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 < infinity. 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 x 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.