The Fourier--Borel transform between spaces of entire functions of a given type and order
Given a Banach space \(E\) and a positive integer \(m\), denote by \({\mathcal P}_ {(s,m(r,q))}(^mE)\) (resp., \({\mathcal P}_{\tilde N,(s,(r,q))}(^mE))\) the space of all \((s,m(r,q))\)-summing (resp., \((s;(r,q))\)-quasi-nuclear) \(m\)-homogeneous polynomials on \(E\). For \(\rho>0\) and \(k\geq 1\), the author introduces the Banach spaces \({\mathcal B}^k_{(s,m(r;q)),\rho}(E)\) (resp., \({\mathcal B}^k_{\tilde N,(s;(r;q)), \rho}(E)\)), consisting of all holomorphic function \(f\) on \(E\) such that \(\hat d^mf(0)\) belongs to \({\mathcal P}_{(s,m(r,q))}(^mE)\) (resp., \({\mathcal P}_{\tilde N,(s,(r,q))} (^mE)\)) and \[ \sum_{j=0}^\infty\rho^{-j}\left({j\over ke}\right)^{j/k}\left\|{\hat d^mf(0)\over j!}\right\|_{(s,m(r,q))}<\infty, \] resp., \[ \sum_{j=0}^\infty\rho^{-j}\left({j\over ke}\right)^{j/k}\left\|{\hat d^mf(0)\over j!}\right\|_{\tilde N,(s,m(r,q))}<\infty. \] Taking the inductive limit over all \(\rho<A\) (resp., projective limit over all \(\rho\geq A\)), he obtains the DF-spaces (resp., Fréchet spaces) \(\text{Exp}^k_{(s,m(r;q)),A}(E)\) and \(\text{Exp}^k_{\tilde N,(s,(r,q)),A}(E)\) (resp., \(\text{Exp}^k_{(s,m(r;q)),0,A}(E)\) and \(\text{Exp}^k_{\tilde N,(s,(r,q)),0,A}(E)\)). The main result of this paper is that when \(E'\) has the \(\lambda\)-approximation property, then the mapping \(T\mapsto FT\) defined by \(FT(\phi)=T(e^\phi)\) establishes an algebraic isomorphism between the spaces \([\text{Exp}^k_{{\tilde N},(s,(r,q)),A}(E)]'\) and \(\text{Exp}^{k'}_{(s',m(r';q'))0, \theta(k)A^{-1}}(E')\) and between \([\text{Exp}^k_{{\tilde N},(s,(r,q)),0, A}(E)]'\) and \(\text{Exp}^{k'}_{(s',m(r';q')),\theta(k)A^{-1}}(E')\). Here, \(\theta(k)={k\over (k-1)^{(k-1)/k}}\). Using a description of the bounded sets in these spaces, the author shows that the first of these algebraic isomorphisms is, in fact, topological when the dual is endowed with the strong topology.
scientific article; zbMATH DE number 5296713