L- and M-weakly compact multilinear operators and their linear adjoints

2025/11/26 by Geraldo Botelho, Botelho, Geraldo, Ariel Monção +1
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.2511.21358

Abstract

Let X₁, …, X_m be Banach spaces and let E₁, …, E_m,F be Banach lattices. Our main results read as follows: (i) The linear adjoint A^* of a continuous multilinear operator A \colon X₁ × ⋯ × X_m → F is M-weakly compact if and only if A is L-weakly compact. (ii) The linear adjoint A^* of a multilinear operator of order bounded variation A \colon E₁ × ⋯ × E_m → F is L-weakly compact if and only if the linearization of A on the positive projective tensor product is M-weakly compact. In our way to prove these results, we develop the basic theory of linear adjoints of multilinear operators between Riesz spaces, we prove that multilinear operators of order bounded variation between Banach lattices are continuous, and we explore different notions of multilinear operators of M-weakly compact-type.

Citations

• Compact positive multilinear operators on Banach lattices
• Compact bilinear operators and paraproducts revisited
• The positive Schur property on positive projective tensor products and spaces of regular multilinear operators
• Generalized adjoints and applications to composition operators

Related

• On the class of order L-weakly and order M-weakly compact operators
• Compact positive multilinear operators on Banach lattices
• Unbounded M-weakly and unbounded L-weakly compact operators
• Lattice copies and applications for weak L- and M-weakly compact operators on Banach lattices
• Generalizations of L- and M-weakly compact operators