Invariance of the hypercyclicity criterion under sequence equivalence and scaling

Determine whether the condition \(\liminf_{p\to\infty} M_p/p!=0\) is preserved under equivalence of weight sequences and whether, for a weight sequence \(\mathbf{M}\) satisfying this condition, the scaled sequence \(\mathbf{M}^c=(c^pM_p)_{p\in\mathbb{N}}\) also satisfies it for some or every \(c\neq 1\).

Background

For a log-convex weight sequence $\mathbf{M}$, the paper characterizes hypercyclicity of the differentiation operator on $H^0_{v_{\mathbf{M}}}(\mathbb{C})$ through the condition $\liminf_{p\to\infty}M_p/p!=0$. The paper observes that this condition is preserved under uniformly bounded multiplicative changes of the sequence, but does not establish its behavior under the weaker equivalence relation $\mathbf{M}\approx\mathbf{N}$.

The paper also studies the scaling operation $\mathbf{M}\mapsto\mathbf{M}^c$, which corresponds to rescaling the associated weight. Examples later in the section show that the condition can fail after a particular scaling, motivating the unresolved question of whether any general preservation principle exists.