On solutions of differential equations in Banach space on the whole real axis

Abstract

We consider an equation of the form $\left(\frac{d}{dt} - A\right)^n\left(\frac{d}{dt} + A\right)^my(t) = 0, \ t \in = (-\infty, \infty)$, \ $n, m \in \mathbb{N}_{0} = \{0\}\cup \mathbb{N}, \ n + m \geq 1 $, where $A$ is the generator of a bounded analytic $C_{0}$-semigroup of linear operators in a Banach space, and describe all its solutions on $(-\infty, \infty)$. It is shown that each such solution is an entire vector-valued function, the set of all the solutions forms an infinite-dimensional space, and for its elements the Phragmen-Lindel\"{o}f principle holds true.