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

Mathematics – Functional Analysis

Scientific paper

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18 pages

Scientific paper

Given an operator ideal, how to consider a natural extension to the polynomial and multilinear settings? How can we evaluate if an extension is more adequate than other? In the last years some efforts have been done in this direction; the interesting concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal were recently introduced by D. Carando, V. Dimant and S. Muro in order to face such questions for polynomials. We propose a different approach by considering pairs $(\mathcal{U}_{k},\mathcal{M}_{k})_{k=1}^{\infty}$, where $(\mathcal{U}_{k})_{k=1}^{\infty}$ is a polynomial ideal and $(\mathcal{M}_{k}%)_{k=1}^{\infty}$ is a multi-ideal, instead of considering just polynomial ideals. Our approach, in particular, makes the pair $(\mathcal{P}%_{k},\mathcal{L}_{k})_{k=1}^{\infty}$ (composed by the ideals of continuous $k$-homogeneous polynomials and continuous $k$-linear operators with the $\sup$ norm) coherent and compatible with the ideal of continuous linear operators with the $\sup$ norm. This result, albeit strongly expected, differs from the original approach, since for real scalars it is surprising that the sequence $(\mathcal{P}_{k})_{k=1}^{\infty}$ is not coherent according to the definition of Carando \emph{et al.}

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