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

TL;DR

This paper investigates conditions under which multilinear operators and polynomials on Banach spaces and lattices attain their norms, extending existing theorems to new settings and classes of operators.

Contribution

It establishes multilinear and polynomial versions of key theorems, providing criteria for norm attainment and weak sequential continuity in Banach spaces and lattices.

Findings

Sufficient conditions for norm attainment of multilinear operators.

Extension of theorems to positive operators on Banach lattices.

Characterization of weakly sequentially continuous operators.

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.