Abstract: Truncated Toeplitz operators and their asymmetric versions are presented and
studied in the context of the Hardy spaces of the half-plane. It is shown,
in particular, that they are equivalent after extension to Toeplitz
operators with 2x2 symbols of a certain form, which allows one to study the
spectral properties of several classes of truncated Toeplitz operators.
This talk is based on joint work with Jonathan Partington.