Growth order of an operator exponential of entire vectors (Q1812405)

Let \(A\) be a closed operator in a Banach space. A vector \(x \in D(A^n)\), \(n=0,1,\dots,\) is called an entire vector of \(A\) if the series \(\exp(\lambda A) x=\sum_{n=0}^{\infty} \lambda^n A^n x/ n!\) is convergent for all \(\lambda \in \mathbb{C}\). The aim of this paper is to describe the dependence of order and type of entire vector functions \(\exp(\lambda A)x\), \(\lambda \in \mathbb{C}\), for nonzero entire vectors \(x\) on the degree of smoothness of \(x\) with respect to \(A\).