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Yu. Tomilov

Publications and source records attributed to Yu. Tomilov.

4 recordsLinked to original sources
arXiv2026

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 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.

math.FA↗
arXiv2018

A general approach to approximation theory of operator semigroups

We develop a general, functional calculus approach to approximation of $C_0$-semigroups on Banach spaces by bounded completely monotone functions of their generators. The approach comprises most of well-known approximation formulas, yields optimal convergence rates, and sometimes even leads to sharp constants. In an important particular case when semigroups are holomorphic, we are able to significantly improve our results for general semigroups. Moreover, we present several second order approximation formulas with rates, which in such a general form appear in the literature for the first time.

math.FA↗
arXiv2016

Operators $L^1 (\mathbb R_+ )\to X$ and the norm continuity problem for semigroups

We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that a) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in [0,1]$ such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in \mathbb R_+$ such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.

math.FA↗