arXiv subjects
Publications and source records attributed to Ariel Monção.
We prove several abstract results giving general conditions under which subspaces of linear or multilinear operators on Banach spaces or Banach lattices are closed. Each of these abstract results is followed by concrete applications, concerning classes of linear/multilinear operators already studied as well as new classes.
Let $X_1, \ldots, X_m$ be Banach spaces and let $E_1, \ldots, E_m,F$ be Banach lattices. Our main results read as follows: (i) The linear adjoint $A^*$ of a continuous multilinear operator $A \colon X_1 \times \cdots \times X_m \to F$ 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 $A \colon E_1 \times \cdots \times E_m \to F$ 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.
We give a necessary condition and a sufficient condition on the Banach lattices E and F so that an operator from E to F is DW-compact whenever its adjoint is DW-compact. We do the same, with different conditions, for DW-DP operators. Moreover, we characterize the Banach lattices E and F for which the adjoint of every DW-compact operator from E to F is DW-compact.