Convergence of monomial expansions in Banach spaces
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 \(v\).
scientific article; zbMATH DE number 6078028