Noncommutative Holomorphic Semicocycles

Abstract

In this paper, we study holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is the 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 noncommutative case nonlinearizable semicocycles are shown to exist. We derive simple conditions for linearizability and show that they are sharp.