Projections in Toeplitz algebra

TL;DR

This paper characterizes projections within Toeplitz algebra on the Hardy space, linking their structure to invariant subspaces and providing new criteria for Toeplitz and Hankel operators to be partial isometries.

Contribution

It offers a complete characterization of when products of Toeplitz and Hankel operators are projections and when the self-commutator of a Toeplitz operator is a projection, advancing understanding of projections in Toeplitz algebra.

Findings

Product of Toeplitz (Hankel) operators is a projection iff it projects onto an invariant subspace.

Provides new proofs for criteria of Toeplitz and Hankel operators being partial isometries.

Characterizes when the self-commutator of a Toeplitz operator is a projection.

Abstract

Motivated by Barr{\'\i}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. This provides a class of nontrivial projections in Toeplitz algebra.