Extension of vector-valued integral polynomials (Q555825)

\textit{P. Kirwan} and \textit{R. Ryan} [Proc. Am. Math. Soc. 126, 1023--1029 (1998; Zbl 0890.46032)] were perhaps the first to discuss extendibility of polynomials on Banach spaces. An \(n\)-homogeneous polynomial \(P:E \to F\) between Banach spaces is said to be {extendible} if for any larger Banach space \(F \supset E,\) there is \(\widetilde{P} \in {\mathcal P}(^nF,X)\) that extends \(P.\) The space of extendible \(n\)-homogeneous polynomials is denoted \(\mathcal{P}_e(^nE,X).\) The basic question studied here is whether \(\widetilde{P}\) has a particular property if \(P\) does, and the particular property in question is some kind of integral condition. For instance, Pietsch-integral polynomials are always extendible to polynomials that are Pietsch-integral on the larger space. On the other hand, although Grothendieck-integral polynomials are extendible, these extensions are not in general Grothendieck-integral. Also, provided that \(E\) does not contain a copy of \(\ell_1,\) the canonical extension \(\widetilde{P}\) of any (Grothendieck or Pietsch)-integral polynomial with representing measure \(G\) is given by \(\widetilde{P}(z) = \int_{B_E^\prime} z(\gamma)^n dG(\gamma).\) In fact, using results by the first author and \textit{I. Zalduendo} [Proc. Am. Math. Soc. 127, 241--250 (1999; Zbl 0908.46031)], the authors obtain an integral representation for \(\widetilde{P}\) that involves the measures that represent \(P.\)