The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra

@misc{sang2026cesàro,
  title = {{The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra}},
  author = {Yuanqi Sang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2608.25740},
  eprint = {2608.25740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25740},
}

TY  - UNPB
TI  - The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra
AU  - Yuanqi Sang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.25740
UR  - https://arxiv.org/abs/2608.25740
AB  - Barr\'ıa and Halmos [\emph{Trans. Amer. Math. Soc.} (1982), (10) Example] asked whether the Cesàro operator belongs to the Toeplitz algebra. We answer this question affirmatively and prove that the Cesàro operator is the product of a Toeplitz operator and a continuous function of a self-adjoint Toeplitz operator. Rhaly operators are a natural generalization of the Cesàro operator. We give necessary and sufficient conditions for a Rhaly operator to belong to the Toeplitz algebra or the Hankel algebra. The main approach is to use the Fejér kernels to get localized estimates for operators. This yields a necessary condition for an operator to belong to the Hankel algebra. As an application, we find a class of Rhaly operators that are asymptotic Toeplitz operators and essentially commute with the unilateral shift, but are not in the Hankel algebra. This answers (17) Question in the same paper.
ER  -