On the Adjoint of Linear Relations in Hilbert Spaces

被引：0
作者：
Adrian Sandovici
机构：
[1] “Gheorghe Asachi”,Department of Mathematics and Informatics
[2] Technical University of Iaşi,undefined
来源：
Mediterranean Journal of Mathematics | 2020年 / 17卷
关键词：
Hilbert space; closed linear relation; Skew–adjoint linear relation; selfadjoint linear relation; normal linear relation; generalized orthogonal projection; 47A06; 47B25; 47B15;
D O I：
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摘要：
Assume that H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {H}}$$\end{document} and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {K}}$$\end{document} are two real or complex Hilbert spaces, A a linear relation from H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {H}}$$\end{document} to K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {K}}$$\end{document}, and B a linear relation from K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {K}}$$\end{document} to H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {H}}$$\end{document}, respectively. Necessary and sufficient conditions for B to be equal to the adjoint of A are provided. Several consequences are also presented. More precisely, new characterizations for closed, skew–adjoint, selfadjoint, normal linear relations, and generalized orthogonal projections are obtained.
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