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Pointwise multipliers on $L^1$ and related spaces


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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Multiplication operators; $L^1$; $C(K)$; $H^1$; Weak compactness; Complete continuity
Language:English
Date:2006
Deposited On:03 Mar 2010 11:19
Last Modified:05 Apr 2016 13:24
Publisher:Università di Lecce
ISSN:1123-2536
Publisher DOI:https://doi.org/10.1285/i15900932v25n1p221

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