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Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 521-527.

Numerical radius inequalities for products of Hilbert space operators

Authors:  Amer Abu-Omar (1) and Fuad Kittaneh (2)
Author institution: (1) Department of Basic Sciences and Mathematics, Philadelphia University, Amman, Jordan
(2) Department of Mathematics, The University of Jordan, Amman, Jordan

Summary:  New numerical radius inequalities for products of two Hilbert space operators are given. Some of our inequalities improve well-known ones. Among other inequalities, it is shown that if $A,B\in\mathcal{B}(\mathcal{H})$, then $w(AB)\leqslant(\Vert A \Vert +D_{A})w(B)$, where $D_{A}=\underset{z\in \mathbb{C}}{\inf } \Vert A-zI \Vert$. Moreover, $w(AB)\leqslant \Vert A \Vert w(B)+({1}/{2})w(AB-BA^{\ast })$. In particular, if $AB=BA^{\ast}$, then $w(AB)\leqslant\Vert A \Vert w(B)$.

Keywords:  numerical radius, operator norm, inequality, normal operator, self-adjoint operator, positive operator

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