Extended Cauchy-Schwarz inequalities for $\sigma$-elementary transformers in Schatten-von Neumann norm ideals

Abstract

Let $q,r,s⩾1$ satisfy $\frac{1}{2q}+\frac{1}{2r}=\frac{1}{s}$ and $X\in {\mathcal{C}}_s(\mathcal{H})$. If $({\lambda }_n{)}_{n=1}^{\infty },(w_n{)}_{n=1}^{\infty }$ are sequences in $(0,+\infty )$ and $({\lambda }_n^{(1-q)/(2q)}{A}_n{)}_{n=1}^{\infty }$, $({\lambda }_n^{1/(2q)}{A}_n^{\ast }{)}_{n=1}^{\infty },(w_n^{-1/(2r)}{B}_n{)}_{n=1}^{\infty }$ and $(w_n^{(r-1)/(2r)}{B}_n^{\ast }{)}_{n=1}^{\infty }$ are strongly square summable, then there exists $\sideset{^{_{\scriptstyle\,\mathcal{C}_{\large s}\!\!}}}{\phantom{}}{\textstyle\sum_{n=1}^{+\infty}}A_nXB_n$ and \begin{equation*} \begin{split} &\bigg\|\!\!\sideset{^{_{{\scriptscriptstyle\,\Large\mathcal{C}_{\!s}\!\!}}}}{\phantom{}} \sum_{\,\,\,n=1}^{\,\,\,\infty}A_nXB_n\bigg\|_s \\ &\leqslant\bigg\|\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} λ_n^{\frac{1}{q}} A_n A_n^* \bigg\|^{\!\frac{1}{2} - \frac{1}{2q}}\! \bigg\|\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty}w_n^{\!-\frac{1}{r}}\! B_n^* B_n \bigg\|^{\!\frac{1}{2} - \frac{1}{2r}}\! \bigg\|\!\bigg(\!\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} λ_n^{\!\frac{1}{q}-1}\! A_n^* A_n\! \bigg)^{\!\!\frac{1}{2q}}\! X\bigg(\!\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} w_n^{1-\frac{1}{r}}\! B_n B_n^*\! \bigg)^{\!\!\frac{1}{2r}}\! \bigg\|_s\!. \end{split} \end{equation*} Equivalent inequalities are also given, together with some applications to families $(A_n{)}_{n=1}^{\infty }$ and $(B_n{)}_{n=1}^{\infty }$ in $\mathcal{B}(\mathcal{H})$ which are not double square summable. The results presented in this article significantly extends the previous results of authors related to $\sigma$-elementary transformers in Schatten-von Neumann ideals.