Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces (Q2781302)

An \(s\)-tensor norm of order \(n\) is an assignment to each normed space \(E\) of a norm \(\alpha\) on the symmetric \(n\)-fold tensor product of \(E\), \(\bigotimes_{n,s}E\), such that \(\varepsilon_s\leq \alpha\leq \pi_s\) and for all continuous linear mappings \(T: E\to F\), \(\|\bigotimes_{n,s}T\|\leq \|T\|^n\) where \(\varepsilon_s\) and \(\pi_s\) are respectively the injective symmetric and projective symmetric norms on \(\bigotimes_{n,s}E\) and \((\bigotimes_{n,s} T)(\sum_i\lambda_i x_i\otimes\ldots\otimes x_i)=\sum_i\lambda_i T(x_i)\otimes\ldots\otimes T(x_i)\). NEWLINENEWLINENEWLINEThe authors introduce an ideal \(\mathcal Q\) of \(n\)-homogeneous polynomials as a subclass of \({\mathcal P}^n\), the class of all \(n\)-homogeneous polynomials, which is closed under addition and scalar multiplication, which is closed under composition on the right with linear operators and which contains all \(n\)-powers of linear forms. A normed ideal of \(n\)-homogeneous polynomials is a natural generalisation of the concept of operator ideal. A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal if \(q\in {\mathcal P}(^nE)\) and \(\|q\|_{Q\max}:=\sup\{\|q|_M\|_Q :M\in \text{FIN}(E)\}<\infty\) implies that \(q\in {\mathcal Q}(^nE)\) and \(\|q\|_Q=\|q\|_{Q\max}\). A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable if \(q_i\in {\mathcal Q}(^nE_i)\) and \(\sup_i \|q_i\|_Q<\infty\) implies that \(\lim_{\mathcal U}q_i\in {\mathcal Q}(^n(E_i)_ {\mathcal U})\) and \(\|\lim_{\mathcal U}q_i\|_Q\leq \sup\|q_i\|_Q\). NEWLINENEWLINENEWLINEThe authors show that for a normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) the following are equivalent: (i) \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal; (ii) \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable; (iii) there is a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), such that \({\mathcal Q}(^nE)\) is isometrically isomorphic to \((\bigotimes_{n,s,\alpha} E)'\) for all \(E\). Using this result, it is shown that if \(({\mathcal Q}, \|\cdot\|_Q)\) is a maximal normed ideal of \(n\)-homogeneous polynomials, \(\mathcal U\) is a local ultrafilter on a Banach space \(E\) and \(q\in {\mathcal P}(^nE)\) then \(q\in {\mathcal Q}(^nE)\) if and only if the ultraiterated Aron-Berner extension, \(\overline{q}^{\mathcal U}\), of \(q\) belongs to \({\mathcal Q}(^nE'')\). When this occurs one also has that \(\|\overline{q}^{\mathcal U}\|_Q=\|q\|_Q\). NEWLINENEWLINENEWLINEGiven a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), and a finitely generated tensor norm of order \(2\), \(\beta\), the authors say that \(q\in{\mathcal P}_{\alpha,\beta}(^nE;F)\) if \(\chi_F\circ q\) is in \((\bigotimes_ {n,s}E\otimes_\beta F)'\). (Here \(\chi_F\) is the natural embedding of \(F\) into \(F''\).) It is shown that the vector-valued ideal \({\mathcal P}_{\alpha,\beta}\) is ultrastable. From this it follows that if \(q\in {\mathcal P}(^nE;F)\) and \(\mathcal U\) is a local ultrafilter then \(q\in {\mathcal P}_{\alpha,\beta} (^nE,F)\) if and only if \(\overline{q}^{\mathcal U}\in {\mathcal P}_{\alpha,\beta}(^nE'',F'')\). Analogous results to some of those mentioned above are also obtained for ideals of \(n\)-linear continuous mappings.