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Livšic theorems for Banach cocycles: existence and regularity

We prove a nonuniformly hyperbolic version Livšic 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 $μ$-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system.