Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol

Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space $H^p$ of the half-plane for $1Toeplitz operators, which allows one to deduce information about their invertibility properties. Shifted model spaces are presented in the context of invariant subspaces, allowing one to deduce new Beurling--Lax theorems.