Submultiplicative functions and spectral theory of Toeplitz operators

This paper is concerned with the application of the theory of submultiplicative functions to the problem of deciding whether a weight satisfies the Muckenhoupt condition and, secondly, to the spectral theory of Toeplitz operators with piecewise continuous symbols. We establish criteria for the membership of integrable functions in the Muckenhoupt class in terms of the indices of suitably constructed submultiplicative functions. These criteria together with localization techniques and Wiener-Hopf factorization methods enable us to describe the spectra of Toeplitz operators with piecewise continuous symbols on L-P spaces with arbitrary Mudcenhoupt weights over arbitrary Carleson Jordan curves.