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

Let $q,r,s⩾1$ satisfy $\frac{1}{2q}+\frac{1}{2r}=\frac{1}{s}$ and $X\in {\mathcal{C}}_s(\mathcal{H})$. If $({\lambda }_n{)}_{n=1}^{\mathrm{\infty }},(w_n{)}_{n=1}^{\mathrm{\infty }}$ are sequences in $(0,+\mathrm{\infty })$ and $({\lambda }_n^{(1-q)\mathrm{/}(2q)}{A}_n{)}_{n=1}^{\mathrm{\infty }}$, $({\lambda }_n^{1\mathrm{/}(2q)}{A}_n^{\ast }{)}_{n=1}^{\mathrm{\infty }},(w_n^{-1\mathrm{/}(2r)}{B}_n{)}_{n=1}^{\mathrm{\infty }}$ and $(w_n^{(r-1)\mathrm{/}(2r)}{B}_n^{\ast }{)}_{n=1}^{\mathrm{\infty }}$ are strongly square summable, then there exists $\text{sideset}{}^{{}_{{\mathcal{C}}_s}}\phantom{}{\sum }_{n=1}^{+\mathrm{\infty }}{A}_{n}X{B}_n$ 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 $(A_n{)}_{n=1}^{\mathrm{\infty }}$ and $(B_n{)}_{n=1}^{\mathrm{\infty }}$ 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.