Pointwise multipliers on $L^1$ and related spaces - Zurich Open Repository and Archive

Jarchow, H; Labuschagne, L (2006). Pointwise multipliers on $L^1$ and related spaces. Note di Matematica, 25(1):221-229.

Abstract

We consider completely continuous and weakly compact multiplication operators on certain classical function spaces, more precisely on Lebesgue spaces $L^1$ on spaces $C(K)$ of continuous functions on a compact Hausdorff space K,and on the Hardy space $H^1$. We will describe such operators in terms of their defining symbols. Our characterizations extend corresponding results known from the literature. In any case, our results reveal the severe restrictions on the symbols of multiplication operators necessary to ensure complete continuity or weak compactness. The apparent simplicity of the obtained descriptions belie the deep and beautiful functional analytic principles that underlie them.

Abstract

We consider completely continuous and weakly compact multiplication operators on certain classical function spaces, more precisely on Lebesgue spaces $L^1$ on spaces $C(K)$ of continuous functions on a compact Hausdorff space K,and on the Hardy space $H^1$. We will describe such operators in terms of their defining symbols. Our characterizations extend corresponding results known from the literature. In any case, our results reveal the severe restrictions on the symbols of multiplication operators necessary to ensure complete continuity or weak compactness. The apparent simplicity of the obtained descriptions belie the deep and beautiful functional analytic principles that underlie them.

Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Multiplication operators; $L^1$; $C(K)$; $H^1$; Weak compactness; Complete continuity English 2006 03 Mar 2010 11:19 05 Apr 2016 13:24 Università di Lecce 1123-2536 https://doi.org/10.1285/i15900932v25n1p221