $ w(A)=\vert \lambda \vert $ with $ \lambda \in \wa $‎, ‎where $ \wa $ 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 will also show some light on the generalized numerical ranges of the elementary operators $\dd$ and $\m$‎.

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