  This paper studies holomorphic semicocycles over semigroups in the unit disk,
which take values in an arbitrary unital Banach algebra. We prove that every
such semicocycle is a solution to a corresponding evolution problem. We then
investigate the linearization problem: which semicocycles are cohomologous to
constant semicocycles? In contrast with the case of commutative semicocycles,
in the non-commutative case non-linearizable semicocycles are shown to exist.
Simple conditions for linearizability are derived and are shown to be sharp.
