Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces
A sequence \(\{x_k\}_1^\infty\) in a Banach space \(X\) is said to have a lower \(q\)-estimate \((1<q<\infty)\) if \(|\sum_1^n a_kx_k|\geq C(\sum_{k=1}^n|a_k|^q)^{1/q}\) for a fixed \(C>0\) and for all \(n\in\mathbb{N}\) and arbitrary scalars \(a_k\). Relations between lower \(q\)-estimates and polynomials behaviour are considered. For example: if in \(X\) there is no weakly null normalized sequence with a lower \(q\)-estimate, then every polynomial of degree less than \(q\) on \(X\) is weakly sequentially continuous. Approximation on subspaces of homogeneous polynomials by subsymmetric ones is studied. Applications to algebras generated by polynomials on \(X\) are given.
scientific article; zbMATH DE number 930197