**Beyond truncated Toeplitz operators**

For an inner function θ ∈ H^{∞ }, the model space K_{θ} is defined by the formula K_{θ}=H^{2}⊖θH^{2}. Truncated Toeplitz operators (TTOs), formally introduced by Sarason more than a decade ago, are compressions to these model spaces of multiplication operators. This area has recently been the focus of extensive research.

In this talk we explore some directions of research that extend the theory of TTOs, namely:

- Considering θ an arbitrary function in the unit ball of H
^{∞}. - Assuming θ is a matrix-valued inner function.
- As a particular aspect of the previous point, investigating maximal algebras of block Toeplitz matrices.

The talk is based on joint works with H. Bercovici, R. Khan, and M.A. Khan.