  A partially hyperbolic dynamical system is said to have the quasi-shadowing
property if every pseudotrajectory can be shadowed by a sequence of points
$(x_n)_{n\in \Z}$ 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.
