A general approach to approximation theory of operator semigroups
This work provides a unified framework for semigroup approximation theory, benefiting researchers in functional analysis and operator theory by offering optimal rates and new formulas.
The paper develops a general functional calculus approach to approximating C0-semigroups on Banach spaces using bounded completely monotone functions of their generators, achieving optimal convergence rates and sharp constants, with improved results for holomorphic semigroups and novel second-order approximation formulas.
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.