Noninvertible cocycles: robustness of exponential dichotomies (Q1760074)

In this paper, considering discrete linear evolution processes (or cocycles) \(\{\mathcal{A}(m,n): m\geqslant n\}\) given by \(\mathcal{A}(m,n)=A_{m-1}\dots A_n\) if \(m>n\) and \(\mathcal{A}(m,m)=\operatorname{Id}\), for all \(m\in \mathbb{Z}\), where \((A_n)_{n\in \mathbb{Z}}\) is a family of bounded linear operators in a Banach space \(X\) and \(\operatorname{Id}\) is the identity operator in \(X\), the authors prove the robustness of uniform and nonuniform exponential dichotomies under small linear perturbations \((A_n+B_n)_{n\in \mathbb{Z}}\), with \((B_n)_{n\in \mathbb{Z}}\) being also a family of small (in some sense) bounded linear operators in \(X\). The results contemplate the case of discrete linear evolution processes which are not necessarily invertible in the stable direction.