Projections in Toeplitz algebra
Functional Analysis
2020-07-28 v1
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.
Cite
@article{arxiv.2007.13319,
title = {Projections in Toeplitz algebra},
author = {Hui Dan and Xuanhao Ding and Kunyu Guo and Yuanqi Sang},
journal= {arXiv preprint arXiv:2007.13319},
year = {2020}
}