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On summability of nonlinear operators

Daniel Pellegrino and Joedson Santos

Vol. 6 (2024), No. 1, 115–135
Abstract

The main motivation of this paper is the following general problem: under what (nontrivial) conditions a vector-valued m-linear operator T : p × × p q satisfies a summability property which originally holds for all scalar-valued m-linear forms T : p × × p ? For instance, under what conditions on T : p × × p q, the famous Bohnenblust–Hille inequality and Hardy–Littlewood inequalities for m-linear forms are lifted to T? We prove a general result for nonlinear operators which solves this problem as a very particular case. Our methods encompass Lipschitz operators, m-linear operators and nonlinear operators under mild assumptions. We show that, even in a very nonlinear environment, if the adjoint of T is almost q-summing, then T has the desired property. A straightforward application of our main result provides a generalization of a theorem of S. Kwapień, stated originally for linear operators.

Keywords
summability property, multiple summing operators, nonlinear summing operators, Kwapień's theorem
Mathematical Subject Classification
Primary: 46G25, 47B10, 47H60, 47H99
Milestones
Received: 23 May 2023
Revised: 31 July 2023
Accepted: 14 August 2023
Published: 20 January 2024
Authors
Daniel Pellegrino
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa
Brazil
Joedson Santos
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa
Brazil