Norm attainment for multilinear operators and polynomials on Banach Spaces and Banach lattices

Abstract

We study norm attainment for multilinear operators and homogeneous polynomials between Banach spaces, as well as for positive multilinear operators between Banach lattices. We establish multilinear and polynomial versions of [23, Theorem B] and [35, Theorem 2.12]. More precisely, we provide sufficient conditions on Banach spaces $X_1,\dots ,X_n$ and $Y$ ensuring that every $A\in \mathcal{L}(X_1,\dots ,X_n;Y)$ (respectively, $P\in \mathcal{P}{(}^n{X}_1;Y)$) is weakly sequentially continuous if and only if it attains its norm. We also obtain analogous results for positive $n$-linear operators and positive $n$-homogeneous polynomials in the setting of Banach lattices.

34 pages