Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:–properties of truncated Toeplitz operators;–Carleson-type embedding theorems for the coinvariant subspaces;–factorizations of pseudocontinuable functions from H1. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.