Tameness in Fréchet spaces of analytic functions
Volume 232 / 2016
Abstract
A Fréchet space $\mathcal {X}$ with a sequence $\{\| \cdot \| _k\}_{k=1}^\infty $ of generating seminorms is called tame if there exists an increasing function $\sigma : \mathbb {N} \rightarrow \mathbb {N}$ such that for every continuous linear operator $T$ from $\mathcal {X}$ into itself, there exist $N_0$ and $C \gt 0$ such that \[ \| T(x)\| _n \leq C\| x\| _{\sigma (n)} \hskip 1em \ \forall x \in \mathcal {X},\, n \geq N_0. \] This property does not depend upon the choice of the fundamental system of seminorms for $\mathcal {X}$ and is a property of the Fréchet space $\mathcal {X}$. In this paper we investigate tameness in the Fréchet spaces $\mathcal {O}(M)$ of analytic functions on Stein manifolds $M$ equipped with the compact-open topology. Actually we will look into tameness in the more general class of nuclear Fréchet spaces with properties $\underline {\rm DN}$ and $\Omega $ of Vogt and then specialize to analytic function spaces. We show that for a Stein manifold $M$, tameness of $\mathcal {O}(M)$ is equivalent to hyperconvexity of $M$.
