Milosevic, S. (2016). Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals. Advances in Operator Theory, 1(2), 147-159. doi: 10.22034/aot.1609.1019

Stefan Milosevic. "Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals". Advances in Operator Theory, 1, 2, 2016, 147-159. doi: 10.22034/aot.1609.1019

Milosevic, S. (2016). 'Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals', Advances in Operator Theory, 1(2), pp. 147-159. doi: 10.22034/aot.1609.1019

Milosevic, S. Norm inequalities for elementary operators related to contractions and operators with spectra contained in the unit disk in norm ideals. Advances in Operator Theory, 2016; 1(2): 147-159. doi: 10.22034/aot.1609.1019

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

Receive Date: 29 September 2016,
Revise Date: 02 December 2016,
Accept Date: 02 December 2016

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.