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

Abstract

If $A,B \in \mathcal {B}(\mathcal {H})$ are normal contractions, then for every $X \in \mathcal {C}_{||| \cdot |||}(\mathcal {H})$ and $\alpha > 0$ holds $$\Big|\Big|\Big| (I - A^{*}A)^{\frac{\alpha}{2}} X(I - B^{*}B)^{\frac{\alpha}{2}} \Big|\Big|\Big| \leqslant \Big|\Big|\Big| \sum_{n=0}^\infty (-1)^{n} \binom{\alpha}{n}A^{n} X B^{n} \Big|\Big|\Big|,$$ which generalizes a result of D.R. Jocić [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.