On the spectral mapping theorem for Weyl spectra of Toeplitz operators

In this paper we study the approximate point spectrum and the surjectivity spectrum of a Toeplitz operators T phi, defined on Hardy spaces H-2(T) of the unit circle T of C, where the symbol phi is an element of L infinity(T), or phi is a continuous symbol. We also extend some results obtained by Farenick and Lee (Trans Am Math Soc 348(10):4153-4174, 1996). In particular, we study Weyl type theorems and the spectral mapping theorem for the Weyl spectra.