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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 592–601, May, 2015.
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Horbachuk, V.M., Horbachuk, M.L. Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator. Ukr Math J 67, 668–679 (2015). https://doi.org/10.1007/s11253-015-1106-0
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DOI: https://doi.org/10.1007/s11253-015-1106-0