Problem
Given a Banach space $E$.
Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$
Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined closed operator.)
Denote the convergence radius by: $$\rho_x:=\left(\limsup_{k\to\infty}\sqrt[k]{\frac{1}{k!}\|A^kx\|}\right)^{-1}$$
Generate a semigroup via Taylor series: $$\rho_x=\infty:\quad e^{tA}x:=\sum_{k=0}^\infty\frac{1}{k!}t^kA^kx$$
Can it happen that it has no entire vectors at all?
Reference
This is the start-up for: Semigroups: Entire Vectors (II)
It is taken from: Engel & Nagel, Exercise 3.12, Page 81