In this short note, we correct and reformulate Theorem 3.1 in the paper published in Proceedings of the Edinburgh Mathematical Society58(3) (2015), 617–629.

In this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two a priori different classes of summing multilinear mappings, and the summability of all multilinear mappings defined on products of Banach spaces. Optimal generalizations of known results are obtained. We also introduce and explore new techniques in the field: for example, a technique to extend coincidence results for linear, bilinear and even trilinear mappings to general multilinear ones.

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

A polynomial P ∈ (kE, F) is left ℓ1-factorable if there are a polynomial Q ∈ (kE, ℓ1) and an operator L ∈ (ℓ1, F) such that P = L ○ Q. We characterise the Radon–Nikodým property by the left ℓ1-factorisation of polynomials on L1(μ). We study the left ℓ1-factorisation of nuclear, compact and Pietsch integral polynomials. For Pietsch integral polynomials, we introduce the left integral ℓ1-factorisation property, obtaining a second polynomial characterisation of the Radon–Nikodým property and showing that it plays a role somehow comparable, in this setting, to nuclearity of operators. A characterisation of 1-spaces is also given in terms of the left compact ℓ1-factorisation of polynomials.

In this paper we provide examples and counterexamples of symmetric ideals of multilinear mappings between Banach spaces and prove that if I1, …, In are operator ideals, then the ideals of multilinear mappings L(I1, …, In) and /I1, …, In/ are symmetric if and only if I1 = … = In.

Let E be a Banach space whose dual E* has the approximation property, and let m be an index. We show that E* has the Radon-Nikodým property if and only if every m-homogeneous integral polynomial from E into any Banach space is nuclear. We also obtain factorization and composition results for nuclear polynomials.

We investigate certain norm and continuity conditions that provide us with ‘uniqe Hahn-Banch Theorems’ from (nc0) to (nℓ∞) and from N(nE) to N(nE″). We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on complex c0 to ℓ∈ but there is no unique norm-preserving extension from (3c0) to (3ℓ∈).