Korean J. Math.  Vol 27, No 4 (2019)  pp.879-898
DOI: https://doi.org/10.11568/kjm.2019.27.4.879

On numerical range and numerical radius of convex function operators

Zaiz Khaoula, Mansour Abdelouahab


In this paper we prove some interesting inclusions concerning the numerical range of some operators and the numerical range of theirs ranges with a convex function. Further, we prove some inequalities for the numerical radius. These inclusions and inequalities are based on some classical convexity inequalities for non-negative real numbers and some operator inequalities.


Numerical range, numerical radius, convex operator function, self-adjoint operator.

Subject classification

47A12, 47A30, 47B15


This work was supported by Laboratory of operator theory, Algeria.

Full Text:



Fadi Alrimawi, Omar Hirzallah, and Fuad Kittaneh, Norm inequalities involving convex and concave functions of operators, Linear and Multilinear Algebra, (2018), (Google Scholar)

Richard Bouldin, The numerical range of a product, ii, Journal of Mathematical Analysis and Appli-cations, 33(1971), 212-219. (Google Scholar)

Sever S. Dragomir, Hermite-hadamards type inequalities for convex functions of selfadjoint operators in hilbert spaces, Linear Algebra and Its Applications, 436(2012), 1503-1515. (Google Scholar)

Sever S. Dragomir, Inequalities for the numerical radius of linear operators in Hilbert spaces, Springer,2013. (Google Scholar)

C.K. Fong and John A.R. Holbrook, Unitarily invariant operator norms, Canad. J. Math., 35(1983), 274-299. (Google Scholar)

Karl E. Gustafson and Duggirala K.M. Rao, Numerical range, Springer, 1997. (Google Scholar)

Paul Richard Halmos, A Hilbert space problem book, Springer Science and Business Media, 1982. (Google Scholar)

Frank Hansen and Gert K. Pedersen, Jensens inequality for operators and Lowners theorem, Mathe-matische Annalen, 258(1982), 229-241. (Google Scholar)

Fumio Hiai, Matrix analysis: matrix monotone functions, matrix means, and majorization, Interdis-ciplinary Information Sciences, 16(2010), 139-248. (Google Scholar)

John A.R. Holbrook, Multiplicative properties of the numerical radius in operator theory, J. reine angew. Math., 237(1969), pp.166-174. (Google Scholar)

Fuad Kittaneh, Notes on some inequalities for hilbert space operators, Publications of the Research Institute for Mathematical Sciences, 24(1988), 283-293. (Google Scholar)

Fuad Kittaneh, Numerical radius inequalities for hilbert space operators, Studia Mathematica, 168(2005), pp.73-80. (Google Scholar)

B. Mond, J.E. Pecaric, Convex inequalities in hilbert space, In Houston Journal of Mathematics, Volume 19, No. 3, 1993, pp. 405-420. (Google Scholar)

J. Pemcarisc, T. Furuta, J. Miscisc Hot, and Y. Seo, Mondpencariec method in operator inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. (Google Scholar)

R.P. Phelps, Convex functions, monotone operators, and differentiability, second ed. Lecture Notes in Mathematics, 1964. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr