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

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 continuous sectorially bounded holomorphic semigroup setting, but intrinsically global in nature. We also discuss examples, sharpness phenomena, and applications to permanence of the Ritt property under convex combinations of operator powers and to domination for operators on Banach lattices by Ritt operators.