Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator - Gorbachuk, V. M.; Gorbachuk, M. L. (Ukrains’kyi Matematychnyi Zhurnal, 2015)
For a strongly continuous one-parameter group {U(t)} t ∈(−∞,∞) of linear operators in a Banach space B with generator A, we prove the existence of a set B_1 dense in B_1 on the elements x of which the function U(t)x admits an extension to an entire BB-valued vector function. The description of the vectors from 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 ∈ B_1 is a necessary and sufficient condition for the existence of the limit _n→ 1(I+tAn)^nx and this limit is equal to U(t)x.