Cohen and multiple Cohen strongly summing multilinear operators

Abstract

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.

Generalized Multiple Summing Multilinear Operators on Banach Spaces

“…. , E m ; F ), was studied in [12]. In our abstract approach, it is recovered by choosing γ s k (·) = ℓ p (·) and γ s (·; N n ) = ℓ p ·; N n , where 1 < p < ∞ and k = 1, .…”

“…Several aspects of the theory of summing multilinear operators show that the class of multiple summing multilinear operators is one of the most suitable and useful approaches to the nonlinear theory of absolutely summing operators. Details can be found, e.g., in [1,5,9,12,16,22,23,19].…”

On the transformation of vector-valued sequences by linear and multilinear operators

“…(vi) Cohen strongly p-summing multilinear operators send absolutely np-summable sequences to Cohen strongly p-summable sequences [1,14,25] (the case n = 1 recovers the ideal of Cohen strongly p-summing linear operators [17]).…”

Factorization of strongly (p,σ)-continuous multilinear operators

“…In [19,20], inclusions between the class of Cohen strongly summing multilinear operators and other classes of operators were systematically analyzed. A related concept and a new generalizations of the concept of Cohen strongly summing multilinear operators have also been recently studied in [8,7,2,3]). For more details concerning the nonlinear theory of summing operators and recent developments and applications we refer to [1,10].…”