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Truncated Toeplitz operators and their asymmetric versions are studied in thecontext of the Hardy space $H^p$ of the half-plane for $1<p<\infty$. It isshown that they are equivalent after extension to $2 \times 2$ matricialToeplitz operators, which allows one to deduce information about theirinvertibility properties. Shifted model spaces are presented in the context ofinvariant subspaces, allowing one to deduce new Beurling--Lax theorems.

Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol