On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

What is an adequate extension of an operator ideal I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} to the polynomial and multilinear settings? This question motivated the appearance of the interesting concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando et al. We propose a different approach by considering pairs (Uk,Mk)k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{U }_{k},\mathcal{M }_{k})_{k=1}^{\infty }$$\end{document}, where (Uk)k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{U }_{k})_{k=1}^{\infty }$$\end{document} is a polynomial ideal and (Mk)k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{M }_{k})_{k=1}^{\infty }$$\end{document} is a multi-ideal, instead of considering just polynomial ideals. It is our belief that our approach ends a discomfort caused by the previous theory: for real scalars the canonical sequence (Pk)k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{P }_{k})_{k=1}^{\infty }$$\end{document} of continuous k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-homogeneous polynomials is not coherent according to the definition of Carando et al. We apply these new notions to test the pairs of ideals of nuclear and integral polynomials and multilinear operators, the factorisation method and different classes that generalise the concept of absolutely summing operator.