Liv\v{s}ic theorems for Banach cocycles: existence and regularity

We prove a nonuniformly hyperbolic version Liv\v{s}ic 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 a $\mu$-continuous solution of the cohomological equation is actually H"older continuous for the uniform hyperbolic system.