Some Results on Complex Toeplitz Eigenvalues

We consider the asymptotics of the eigenvalues of Toeplitz matrices generated by a complex valued symbol f which is essentially bounded. We prove that if the essential range of f has empty interior and does not separate the complex plane, then the corresponding Toeplitz eigenvalues have the canonical distribution in the sense of Widom. The assumptions on the symbol only concern the topological structure of its essential range, and are completely independent of its smoothness properties. We also prove some clustering properties of eigenvalues and some new distribution results.