Almost summing and multiple almost summing multilinear operators on l p spaces

Abstract

We use the Maurey-Rosenthal factorization theorem to give a characterization of almost summing and multiple almost summing multilinear operators on a cartesian product of lp spaces. As applications we give the necessary and sufficient conditions for the multiplication operator on lp spaces be almost summing and multiple almost summing.Mathematics Subject Classification. Primary 46G25; Secondary 46B25, 46C99.

Constructing hyper-ideals of multilinear operators between Banach spaces

“…The ideal of almost summing linear operators was introduced in [14] and several classes of almost summing multilinear operators have been studied, see, e.g. [4,6,19,21,25]. Such classes often fail to be hyper-ideals: Example 2.9.…”

Multiple almost summing operators

“…The concept of multiple m-linear almost summing was initially introduced in [11] and explored in other papers of which we highlight [18,19]. Let…”

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

“…As usual, it is the inequality that is used in the computations, so it has been taken for grant that almost summing operators are defined by inequality (4). Following this understanding, the classes of almost summing linear and multilinear operators and their relatives have been extensively studied (see, e.g., [8,9,12,27,29] and a very recent contribution in [33]) using inequality (4) or its multilinear version as definition. As to the transformation of vector-valued sequences, since ℓ u 2 (·) < ℓ w 2 (·) and Rad(·) ≺ RAD(·), by Corollary 1.6(iii) the following are equivalent for an n-linear operator…”