[Paper Review] Coincidences for multiple summing mappings

Geraldo Botelho, Daniel Pellegrino|ArXiv.org|Sep 24, 2008

Advanced Banach Space Theory4 references5 citations

TL;DR

This paper establishes coincidence results for multiple summing multilinear mappings between Banach spaces with specific cotype properties. It proves that when the domain spaces have cotype 2 or q > 2, the spaces of multiple (q;p₁,…,pₙ)-summing mappings coincide for certain ranges of p and q, extending classical results on absolutely summing operators to the multilinear setting via factorization and Hölder-type estimates in sequence spaces.

ABSTRACT

In this note we prove new coincidence results for multiple summing mappings, related to the cotypes of the Banach spaces involved.

Motivation & Objective

• To investigate the coincidence of multiple summing multilinear mappings in Banach spaces with specific cotype properties.
• To extend classical results on absolutely summing operators to the multilinear setting using cotype assumptions.
• To determine conditions under which different classes of multiple summing mappings coincide, particularly for cotype 2 and q > 2 spaces.
• To provide a unified framework for understanding the interplay between cotype and the coincidence of multiple summing ideals.

Proposed method

• Utilizes the notion of multiple (q;p₁,…,pₙ)-summing mappings defined via uniform estimates over finite families of vectors.
• Applies the factorization of weakly p-summable sequences using the property that l₁^w(E) = l_r' l_r^w(E) when E has cotype q > 2.
• Employs Lemma 1 to show that products of sequences in l_p spaces belong to l_p, enabling control over multilinear terms.
• Applies Hölder's inequality to the product of coefficients and multilinear evaluations to prove integrability in l₁.
• Relies on known results from the literature (e.g., [2, Proposition 6], [4]) to relate weak and absolutely summing sequences.
• Uses duality and interpolation-type reasoning via conjugate exponents to extend results across different p and q ranges.

Experimental results

Research questions

• RQ1 Under what conditions do multiple (q;p₁,…,pₙ)-summing mappings coincide for different values of q and p?
• RQ2 How does the cotype of Banach spaces influence the coincidence of multiple summing mapping ideals?
• RQ3 Can the classical coincidence result for absolutely summing operators be extended to the multilinear setting?
• RQ4 What is the role of the generalized Hölder inequality in controlling multilinear terms in sequence spaces?
• RQ5 How do the results compare with independent findings in the literature, such as those by Popa [5]?

Key findings

• If E₁,…,En have cotype 2, then Π₂ⁿ(E₁,…,En;F) ⊂ Π₁ⁿ(E₁,…,En;F), extending the classical inclusion for linear operators.
• For 1 ≤ p ≤ r < 2, if E₁,…,En have cotype 2, then Π_pⁿ(E₁,…,En;F) = Π_rⁿ(E₁,…,En;F), showing full coincidence in this range.
• If E₁,…,En have cotype q > 2, then Π_sⁿ(E₁,…,En;F) ⊂ Π₁ⁿ(E₁,…,En;F) for all 1 ≤ s < q/(q−1), generalizing the linear case.
• For 1 ≤ s ≤ p < q/(q−1), if E₁,…,En have cotype q > 2, then Π_sⁿ(E₁,…,En;F) = Π_pⁿ(E₁,…,En;F), establishing full coincidence in this range.
• When F also has cotype 2, the coincidence extends to the full range 1 ≤ p ≤ r ≤ 2, so Π_pⁿ(E₁,…,En;F) = Π_rⁿ(E₁,…,En;F).
• The results are shown to be consistent with independent work by D. Popa [5], suggesting robustness of the coincidence phenomena in this setting.