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A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space E, the infumum of the q such that the identity map idE\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$id_{E}$$\end{document} is absolutely q,1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( q,1\right) $$\end{document}-summing is precisely cotE\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cot E$$\end{document}. In the same direction, the Dvoretzky–Rogers Theorem asserts idE\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$id_{E}$$\end{document} fails to be absolutely p,p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( p,p\right) $$\end{document}-summing, for all p≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\ge 1$$\end{document}. In this note, among other results, we unify both theorems by charactering the parameters q and p for which the identity map is absolutely q,p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( q,p\right) $$\end{document}-summing. We also provide a result that we call strings of coincidences that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending a classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień.
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Mar 5, 2020
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