Convergence of monomial expansions in Banach spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.46016
author: Andreas Defant
author: Pablo Sevilla-Peris
published in: The Quarterly Journal of Mathematics
publication date: 2012-09-05T00:00:00Z
full work available at URL: http://hdl.handle.net/10251/79774
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 6078028
MaRDI profile type: Publication
OpenAlex ID: W1981918930
DOI: 10.1093/QMATH/HAQ053
title: Convergence of monomial expansions in Banach spaces
review text: A holomorphic function \(f\) from an open subset \(U\) of a Banach sequence space \(E\) into a Banach space \(Y\) has a monomial expansion \(f(z)=\sum_{\alpha\in {\mathbb N}_0^{({\mathbb N})}} c_\alpha(f)z^\alpha\), defined for each \(z\) in \(E\) with finite support. Given a family of holomorphic functions \({\mathcal F}(U,Y)\) from \(U\) into \(Y\), the authors let NEWLINE\[NEWLINE\text{mon}\, {\mathcal F}(U,Y)=\{z\in E: \sum_\alpha \|c_\alpha(f)\||z|^\alpha<\infty \text{ for all } f\in {\mathcal F}(U,Y)\}.NEWLINE\]NEWLINE They then proceed to determine upper and lower bounds for \(\text{mon}\, {\mathcal F} (B_{\ell_r},\ell_q)\) when \({\mathcal F}(B_{\ell_r},\ell_q)\) is a set of bounded holomorphic functions from \(B_{\ell_r}\) into \(\ell_q\) which contains all bounded linear operators and for \({\mathcal P}_p(^m\ell_r,\ell_q)\), the set of all continuous \(m\)-homogeneous polynomials from \({\ell_r}\) into \(\ell_q\) which take values in \(\ell_p\) (\(p\leq q\)). These bounds depend on the relative values of \(r\), \(q\), \(m\), \(p\) and \(2\). In the final section, the authors relate the supremum of those \(r\) so that \(\ell_r\) is contained in \(\text{mon}\{v\circ Q: Q\in {\mathcal P}(\ell_\infty X)\}\) for fixed \(v: X\to Y\) linear with the width of a Bohr strip for two Dirichlet series defined using \(v\).
reviewed by: Christopher Boyd
description: scientific article; zbMATH DE number 6078028