A discrete Mellin calculus in the Toeplitz algebra

RESUMO

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 \[ W(\kappa)_{jk} = \frac{1}{j+1}\, \kappa\!\left(\frac{k+1}{j+1}\right). \] For a natural algebra of kernels $A$, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, \[ W(\kappa)W(\eta) - W(\kappa \star \eta) \in S_2, \qquad \kappa,\eta \in A. \] We further show that every operator $W(\kappa)$, $\kappa \in A$, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel $\kappa = \mathbf 1_{(0,1]}$, this resolves the Barr\'ıa--Halmos question as a special case of the general framework.