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 $1<p<\mathrm{\infty }$. It is shown that they are equivalent after extension to $2\times 2$ matricial Toeplitz 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.