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We give the necessary and sufficient conditions for a bounded bilinear operator on X×c0Y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X\times c_{0}\left( {\mathcal {Y}}\right) $$\end{document} to be nuclear. As application, we find the necessary and sufficient conditions for bilinear multiplication operators MV:EX×c0Y→FZ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{{\mathcal {V}}}:E\left( {\mathcal {X}}\right) \times c_{0}\left( {\mathcal {Y}}\right) \rightarrow F\left( {\mathcal {Z}}\right) $$\end{document} defined by MVx,y=Vnxn,ynn∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_{{\mathcal {V}}}\left( x,y\right) =\left( V_{n}\left( x_{n},y_{n}\right) \right) _{n\in {\mathbb {N}}}$$\end{document} to be nuclear. For various other similar type of operators, we give the necessary and sufficient conditions to be nuclear.
Annals of Functional Analysis – Springer Journals
Published: Jul 1, 2023
Keywords: Nuclear operators; Multiplication operator; Vector valued Banach sequence spaces; 47B10; 47L20; 46B45; 46A45
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