On Orthogonally Additive Operators in Köthe–Bochner Spaces

In this article we consider some classes of orthogonally additive operators in Köthe–Bochner spaces in the setting of the theory of lattice-normed spaces and dominated operators. The our first main result asserts that the C-compactness of a dominated orthogonally additive operator S:E(X)→F(Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:E(X)\rightarrow F(Y)$$\end{document} implies the C-compactness of its exact dominant ||S||:E→F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\!|S|\!|:E\rightarrow F$$\end{document}. Then we show that a dominated orthogonally additive operator S:E(X)→F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:E(X)\rightarrow F$$\end{document} from a Köthe–Bochner space to a Banach lattice F with an order continuous norm is narrow if and only if its exact dominant ||S||:E→F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\!|S|\!|:E\rightarrow F$$\end{document} is. Finally we prove that every laterally-to-norm continuous dominated orthogonally additive operator from E(X) to a sequence Banach lattice F is narrow.