Advances in Operator Theory

Class of operators with superiorly closed numerical ranges

Abderrahim Baghdad and Mohamed Chraibi Kaadoud

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‎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 $\mathcal{M}_{2,A,B}$‎.

Article information

Adv. Oper. Theory, Volume 4, Number 3 (2019), 673-687.

Received: 25 June 2018
Accepted: 25 January 2019
First available in Project Euclid: 2 March 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A12: Numerical range, numerical radius
Secondary: 47B15‎ ‎47B20‎ ‎47B47‎

compact operator‎‎ ‎‎spectrum‎ numerical range ‎ ‎spectral radius ‎ ‎numerical radius


Baghdad, Abderrahim; Kaadoud, Mohamed Chraibi. Class of operators with superiorly closed numerical ranges. Adv. Oper. Theory 4 (2019), no. 3, 673--687. doi:10.15352/aot.1806-1387.

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