[Paper Review] Absolutely summing operators revisited: new directions in the nonlinear theory

A. Thiago Lopes Bernardino, Daniel Pellegrino+2 more|arXiv (Cornell University)|Sep 22, 2011

Advanced Banach Space Theory57 references3 citations

TL;DR

This paper revisits the theory of absolutely summing operators by introducing new multilinear and polynomial extensions—specifically multiple mixing summing and multiple (p;q;r₁,…,rₙ)-summing operators—using coherence and compatibility as guiding principles. It establishes coincidence theorems, inclusion results, and structural properties, showing that these new classes generalize classical ideals while preserving essential features, offering a more adequate framework than prior approaches.

ABSTRACT

In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Besides this subject's intrinsic mathematical interest, the main motivation is our belief (based on facts that shall be presented) that some of the already existing approaches are not adequate.

Motivation & Objective

• To address perceived inadequacies in existing multilinear and polynomial generalizations of absolutely summing operators.
• To propose new classes of multilinear and polynomial operators—multiple mixing summing and multiple (p;q;r₁,…,rₙ)-summing—based on coherence and compatibility.
• To establish structural properties such as coherence and compatibility for the new operator ideals.
• To investigate coincidence theorems, inclusion theorems, and holomorphic extensions in the nonlinear setting.
• To provide a more mathematically sound and conceptually consistent framework for nonlinear operator ideals.

Proposed method

• Introduces the concept of multiple (s,p;p₁,…,pₙ)-mixing summing multilinear operators via symmetric multilinear extensions.
• Defines the norm ‖·‖_mx(s,p) for polynomials using the associated symmetric multilinear mapping.
• Applies Hölder’s inequality and ε-approximation techniques to derive norm estimates and prove boundedness.
• Uses the notion of coherent and compatible ideals to ensure consistency across different degrees of multilinearity.
• Establishes coincidence theorems by showing that if the full space of multilinear operators equals the ideal, then so do all subspaces of lower degree.
• Leverages known results from linear theory and extends them to nonlinear settings through structural analysis.

Experimental results

Research questions

• RQ1 Are existing multilinear and polynomial extensions of absolutely summing operators conceptually adequate, or do they fail to preserve essential properties of the original ideals?
• RQ2 Can new classes of multilinear and polynomial operators be defined such that they are coherent and compatible across different degrees?
• RQ3 What are the conditions under which coincidence theorems hold for multiple mixing summing and (p;q;r₁,…,rₙ)-summing operators?
• RQ4 How can inclusion theorems and holomorphic extensions be generalized in this new framework?
• RQ5 Can the new classes be shown to generalize classical absolutely summing operators while maintaining structural integrity?

Key findings

• The family of polynomial ideals (P_mx(s,p)^n, ‖·‖_mx(s,p))_n=1^∞ is coherent, ensuring consistency across different degrees of homogeneity.
• For each n, the ideal (P_mx(s,p)^n, ‖·‖_mx(s,p)) is compatible with the operator ideal (Π_mx(s,p), π_mx(s,p)).
• If L(E₁,…,Eₙ;F) = Π_mx(s,q;p₁,…,pₙ)(E₁,…,Eₙ;F), then the same equality holds for all subcollections of indices, establishing a coincidence result.
• For any P ∈ P_mx(s,p)^n(E;F) and a ∈ E, the contraction P_a belongs to P_mx(s,p)^{n-1}(E;F) with ‖P_a‖_mx(s,p) ≤ ‖P‖_mx(s,p)‖a‖.
• The proof of the norm estimate uses ε-approximation and Hölder’s inequality to derive the bound lim_{ε→0} (1+ε)‖A‖_mx(s,q;p₁,…,pₙ) ∏‖(x_i^{(j)})‖_w,p_j.
• The framework provides a solid foundation for further study of inclusion theorems, holomorphic mappings, and variants of mixing summability.