Livsic theorems for Banach cocycles: Existence and regularity

We prove a nonuniformly hyperbolic version Livgic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, we also prove that a mu-continuous solution of the cohomological equation is actually Holder continuous for the uniform hyperbolic system, where a map is called mu-continuous if there exists a sequence of compact subsets whose union is of mu-full measure, such that the restriction of the map to each of these compact subsets is continuous. (C) 2020 Elsevier Inc. All rights reserved.