Abstract
We demonstrate that an idea related to the Central Limit Theorem and approx- imations by accompanying laws in probability theory is useful to get optimal conver- gence rates in some approximation formulas for operators. As examples we provide a bound for Euler approximations of bounded holomorphic semigroups; a bound for error in approximation of a power of operators by accompanying exponents, which is a useful tool in analysis of the Trotter-Kato formula, and can be considered as an extended version of Chernoff's ' √ n lemma'.
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