Toeplitz代数中的离散Mellin演算
A discrete Mellin calculus in the Toeplitz algebra
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中文总结 AI 辅助
本文通过离散Mellin演算证明经典Cesàro算子属于Toeplitz代数，并解决Barría和Halmos提出的问题。
中文摘要 AI 辅助
我们证明经典Cesàro算子属于Toeplitz代数，从而为Barría和Halmos提出的一个问题提供了独立的解答。我们的方法基于采样比率矩阵\\[ W(\kappa)_{jk} = \frac{1}{j+1}\\, \kappa\\!\left(\frac{k+1}{j+1}\right) \\]的离散Mellin演算。对于核的自然代数$A$，我们证明该量化在模Hilbert--Schmidt算子意义下是乘性的，即\\[ W(\kappa)W(\eta) - W(\kappa \star \eta) \in S_2, \qquad \kappa,\eta \in A. \\]我们进一步证明，每个算子$W(\kappa)$（其中$\kappa \in A$）都属于Toeplitz代数的换位子理想。由于Cesàro算子对应于核$\kappa = \mathbf 1_{(0,1]}$，这作为一般框架的特例解决了Barría--Halmos问题。
英文摘要
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.
发表机构
- Università degli Studi di Milano（米兰大学）
- Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）
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