Cotype 2 estimates for spaces of polynomials on sequence spaces
For a Banach space \(E\) let \({\mathcal P}(^mE)\) be the Banach space of all scalar valued \(m\)-homogeneous continuous polynomials on \(E\) equipped with the topology of uniform convergence on bounded subsets of \(E\). If \(X\) is a Banach sequence space, then \(X_n\) is defined as the subspaces spanned by the sequences \(e_k:=(\delta_{kj})_{j \in {\mathbb{N}}}\), \(1\leq k \leq n\). In the main part of this paper, the authors show that the following conjecture is true for spaces \(X\) which are either 2-convex or 2-concave and have non-trivial convexity. Conjecture. Let \(X\) be a symmetric real Banach sequence space and \(m \geq 2\). Then \[ {\mathbb{C}}_2({\mathcal P}(^mX_n)) \asymp (n^{1/2})^{m-1}{\mathbb{C}}_2(X'_n), \] where \({\mathbb{C}}_2(E)\) denotes the cotype 2 constant of the Banach space \(E\). For this purpose, they prove that the space \({\mathcal P}(^mX_n) = \otimes^{m,s}_{\epsilon_s}X'_n\) (the \(m\)-fold symmetric tensor product of \(X'_n\) equipped with the symmetric injective tensor norm) can be replaced by \(\otimes^{m}_{\epsilon}X'_n\). Several steps are necessary to verify the result mentioned above (Sections 5 and 6 of this paper). In Section 7 it is shown that these results can be transfered to complex Banach spaces. In Section 8 the authors apply their results to special sequence spaces, that is, to \(\ell_p\) (\(1 \leq p \leq \infty\)), Orlicz spaces, and the Lorentz spaces \(d(w,p)\) and \(\ell_{p,q}\). In an Appendix a result is presented, which is not in the main scope of this paper, but it is based on techniques used in this paper and improves a result on cotype 2 estimates for injective tensor products of \(\ell_{p}^n\)'s given by \textit{B. Carl} and \textit{A. Defant} [J. Approximation Theory 88, No.~2, 228-256 (1997; Zbl 0892.46075)].
topology of uniform convergence on bounded subsets
scientific article; zbMATH DE number 1760720
scientific article