[Paper Review] Some remarks on absolutely summing multilinear operators

A. Thiago Lopes Bernardino, Daniel Pellegrino|arXiv (Cornell University)|Jan 11, 2011

Advanced Banach Space Theory22 references3 citations

TL;DR

This paper resolves a gap in recent work on multilinear absolutely summing operators by showing that Pietsch's composition theorem naturally extends to the multilinear and polynomial settings, a fact overlooked in prior literature. The key contribution is a new inclusion result: if one factor space has cotype 2, then absolutely $(s;s,t,...,t)$-summing operators are also absolutely $(p;p,t,...,t)$-summing for $1 \leq p \leq s \leq 2$, generalizing earlier results and providing a partial solution to a problem posed by Popa.

ABSTRACT

This short note has a twofold purpose: (i) to solve the question that motivates a recent paper of D. Popa on multilinear variants of Pietsch's composition theorem for absolutely summing operators. More precisely, we remark that there is a natural perfect extension of Pietsch's composition theorem to the multilinear and polynomial settings. This fact was overlooked in the aforementioned paper; (ii) to investigate extensions of some results of the aforementioned paper for particular situations, mainly by exploring cotype properties of the spaces involved.

Motivation & Objective

• To resolve a key oversight in a recent paper by D. Popa concerning multilinear extensions of Pietsch's composition theorem.
• To establish a natural multilinear generalization of Pietsch's composition theorem, showing that $\Pi_q \circ \delta_p^n \subset \Pi_r^n$ holds under broader conditions than previously known.
• To investigate how cotype properties of Banach spaces influence the inclusion relations between absolutely summing multilinear operators.
• To provide a partial solution to Problem 1.4 from Popa's paper using a simple but overlooked observation on the inclusion $\delta_p^n \subset \Pi_r^n$ when $r \leq p$.

Proposed method

• Leverages recent results from Blasco et al. on cotype and weakly summable sequences to derive inclusion theorems for multilinear operators.
• Applies the duality between weak $\ell_p$-summability and cotype, using the fact that $\ell_p^w(X) = \ell_r \ell_q^w(X)$ when $X$ has cotype $s$ and $1/p = 1/r + 1/q$ with $p < s^*$.
• Uses the characterization of $q$-dominated operators as $\delta_q^n = \Pi_{q/n; q}^n$ to relate different classes of multilinear operators.
• Applies the inclusion $\delta_p^n \subset \Pi_r^n$ for $r \leq p$ to derive a partial solution to Problem 1.4, valid for $n \geq p/r$.
• Combines known results on cotype 2 spaces (e.g., $\Pi_2^{\text{mult},n} = \Pi_r^{\text{mult},n}$ for $r \leq 2$) with composition theorems to strengthen inclusion chains.
• Employs the duality between $\ell_p^w(X)$ and $\ell_q^w(X^*)$ to analyze the behavior of multilinear operators under composition with absolutely summing linear maps.

Experimental results

Research questions

• RQ1 Does Pietsch's composition theorem for absolutely summing operators extend naturally to the multilinear and polynomial settings, as suggested by the structure of the classes involved?
• RQ2 Under what conditions on the cotype of the domain spaces does the inclusion $\Pi_{(s;s,t,...,t)}^n \subset \Pi_{(p;p,t,...,t)}^n$ hold for $1 \leq p \leq s \leq 2$?
• RQ3 Can the inclusion $\Pi_q \circ \delta_p^n \subset \Pi_r^n$ be established for $r > 2$, and if so, for which values of $n$?
• RQ4 Is there a simple, overlooked argument that can resolve Problem 1.4 from Popa's paper, particularly for $n \geq p/r$?
• RQ5 How do cotype properties of the spaces $X_j$ influence the relationship between multiple and absolutely summing multilinear operators?

Key findings

• If $X_1$ has cotype 2 and $1 \leq p \leq s \leq 2$, then every absolutely $(s;s,t,...,t)$-summing multilinear operator from $X_1 \times \cdots \times X_n$ to $Z$ is also absolutely $(p;p,t,...,t)$-summing for all $t \geq 1$ and all $X_2, \dots, X_n, Z$.
• In particular, under the same hypotheses, every absolutely $(s;s,\dots,s)$-summing multilinear operator is absolutely $(p;p,\dots,p)$-summing.
• A similar result holds when $X_j$ has cotype 2 for any $j \neq 1$, generalizing results of Junek, Botelho, and Popa.
• If $X_1$ has cotype $s > 2$ and $1 \leq p \leq q < s^*$, then $\Pi_{(q;q,t,...,t)}^n \subset \Pi_{(p;p,t,...,t)}^n$ for all $t \geq 1$, extending the inclusion to higher cotype spaces.
• A simple argument shows that $\delta_p^n \subset \Pi_r^n$ whenever $r \leq p$, which implies that $\Pi_q \circ \delta_p^n \subset \Pi_r^n$ for all $n \geq p/r$, providing a partial solution to Problem 1.4.
• When $X_j$ has cotype 2 for some $j$ and $Y$ has cotype 2, then $\Pi_q(Y;Z) \circ \bigcup_{p \geq 1} \delta_p^n(X_1,\dots,X_n;Y) \subset \Pi_r^n(X_1,\dots,X_n;Z)$ for all $q \in [1,\infty)$ and $r \in [1,2]$, strengthening known inclusion chains.