A note on multiple summing operators and applications

Abstract

We prove a new result on multiple summing operators and among other results applications, we provide a new extension of Littlewood's 4/3 inequality to m-linear forms.2010 Mathematics Subject Classification. Primary 46B25, 47H60.

Summing multilinear operators and sequence classes

“…Pietsch systematized the theory of ideals of linear operators (operator ideals) in [57] and, shortly after, sketched the theory of ideals of multilinear operators in [58]. This field has proved to be quite fruitful and a large number of contributions have appeared in the last decades, see, e.g., [1,3,4,10,14,16,23,24,25,30,39,51,62]. Several authors have devoted their efforts to study multilinear generalizations of the ideals of absolutely (q; p)-summing linear operators to the multilinear context.…”

“…has led to very interesting results, including applications to the study of classical inequalities, such as Bohnenblust-Hille's inequality and Hardy-Littlewood's inequality, see, e.g., [3,4,5,6,7,8,9,15,16,51]. This is the anisotropic case, whereas the other cases are the isotropic ones.…”

Summing multilinear operators by blocks: the isotropic and anisotropic cases

“…Absolutely summing multilinear operators between Banach spaces, as well as closely related classes of multilinear operators, have been studied since the 1983 seminal paper [53] by A. Pietsch. A huge amount of research has been done in the subject since then; to avoid a long list of references we just refer the reader to recent developments that can be found in, e.g., [1,3,5,16,21,22,23,30,37,47,42].…”

“…. This is called the anisotropic approach and has already been considered in [2,3,5,6,8,9,10,15,16,47]. Of course, the case q = q 1 = • • • = q n recovers the isotropic case.…”

Operadores multilineares somantes por blocos e definidos por transformações de sequências vetoriais

“…A teoria de ideais de operadores lineares entre espaços de Banach foi proposta por A. Pietsch em [71] para o caso linear e, pouco tempo depois, foi generalizada por ele mesmo para o contexto multilinear em [72]. Essa área de estudo tem se mostrado bastante frutífera e ganhado muitas contribuições nas últimas quatro décadas (veja, por exemplo, [1,3,4,10,14,16,23,24,25,36,47,61,76] e também os trabalhos referidos nesses artigos). Por exemplo, uma das linhas de desenvolvimento da teoria é criar classes de operadores multilineares cujos casos lineares são ideais de operadores.…”