Asymmetric truncated Toeplitz operators and conjugations

Abstract: Truncated Toeplitz operators in a model space are C--symmetric with respect to a natural conjugation in that space. We show that this and another conjugation associated to an orthogonal decomposition possess unique properties and we study their relations with asymmetric truncated Toeplitz operators in terms of C--symmetry. New connections with Hankel operators are established through this approach.