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This paper is a continuation of our study of a class of Toeplitz-likeoperators with a rational symbol which has a pole on the unit circle. Adescription of the spectrum and its various parts, i.e., point, residual andcontinuous spectrum, is given, as well as a description of the essentialspectrum. In this case, the essential spectrum need not be connected in${\mathbb C}$. Various examples illustrate the results.

A Toeplitz-like operator with rational symbol having poles on the unit circle II: the spectrum