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

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions