Quasinormable weighted Fréchet spaces of entire functions (Q2389177)

The quasinormability of weighted Fréchet spaces of entire functions is studied. Given an increasing sequence \(W=(w_n)_n\) of strictly positive continuous weights on \(\mathbb C\), denote by \(HW(\mathbb C)\) the intersection of the Banach spaces \(Hw_n(\mathbb C)\) of all entire functions \(f\) such that \(w_n| f| \) is bounded in \(\mathbb C\), and denote by \(HW_0(\mathbb C)\) the intersection of the Banach spaces \(H(w_n)_0(\mathbb C)\) of all entire functions \(f\) such that \(w_n| f| \) vanishes at infinity on \(\mathbb C \). Given constants \(A>0\) and \(a>0\), we say that a continuous, radial, strictly positive weight \(w(r):=v(r)e^{-ar}\) (\(r \geq 0\)) belongs to the class \((E)_{a,A}\) introduced by \textit{K.\,D.\thinspace Bierstedt}, \textit{J.\,Bonet} and \textit{J.\,Taskinen} in [Monatsh.\ Math.\ 154, No.\,2, 103--120 (2008; Zbl 1182.46013)] if the following hold: \(v\) is differentiable, strictly increasing, and \( {rv'(r)} \leq {Aav(r)}\) for all \(r\geq 0\). Under the assumption that the system of weights belongs to the class \((E)_{a,A}\), it is proven, among other characterizations in terms of the weights, that the following are equivalent: (1) \(HW(\mathbb C)\) is quasinormable, (2) \(HW_0(\mathbb C)\) is quasinormable, (3) \(HW_0(\mathbb C)\) is quasinormable by operators, and (4) for every \(m \in \mathbb N\), there is \(n > m\) such that, for every \(p \geq m\) and \(\varepsilon > 0\), there is \(\lambda >0\) such that \( \left( 1/w_n \right)\tilde{} \leq (\lambda/w_p) + (\varepsilon/w_m) \) on \(\mathbb C\), where \(\tilde{w}\) stands for the associated growth defined for a weight \(w\) by \(\tilde{w}(z) :=\sup\{| g(z)| : \text{ }g\) is entire and \(| g | \leq v \)