Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

Abstract

For a strongly continuous one-parameter group $\{U(t)\} t ∈(−∞,∞)$ of linear operators in a Banach space $\mathfrak{B}$ with generator $A$, we prove the existence of a set $\mathfrak{B}_1$ dense in $\mathfrak{B}_1$ on the elements $x$ of which the function $U(t)x$ admits an extension to an entire B$\mathfrak{B}$-valued vector function. The description of the vectors from $\mathfrak{B}_1$ for which this extension has a finite order of growth and a finite type is presented. It is also established that the inclusion $x ∈ \mathfrak{B}_1$ is a necessary and sufficient condition for the existence of the limit ${ \lim}_{n\to 1}{\left(I+\frac{tA}{n}\right)}^nx$ and this limit is equal to $U(t)x$.