Ideals of mid p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{p}}$$\end{document}-summing operators: a tensor product approach

In this article, we study the ideals of mid p-summing operators and obtain a representation by tensor norms for these operator ideals. These tensor norms are defined using a particular kind of sequential dual of the class of mid p-summable sequences. As a consequence, we prove a characterization of the adjoints of weakly and absolutely mid p-summing operators in terms of the operators that are defined by the transformation of dual spaces of certain vector-valued sequence spaces. Further, we study the natural norms induced on ℓp⊗X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{p}\otimes X$$\end{document} from the class of mid p-summable sequences and its dual sequence class. These norms connect the ideals of mid p-summing operators and the associated tensor norms to the calculus of traced tensor norms which provides several interesting results about these ideals and the norms associated with them.