Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals

Document Type: Original Article

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Abstract

If $A,B\in{\mathcal B}({\mathcal H})$ are normal contractions, then for every $X\in {\mathcal C}_{\left|\!\!\;\left|\!\!\;\left|\cdot\right|\!\!\;\right|\!\!\;\right|}({\mathcal H})$ and $\alpha > 0$ holds
\begin{equation}
\biggl\vert\!\biggl\vert\!\biggl\vert \Bigl(I - A^*A\Bigr)^{\frac{\alpha}{2}} X \Bigl(I - B^*B\Bigr)^{\frac{\alpha}{2}} \biggr\vert \!\biggr\vert \!\biggr\vert \leqslant
\biggl\vert\!\biggl\vert\!\biggl\vert \sum_{n=0}^\infty (-1)^n\binom{\alpha}{n}A^n X B^n \biggr\vert \!\biggr\vert \!\biggr\vert,
\end{equation}
which generalizes a result of D.R. Joci'c [Proc. Amer. Math. Soc. 126 (1998), no. 9, 2705--2713] for $\alpha$ not being an integer. Similar inequalities in the Schatten $p$-norms, for non-normal $A,B$ and in the $Q$-norms if one of $A$ or $B$ is normal, are also given.

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