Cauchy-Schwarz and means inequalities for elementary operators into norm ideals

Abstract:

The Cauchy-Schwarz norm inequality for normal elementary operators \[ \left \Vvert \sum _{n=1}^\infty A_nXB_n \right \Vvert \leq \left \Vvert (\sum _{n=1}^\infty A_n^*A_n)^{1/2}X (\sum _{n=1}^\infty B_n^*B_n)^{1/2} \right \Vvert , \] implies a means inequality for generalized normal derivations \[ \left \Vvert \frac {AX+XB}2 \right \Vvert \leq \Vvert X \Vvert ^{1-\frac 1r} \left \Vvert \frac {|A|^rX+X|B|^r}2 \right \Vvert ^\frac 1r,\] for all $r\ge 2$, as well as an inequality for normal contractions $A$ and $B$ \[ \left \Vvert (I-A^*A) ^\frac 12X(I-B^*B)^\frac 12\right \Vvert \leq \Vvert X-AXB\Vvert , \] for all $X$ in $B(H)$ and for all unitarily invariant norms $\Vvert \cdot \Vvert .$