Properties of generalized Toeplitz operators
by
Pedro J. Paúl
Universidad de Sevilla
Coauthors: Carmen H. Mancera

An operator X :H₁ --> H₂ is said to be a generalized Toeplitz operator with respect to given contractions T₁ and T₂ if X=T₂XT₁^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia s, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators as well as some clarifying examples. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, the non-existence of compact classical Toeplitz operators due to Brown and Halmos, and spectral properties that complement the work done by Sz.-Nagy and Foia s. As a by-product we prove that the spectrum of a function \phi in H^\infty equals the approximate point spectrum of its Toeplitz operator.

Date received: March 6, 2001