A unified Pietsch domination theorem

Abstract

In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc.

Composition results for strongly summing and dominated multilinear operators

“…For the class of strongly -summing operators we have again a Grothendieck-Pietsch domination theorem, see [16,Proposition 1.2] and for more general situations [6,31].…”

Domination spaces and factorization of linear and multilinear summing operators

“…Actually, even the most abstract approaches to the problem are based in this kind of expressions (see [2,16,17]). In [2] the R-S-abstract p-summing mappings were introduced with the aim of unifying the wide variety of summing classes of mappings that can be found in the literature.…”

“…Actually, even the most abstract approaches to the problem are based in this kind of expressions (see [2,16,17]). In [2] the R-S-abstract p-summing mappings were introduced with the aim of unifying the wide variety of summing classes of mappings that can be found in the literature. These abstract approaches provide domination results that recover all the known related dominations for classes of summing mappings in the settings of linear and multilinear operators, homogeneous polynomials, Lipschitz mappings and subhomogeneous mappings among others.…”

“…To do so, we introduce a general family of summability properties for linear and multilinear mappings, that are characterized by means of domination inequalities that always lead to factorization theorems. The related class of summing operators is defined by means of a homogeneous mapping Φ, and can be seen as a subclass of R-S-abstract p-summing mappings in the sense of [2], that includes the classes of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p 1 , . .…”

When is the Haar measure a Pietsch measure for nonlinear mappings?

“…The measure µ above is also called a Pietsch measure for v. The Pietsch Domination Theorem has analogs in different contexts, including versions for classes of absolutely summing nonlinear operators (see, for example, [1,6,7,9,10,15,22]). Recently, in [4,21,23] the concept of abstract R-S-abstract p-summing mapping was introduced in such a way that several previous known versions of the Pietsch Domination Theorem can be regarded as particular instances of one single result.…”