On the nuclearity of integral operators

被引:0
作者
José C. Ferreira
Valdir A. Menegatto
Claudemir P. Oliveira
机构
[1] ICMC-USP – São Carlos,Departamento de Matemática
[2] Universidade Federal de Itajubá,ICE
来源
Positivity | 2009年 / 13卷
关键词
45905; 45H05; 47B34; Positive definiteness; nuclear operators; integral operators; Mercer’s theorem; trace;
D O I
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中图分类号
学科分类号
摘要
Let X be a nonempty measurable subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^m$$\end{document} and consider the restriction of the usual Lebesgue measure σ of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^m$$\end{document} to X. Under the assumption that the intersection of X with every open ball of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^m$$\end{document} has positive measure, we find necessary and sufficient conditions on a L2(X)-positive definite kernel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K : X \times X \rightarrow \mathbb{C}$$\end{document} in order that the associated integral operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K} : L^2(X) \rightarrow L^2(X)$$\end{document} be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(X \times X)$$\end{document}.
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页码:519 / 541
页数:22
相关论文
共 5 条
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