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arXiv · 2607.10507

Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators

A. Borichev·A. Gomilko·Yu. Tomilov
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Abstract

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.

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A. Borichev, A. Gomilko, Yu. Tomilov. 2026-07-11. Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators. https://arxiv.org/abs/2607.10507

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