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Operators for Matrix-Valued Riesz Bases Over LCA Groups
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DOI：10.1007/s00025-025-02541-z.png)
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The image of a given orthonormal basis for a separable Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document} under a bijective, bounded, and linear operator acting on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document} is called a Riesz basis of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}. Contrary to what happens with Riesz bases (in the usual sense) in separable Hilbert spaces, it is not true in general that the image of a matrix-valued orthonormal basis under a bounded, linear, and bijective operator on L2(G,Csxr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(G, \mathbb {C}<^>{s\times r})$$\end{document} is also a basis and frame for the space L2(G,Csxr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(G, \mathbb {C}<^>{s\times r})$$\end{document}, where G is a sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-compact and metrizable locally compact abelian (LCA) group. We give some classes of operators for the construction of matrix-valued Riesz bases from orthonormal bases of the space L2(G,Csxr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(G, \mathbb {C}<^>{s\times r})$$\end{document}. Motivated by a result due to Holub, we show that a bounded, linear, and bijective operator acting on L2(G,Csxr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(G, \mathbb {C}<^>{s\times r})$$\end{document} which is adjointable with respect to the matrix-valued inner product is positive if and only if it maps a matrix-valued Riesz basis of the space L2(G,Csxr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2(G, \mathbb {C}<^>{s\times r})$$\end{document} to its dual Riesz basis.
Keyword:
Riesz basis
orthonormal basis
frames
positive operator
locally compact group
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1.2
论文数:
157
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0
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Hilbert-Schmidt frames and Riesz bases with respect to tensor product of Hilbert spaces希尔伯特-施密特框架和关于希尔伯特空间张量积的里斯基。
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