Holomorphy types and the Fourier-Borel transform between spaces of entire functions of a given type and order defined on Banach spaces (Q2879619)

The aim of the paper is to extend the results of the first author's paper [Port. Math. (N.S.) 65, No. 2, 285--309 (2008; Zbl 1152.46033)] from the spaces of entire functions of given type and order and their duals defined in terms of classes of \((s;(r,q))\)-quasi-nuclear polynomials to the analogous objects defined by \(\pi_1\)-holomorphy types. They start with a self-contained introduction of the basic concepts from Nachbin's usual holomorphy types as sequences \(\Theta\equiv [P\Theta(\strut^j E; F), \| . \|_\Theta]_{j=0}^\infty\) by spaces of polynomials \(E\to F\) with degree \(j\), giving rise to the Banach spaces \({\mathcal B}_{\Theta,\rho}^k(E)\) of the holomorphic functions \(f:E\to\mathbb C\) such that \(\| f\|_{\Theta,k,\rho} := \sum_{j=0}^\infty \rho^{-j} \left[j/(k e)\right]^{j/k} \| j!^{-1} \widehat{d}^j f(0) \|_\Theta <\infty\). The chief role is played by the inductive limit DF-spaces \(\text{Exp}_{\Theta,A}^k (A) := \bigcup_{\rho<A} {\mathcal B}_{\Theta,\rho}^k(E)\) and the projective limit Fréchet spaces \(\text{Exp}_{\Theta,0,A}^k (A) := \bigcap_{\rho>A} {\mathcal B}_{\Theta,\rho}^k(E)\), respectively. On the basis of a fine characterization of the bounded subsets in \(\text{Exp}_{\Theta,A}^k\) resp. \(\text{Exp}_{\Theta,0,A}^k\), the authors show that the Fourier-Borel transform \({\mathcal F}T(\varphi):=T(e^\varphi)\) \((\varphi\!\in\! E^\prime, \;T\!\in\! [\text{Exp}_{\Theta,A}^k]^\prime)\) establishes an algebraic isomorphism \([\text{Exp}_{\Theta,A}^k]^\prime \to \text{Exp}_{\Theta^\prime,0,1/[\lambda(k)A]}^{k^\prime}\) with \(\lambda(k)\!:=\! k (k \!-\! 1)^{1/k \!-\! 1}\) and the conjugate values \(1/\Theta \!+\! 1/\Theta^\prime \!=\! 1\), \(1/k \!+\! 1/k^\prime \!=\! 1\) whenever \(\Theta\) is a \(\pi_1\)-holomorphy type, i.e., if \(\Theta=({\mathcal P}_\Theta(\strut^j E; F))_{j=0}^\infty\) satisfies \(\| \phi^j \otimes b \|_\Theta = \| \phi\|^j \| b\|\) for all \(\phi\in E^\prime\), \(b\in F\), \(j=0,1,\ldots\), and the finite type elements form a dense submanifold in each space \([{\mathcal P}_\Theta(\strut^j E; F), \|.\|_\Theta]\). Analogous results are also obtained for the spaces \([\text{Exp}_{\Theta,0,A}^k]^\prime\) and sufficient conditions are established for the Fourier-Borel transforms to be a topological isomorphism.