Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n{)}_{n\in \mathbb{Z}}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction.