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

TL;DR

This work extends Cauchy–Schwarz norm inequalities for σ-elementary transformers in Schatten–von Neumann ideals to the setting of asymmetrically weighted double square summable operator sequences. By developing a Hilbert C*-module framework and a suite of weighted, equivalent CS inequalities, it removes commutativity requirements and yields explicit existence and norm bounds for sums of the form $\sum_{n} \lambda_n^{\frac{1}{2}-\frac{1}{2q}} w_n^{\frac{1}{2r}} A_n X B_n$ with $X$ in ${\EuScriptBold C}_{s}(\mathcal{H})$. The main results are complemented by applications to a.w.s.s. operator families, including rank-one constructions based on orthonormal bases, which provide concrete bounds and illustrate connections to hypercontractivity phenomena. The findings broaden the applicability of CS-type norm inequalities in noncommutative operator ideals and offer tools for analyzing asymmetric operator sequences in Schatten norms.

Abstract

Let $q, r, s \geqslant 1$ satisfy $\frac{1}{2q} + \frac{1}{2r} = \frac{1}{s}$ and $X \in \mathcal{C}_s(\mathcal{H})$. If $(λ_n)_{n=1}^{\infty}, (w_n)_{n=1}^{\infty}$ are sequences in $(0,+\infty)$ and $(λ_n^{(1-q)/(2q)} A_n)_{n=1}^{\infty}$, $(λ_n^{1/(2q)} A_n^*)_{n=1}^{\infty}, (w_n^{-1/(2r)} B_n)_{n=1}^{\infty}$ and $(w_n^{(r-1)/(2r)} B_n^*)_{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 $σ$-elementary transformers in Schatten-von Neumann ideals.

Theorems & Definitions (13)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Definition 2.7
• Theorem 2.8
• Theorem 3.1
• Corollary 3.2