Advances in Operator Theory

Class of operators with superiorly closed numerical ranges

Abderrahim Baghdad and Mohamed Chraibi Kaadoud

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $\mathcal{M}_{2,A,B}$‎.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aot/1551495627

Digital Object Identifier
doi:10.15352/aot.1806-1387

Mathematical Reviews number (MathSciNet)
MR3919038

Zentralblatt MATH identifier
07056792

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

Keywords
compact operator‎‎ ‎‎spectrum‎ numerical range ‎ ‎spectral radius ‎ ‎numerical radius

Citation

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. https://projecteuclid.org/euclid.aot/1551495627


Export citation

References

  • M. Barraa, Convexoid and generalized derivations, Linear Algebra Appl. 350 (2002), 289–292.
  • F. F. Bonsall and J. Ducan, Numerical ranges of operators on normed spaces and elements of normed algebras, London Math. Soc. Lecture Note Series. Cambridge Univ. Press. Cambridge, (1971).
  • A. Bouchen and M. K. Chraibi, Inégalité de Putnam et opérateurs élémentaires, Afr. Mat. 24 (2013), 355–365.
  • R. Bouldin, The numerical range of a product, J. Math. Anal. Appl. 32 (1970), 459–467.
  • M. Boumazgour and H. A. Nabwey, A note concerning the numerical range of a basic elementary operator, Ann. Funct. Ann. 7 (2016), no. 3, 434–441.
  • A. Brown and C. Pearcy, Spectra of tensor products, Proc. Amer. Soc. 17 (1966), 162–166.
  • M. E. Chad, Ranges and kernels of derivations, Turkish J. Math. 41 (2017), 508–514.
  • L. A. Fialkow, Structural properties of elementary operators, in Elementary Operators and Applications (Blaubeuren, 1991), 55–113. World Scientific, River Edge, NJ, USA, 1992.
  • K. E. Gustafson and D. K. M. Rao, Numerical range: The field of values of linear operators and matrices, New York, NY, USA, (1997).
  • P. R. Halmos, Hilbert space problem book, Van Nostrand, (1967).
  • M. C. Kaadoud, Domaine Numérique de l'Opérateur Produit $\m$ et de la dérivation généralisée $ \delta_{2,A,B} $, Extracta Math. 17 (2002), no. 1, 59–68.
  • M. C. Kaadoud, Domaine Numérique du Produit et de la bimultiplication $\m$, Proc. Amer. Math. Soc. 132 (2004), 2421–2428.
  • J. Kyle, Numerical range of derivations, Proc. Edinbrugh Math. Soc. 21 (1978), no. 1, 33–39.
  • G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41.
  • A. Seddik, The numerical range of elementary operators II, Linear Algebra Appl. 338 (2001), 239–244.
  • A. Seddik, The numerical range of elementary operators, Integral Equations and Operator Theory, 43 (2002), 248–252.
  • J. G. Stampfli and J. P. Williams, Growth condition and the numerical range in Banach algebra, Tohuku Math. J. 20 (1968), 417–424.