Bounded Toeplitz Products on the Hardy Space

TL;DR

This paper characterizes when the product of two Toeplitz operators on the Hardy space is bounded, linking it directly to the boundedness of the product of their symbols under certain conditions.

Contribution

It provides a new characterization of the boundedness of Toeplitz operator products via their symbols, addressing a longstanding open problem.

Findings

Product of Toeplitz operators is bounded iff the product of their symbols is bounded under certain assumptions.

Establishes a direct link between operator boundedness and symbol boundedness for Toeplitz products.

Advances understanding of Toeplitz operator algebra on the Hardy space.

Abstract

A Toeplitz operator on the Hardy space of the unit circle is bounded if and only if its symbol is bounded. For two Toeplitz operators, there are no known function-theoretic conditions for their symbols, which are equivalent to the product of the Toeplitz operators being bounded. In this paper, we provide a solution to this problem, by showing under certain assumptions that the product of two Toeplitz operators is bounded if and only if the product of their symbols is bounded.