Properties of generalized Toeplitz operators

An operator X :H-1 --> H2 is said to be a generalized Toeplitz operator wit h respect to given contractions T-1 and T-2 if X = T2XT1*. The purpose of t his line of research, started by Douglas, Sz.-Nagy and Foias, and Ptak and Vrbova, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropria te extensions of a number of results about Toeplitz operators. Namely, Wint ner's theorem of invertibility of analytic ToepLitz operators, Widom and De vinatz's invertibility criteria for Toeplitz operators with unitary symbols , Hartman and Wintner's theorem about Toeplitz operator having a Fredholm s ymbol, Hartman and Wintner's estimate of the norm of a compactly perturbed ToepIitz operator, and the nonexistence of compact classical Toeplitz opera tors due to Brown and Halmos.