“…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…”