Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

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properties of truncated Toeplitz operators;

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Carleson-type embedding theorems for the coinvariant subspaces;

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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.