Properties of iteration of toeplitz operators with toeplitz preconditioners

We consider the problems of preconditioning and iterative inversion of Toeplitz operators on sequences of complex numbers. We divide the preconditioned operator into two parts, of which one is compact and the other is regarded as a small perturbartion. It will be shown that the Krylov subspace methods (such as GMRES) will perform initially at superlinear speed when applied to such a preconditioned system. However, with large iteration numbers, the speed will settle down to linear order. Most of our results are stated in terms of the symbol of the Toeplitz operator in question.