{"title":"利用线性分数阶算子函数和加权平均逼近算子半群","authors":"J. L. Rogava","doi":"10.1134/S0016266322020058","DOIUrl":null,"url":null,"abstract":"<p> 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 <span>\\(O(n^{-2}\\ln(n))\\)</span>. For a self-adjoint positive definite operator <span>\\(A\\)</span> decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup <span>\\(\\exp(-tA)\\)</span> (<span>\\(t\\geq0\\)</span>) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals <span>\\(O(n^{-1/2}\\ln(n))\\)</span>. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"56 2","pages":"116 - 129"},"PeriodicalIF":0.6000,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages\",\"authors\":\"J. L. Rogava\",\"doi\":\"10.1134/S0016266322020058\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> 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 <span>\\\\(O(n^{-2}\\\\ln(n))\\\\)</span>. For a self-adjoint positive definite operator <span>\\\\(A\\\\)</span> decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup <span>\\\\(\\\\exp(-tA)\\\\)</span> (<span>\\\\(t\\\\geq0\\\\)</span>) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals <span>\\\\(O(n^{-1/2}\\\\ln(n))\\\\)</span>. </p>\",\"PeriodicalId\":575,\"journal\":{\"name\":\"Functional Analysis and Its Applications\",\"volume\":\"56 2\",\"pages\":\"116 - 129\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Functional Analysis and Its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S0016266322020058\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266322020058","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages
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 \(O(n^{-1/2}\ln(n))\).
期刊介绍:
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.