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 $$ \mathfrak{B} $$ on the elements x of which the function U(t)x admits an extension to an entire $$ \mathfrak{B} $$ -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.