Ideals of homogeneous polynomials
type: Item
instance of: scholarly article
title: Ideals of homogeneous polynomials
zbMATH Open document ID: 1270.46038
DOI: 10.2977/PRIMS/93
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2013-01-17T00:00:00Z
review text: An ideal \({\mathcal Q}\) of homogeneous polynomials is an assignment to each positive \(n\) and each pair of Banach spaces \(E\) and \(F\) of a subspace \({\mathcal Q}(^nE,F)\) of \({\mathcal P}(^nE,F)\), the space of all continuous \(n\)-homogeneous polynomials from \(E\) into \(F\) such that (i) \({\mathcal Q}(E,F)\) contains all finite rank operators from \(E\) into \(F\); (ii) for any pair of Banach spaces \(G\) and \(H\) and any pair of continuous linear operators \(u: G\to E\) and \(v: F \to H\), the composition \(v\circ P\circ u\) belongs to \({\mathcal Q}(^nG,H)\) whenever \(P\) belongs to \({\mathcal Q}(^nE,F)\). Given an operator ideal \({\mathcal I}\), the authors introduce the ideal of \({\mathcal I}\)-bounded polynomials, \({\mathcal P}_{\mathcal I}\), as those homogeneous polynomials \(P: E\to F\) with the property that for every \(x\) in \(E\) there is a neighbourhood \(V_x\) of \(x\), a Banach space \(Z_x\) and \(T_x\in {\mathcal I}(Z_x,F)\) such that \(P(V_x)\subseteq T_x(B_{Z_x})\). Endowed with a suitable norm, \({\mathcal P}_{\mathcal I}\) becomes a Banach polynomial ideal whenever \({\mathcal I}\) is a Banach ideal. The stability of \({\mathcal I}\)-bounded polynomials under composition with arbitrary polynomials is discussed and a condition on an operator ideal \({\mathcal I}\) is given that ensures that the ideal of \({\mathcal I}\)-bounded polynomials, \({\mathcal P}_{\mathcal I}\), coincides with the polynomial ideal \({\mathcal I} \circ {\mathcal P}\) of all homogeneous polynomials that can be written as a composition \(T\circ Q\), where \(Q\) is a homogeneous polynomial and \(T\) is a linear operator belonging to \({\mathcal I}\).
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 47L22
zbMATH DE Number: 6126569
zbMATH Keywords: polynomial ideal
zbMATH Keywords: \(p\)-compact polynomials
zbMATH Keywords: composition ideal
author: Richard M. Aron
author: Pilar Rueda
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.2977/prims/93
OpenAlex ID: W2014681918
description: scientific article; zbMATH DE number 6126569