Complex symmetric Toeplitz operators on the Dirichlet space

Abstract We study when a Toeplitz operator T ϕ on the Dirichlet space of the unit disk is complex symmetric with respect to a class of conjugations and find surprisingly that the case of complex symmetries of Toeplitz operators according to these conjugations is very few. We also show that if T ϕ is complex symmetric, then the curve ϕ | T ( T ) must be nowhere winding. Furthermore, the spectrum and invertibility of complex symmetric Toeplitz operators are described.