For an equation of the form $({d}/{dt} - A)^{n}({d}/{dt} + A)^{m}y(t)  = 0$, $(n, m \in \mathbb{N}_{0} = \{0\}\textstyle\bigcup \mathbb{N}, n + m \geq 1)$ on the semiaxis or the whole real axis, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup of linear operators on a Banach space, all its solutions are described. It is shown that any solution of the equation under consideration on $(0,\infty)$ is an analytic vector-valued function on this semiaxis, and every its solution on $(-\infty,\infty)$ admits an extension to an entire vectorvalued function. In both cases, an analogue of the Phragmén-Lindelöf principle for the solutions is established.