  In this article, we consider a class of finite rank perturbations of Toeplitz
operators that have simple eigenvalues on the unit circle. Under a suitable
assumption on the behavior of the essential spectrum, we show that such
operators are power bounded. The problem originates in the approximation of
hyperbolic partial differential equations with boundary conditions by means of
finite difference schemes. Our result gives a positive answer to a conjecture
by Trefethen, Kreiss and Wu that only a weak form of the so-called Uniform
Kreiss-Lopatinskii Condition is sufficient to imply power boundedness.
