贝尔林 - 加藤理论、哈代 - 索伯列夫演算与里特算子
Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators
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中文总结 AI 辅助
该研究为有界算子发展离散贝尔林 - 加藤理论，将其与斯托尔兹域上哈代 - 索伯列夫演算联系，证明有界算子是里特算子的充要条件，基于对数公式等构造理论，推导相关结果并讨论应用。
中文摘要 AI 辅助
我们为有界算子发展了一种离散的贝尔林 - 加藤理论，并将其与斯托尔兹域上的哈代 - 索伯列夫泛函演算联系起来。核心类别是里特算子。我们证明有界算子是里特算子当且仅当它在斯托尔兹域上允许有界的哈代 - 索伯列夫演算。构造基于对数再生公式和相关对数核的一致界。接着我们推导加藤型结果，并通过根据算子幂表述的贝尔林 - 加藤缺陷来刻画里特性质。这产生了一个在精神上与连续扇形有界全纯半群设置平行，但本质上是全局的离散理论。我们还讨论了例子、尖锐性现象以及在算子幂的凸组合下里特性质的持久性和巴拿赫格上算子由里特算子控制方面的应用。
英文摘要
We develop a discrete Beurling--Kato theory for bounded operators and relate it to a Hardy--Sobolev functional calculus on Stolz domains. Our central class is that of Ritt operators. We prove that a bounded operator is Ritt if and only if it admits a bounded Hardy--Sobolev calculus on a Stolz domain. The construction is based on a logarithmic reproducing formula and uniform bounds for the associated logarithmic kernels. We then derive Kato-type results and characterise the Ritt property by Beurling--Kato defects formulated in terms of powers of the operator. This yields a discrete theory parallel in spirit to the sectorially bounded holomorphic semigroup setting, but intrinsically global in nature. We also discuss examples and sharpness phenomena, and prove that the Ritt property is preserved under convex combinations of powers and under positive domination on Banach lattices.
