Inequalities related to Schatten norm

Abstract

In this paper, we investigate the known operator inequalities for the $p$-Schatten norm and obtain some refinements of these inequalities when parameters taking values in different regions.
Let $A_{1},\cdots,A_{n},B_{1},\cdots,B_{n}\in B_{p}(H)$ such that $\Sigma^{n}_{i,j=1}A^{*}_{i}B_{j}=0$. Then $ p\geq 2,~p\leq \lambda$ and $\mu \geq 2$,\begin{eqnarray} &&2^{1/p-\mu/4}n^{3/p-
\mu/4-1/2}(\sum^{n}_{i=1}\|A_{i}\|^{4/\mu}_{p}+
\sum^{n}_{i=1}\|B_{i}\|^{4/\mu}_{p})^{\mu/4}
\nonumber\\ &\leq& n^{2/p-1/2}\|\sum^{n}_{i=1}|A_{i}|^{2}+\sum^{n}_{i=1}|B_{i}|^{2}\|^{1/2}_{p/2} \nonumber\\ &\leq& n^{2/p-2/\lambda}(\sum^{n}_{i,j=1}\|A_{i}\pm B_{j}\|^{\lambda}_{p})^{1/\lambda}. \nonumber \end{eqnarray}
For $0<p\leq 2,~p\geq\lambda>0$ and $0<\mu\leq 2$, the inequalities are reversed.
Moreover, we get some applications of our results.