Dominated multilinear operators defined on tensor products of Banach spaces

Abstract

In this paper, we introduce and study new classes of dominated multilinear operators, which we call $(p;p_{1},\ldots ,p_{n};G_{1},\ldots ,G_{n})$-dominated and $({\tilde{p}};p_{1},\ldots ,p_{n};G_{1},\ldots ,G_{n})$-dominated multilinear operators defined on the tensor product of Banach spaces. Some characterizations of this type of operators are given and we prove some important coincidence results. As an application, we characterize $(p;p_{1},\ldots ,p_{n})$-dominated multilinear operators on ${\mathcal {C}}(K,G)$ and $(p;p_{1},\ldots ,p_{n})$-dominated multilinear operators in the sense of Dinculeanu on ${\mathcal {C}}(K,G)$, where K is a compact Hausdorff space and G a Banach space. We also treat the connection between an operator T and its associated operators $T^{t},{\tilde{T}}$ and $T^{\#}$ for certain classes.