MULTIPLICATION OPERATORS IN
MEASURABLE SECTIONS SPACES
Marat Pliev
Southern Mathematical Institute of RAS
Vladikavkaz, 362027, RUSSIA

Abstract

In this article we consider linear operators in measurable section spaces. Let $\mathcal{X}$ be a liftable measurable bundle of Banach spaces and $E$ be an order continuous Köthe function space over a finite measure space $(A,\Sigma,\mu)$. We prove that a linear continuous operator $T$ in a measurable sections space $E(\mathcal{X})$ is a multiplication operator (by a function in $L_{\infty}(\mu)$) if and only if the equality $T(g\langle f,\phi^{\star}\rangle\phi)=g\langle T(f),\phi^{\star}\rangle\phi)$ holds for every $g\in L_{\infty}(\mu),f\in E(\mathcal{X})$, $\phi\in L_{\infty}(\mathcal{X})$ and $\phi^{\star}\in L_{\infty}(\mathcal{X}^{\star})$.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 5
Year: 2019

DOI: 10.12732/ijam.v32i5.11

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