Noninvertible cocycles: robustness of exponential dichotomies
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.
scientific article; zbMATH DE number 6104843