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 Hp of the half-plane for 1 < p < ∞. The question of uniqueness of the symbol is solved via the characterization of the zero operator. It is shown that asymmetric truncated Toeplitz operators are equivalent after extension to 2 × 2 matricial Toeplitz operators, which allows one to deduce criteria for Fredholmness and invertibility. Shifted model spaces are presented in the context of invariant subspaces, allowing one to derive new Beurling–Lax theorems.