Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

被引：0
作者：
V. M. Horbachuk
M. L. Horbachuk
机构：
[1] Ukrainian National Technical University,“Kyiv Polytechnic Institute”
[2] Ukrainian National Academy of Sciences,Institute of Mathematics
来源：
Ukrainian Mathematical Journal | 2015年 / 67卷
关键词：
Banach Space; Linear Operator; Cauchy Problem; Entire Function; Vector Function;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
For a strongly continuous one-parameter group {U(t)}t ∈(−∞,∞) of linear operators in a Banach space B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document} with generator A, we prove the existence of a set B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document}1 dense in B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document} on the elements x of which the function U(t)x admits an extension to an entire B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document}-valued vector function. The description of the vectors from B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document}1 for which this extension has a finite order of growth and a finite type is presented. It is also established that the inclusion x ∈B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{B} $$\end{document}1 is a necessary and sufficient condition for the existence of the limit limn→1I+tAnnx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ { \lim}_{n\to 1}{\left(I+\frac{tA}{n}\right)}^nx $$\end{document} and this limit is equal to U(t)x.
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页码：668 / 679
页数：11
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