Ideals of homogeneous polynomials
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 \({\mathcal I}\).
scientific article; zbMATH DE number 6126569
scientific article