Year: 2026
Type: preprint
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 \[ W(κ)_{jk} = \frac{1}{j+1}\, κ\!\left(\frac{k+1}{j+1}\right). \] For a natural algebra of kernels $A$, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, \[ W(κ)W(η) - W(κ\star η) \in S_2, \qquad κ,η\in A. \] We further show that every operator $W(κ)$, $κ\in A$, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel $κ= \mathbf 1_{(0,1]}$, this resolves the Barr\'ıa--Halmos question as a special case of the general framework.
Source: arXiv (Cornell University)
Authors: Bellavita, Carlo, Stylogiannis, Georgios
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212165213