# 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 $w(AB)\leqslant\Vert A \Vert w(B)$.

**DOI: **http://dx.doi.org/10.7900/jot.2013jun12.1990

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

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