Toeplitz operators on Hardy spaces

In the present thesis we establish Banach space counterparts for several results known for Toeplitz operators on Hardy-Hilbert spaces. We use methods of Didas, Eschmeier and Everard to construct Toeplitz projections for Toeplitz operators acting on a general class of Hardy-type spaces. These Toeplitz projections provide a general framework for Brown-Halmos type characterizations of Toeplitz operators and allow us to prove a Banach space version of a classical spectral inclusion theorem of Hartman and Wintner. Furthermore we show that a multivariable spectral mapping theorem of Eschmeier for Toeplitz tuples on Hardy-Hilbert spaces over strictly pseudoconvex domains with smooth boundary remains true in the corresponding Banach space setting. As an application we derive a spectral mapping theorem for truncated Toeplitz systems which generalizes a one dimensional result proved by Bessonov for the Hardy-Hilbert space on the unit disc