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A discrete Mellin calculus in the Toeplitz algebra
Abstract
We prove that the classical Cesàro operator belongs to the Toeplitz algebra, providing an independent solution to a question raised by Barría and Halmos. Our approach is based on a discrete Mellin calculus for the sampled-ratio matrices W(κ)jk = 1/(j+1) κ ((k+1)/(j+1)). For a natural algebra of kernels A, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, W(κ)W(η) − W(κ⋆ η) ∈ S2, κ,η∈ A. We further show that every operator W(κ), κ∈ A, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel κ= 1(0,1], this resolves the Barr\'ıa--Halmos question as a special case of the general framework.
Related
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BibTeX
@misc{bellavita2026discrete,
title = {{A discrete Mellin calculus in the Toeplitz algebra}},
author = {Carlo Bellavita and Georgios Stylogiannis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09453},
eprint = {2609.09453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09453},
}
RIS
TY - UNPB
TI - A discrete Mellin calculus in the Toeplitz algebra
AU - Carlo Bellavita
AU - Georgios Stylogiannis
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09453
UR - https://arxiv.org/abs/2609.09453
AB - We prove that the classical Cesàro operator belongs to the Toeplitz algebra, providing an independent solution to a question raised by Barría and Halmos. Our approach is based on a discrete Mellin calculus for the sampled-ratio matrices W(κ)_jk = 1/(j+1) κ ((k+1)/(j+1)). For a natural algebra of kernels A, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, W(κ)W(η) − W(κ⋆ η) ∈ S_2, κ,η∈ A. We further show that every operator W(κ), κ∈ A, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel κ= 1_(0,1], this resolves the Barr\'ıa--Halmos question as a special case of the general framework.
ER -