Abstract
We employ tools and techniques from multivariable operator theory to obtain new proofs and extensions of well known inequalities regarding the norm and the numerical radius of elementary operators defined on the C⁎–algebra of all bounded operators on Hilbert space, or on the ⁎–ideal of Hilbert-Schmidt operators. In the process, we provide new insights on the study of Heinz-type inequalities related to the arithmetic-geometric mean inequality, and generalize results of several authors, including R. Bhatia, G. Corach, C. Davis, F. Kittaneh, and M.S. Moslehian. To estimate the norm, our approach exploits, in particular, the Spectral Mapping Theorem for the Taylor spectrum, and Ky Fan's Dominance Theorem. For the numerical radius, we use S. Hildebrandt's description of the numerical range of an operator in terms of the norm of its translates.
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