On Cohen almost summing multilinear operators

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

“…We denote by D n p the ideal of Cohen strongly p-summing n-linear operators (see Introduction (vi)), as usual we write D p in the linear case, and by D n as we denote the ideal of Cohen almost summing n-linear operators (see Example 2.8(c)). One of the main results of [13], namely [13,Theorem 3.4], asserts that D n p ⊆ D n as , a result we improve next. Moreover, the proof we give for the aforementioned inclusion is shorter than the original proof.…”

“…Two main differences arise: linear stability does not imply multilinear stability and the proofs of multilinear stability use ad hoc arguments to each case. As an application of our multilinear stability results we improve a result of [13]. Definition 3.1.…”

Hyper-ideals of multilinear operators and two-sided polynomial ideals generated by sequence classes

“…Given 1 ≤ p < ∞, since the sequence classes ℓ p (•) and ℓ p • are multilinearly stable, from Theorem 2.3 we have that the class L ℓp(•);ℓp • is a Banach hyper-ideal. In the linear case, this class coincides with the Banach operator ideal of Cohen strongly p-summing operators, but in the multilinear case it does not coincide with the well studied class of Cohen strongly p-summing operators (see [2,15,16,17,25,26,31]). In fact, according to [17], an n-linear operator is Cohen strongly p-summing if and only if it sends sequences in ℓ np (•) to sequences in ℓ p • , and not sequences in ℓ p (•) to sequences in ℓ p • .…”

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].…”