Symbols of truncated Toeplitz operators

TL;DR

This paper explores properties of truncated Toeplitz operators, Carleson embeddings, and factorizations in Hardy spaces, revealing their deep interconnections and addressing longstanding questions in the field.

Contribution

It introduces a new factorization-based approach that links three key problems in Hardy spaces, providing answers to open questions about truncated Toeplitz operators.

Findings

Properties of truncated Toeplitz operators clarified

Carleson-type embedding theorems established

Factorizations of pseudocontinuable functions developed

Abstract

We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: 1) properties of truncated Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. 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 Donald Sarason.