Pointwise multipliers of Calderón‐Lozanovskiǐ spaces

Abstract

AbstractSeveral results concerning multipliers of symmetric Banach function spaces are presented firstly. Then the results on multipliers of Calderón‐Lozanovskiǐ spaces are proved. We investigate assumptions on a Banach ideal space E and three Young functions φ1, φ2 and φ, generating the corresponding Calderón‐Lozanovskiǐ spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E_{\varphi _1}, E_{\varphi _2}, E_{\varphi }$\end{document} so that the space of multipliers \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M(E_{\varphi _1}, E_{\varphi })$\end{document} of all measurable x such that x y ∈ Eφ for any \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$y \in E_{\varphi _1}$\end{document} can be identified with \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E_{\varphi _2}$\end{document}. Sufficient conditions generalize earlier results by Ando, O'Neil, Zabreǐko‐Rutickiǐ, Maligranda‐Persson and Maligranda‐Nakai. There are also necessary conditions on functions for the embedding \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M(E_{\varphi _1}, E_{\varphi }) \subset E_{\varphi _2}$\end{document} to be true, which already in the case when E = L1, that is, for Orlicz spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M(L^{\varphi _1}, L^{\varphi }) \subset L^{\varphi _2}$\end{document} give a solution of a problem raised in the book 26. Some properties of a generalized complementary operation on Young functions, defined by Ando, are investigated in order to show how to construct the function φ2 such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M(E_{\varphi _1}, E_{\varphi }) = E_{\varphi _2}$\end{document}. There are also several examples of independent interest.