Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages

Pub Date : 2022-10-10 DOI:10.1134/S0016266322020058
J. L. Rogava
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Abstract

An analytic semigroup of operators on a Banach space is approximated by a sequence of positive integer powers of a linear-fractional operator function. It is proved that the order of the approximation error in the domain of the generating operator equals \(O(n^{-2}\ln(n))\). For a self-adjoint positive definite operator \(A\) decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup \(\exp(-tA)\) (\(t\geq0\)) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals \(O(n^{-1/2}\ln(n))\).

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利用线性分数阶算子函数和加权平均逼近算子半群
用线性分数阶算子函数的正整数幂序列来逼近巴拿赫空间上算子的解析半群。证明了生成算子域内的近似误差阶为\(O(n^{-2}\ln(n))\)。对于分解为多个自伴随正定算子和的自伴随正定算子\(A\),也考虑了半群\(\exp(-tA)\) (\(t\geq0\))的加权平均逼近。证明了算子范数中近似误差的阶为\(O(n^{-1/2}\ln(n))\)。
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