Holomorphy types and the Fourier-Borel transform between spaces of entire functions of a given type and order defined on Banach spaces - Fávaro, Vinícius V.; Jatobá, Ariosvaldo M. (MATHEMATICA SCANDINAVICA, 2012)
Let $E$ be a Banach space and $Θ$ be a $π_{1}$-holomorphy type. The main purpose of this paper is to show that the Fourier-Borel transform is an algebraic isomorphism between the dual of the space ${\operatorname{Exp}}_{Θ,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $Θ$-type strictly less than $A$ and the space ${\operatorname{Exp}}_{Θ^{′},0,(λ (k) A)^{-1}}^{k^{′}}(E^{′})$ of entire functions on $E^{′}$ of order $k^{′}$ and $Θ^{′}$-type less than or equal to $(λ(k)A)^{-1}$. The same is proved for the dual of the space ${\operatorname{Exp}}_{Θ,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $Θ$-type less than or equal to $A$ and the space ${\operatorname{Exp}}_{Θ^{′}, (λ(k)A)^{-1}}^{k^{′}}( E^{′})$ of entire functions on $E^{′}$ of order $k^{′}$ and $Θ^{′}$-type strictly less than $(λ(k)A)^{-1}$. Moreover, the Fourier-Borel transform is proved to be a topological isomorphism in certain cases.