Multiplication and composition operators on Lorentz--Bochner spaces
type: Item
instance of: scholarly article
title: Multiplication and composition operators on Lorentz--Bochner spaces
zbMATH Open document ID: 1165.47023
author: Gopal Datt
author: Satish Verma
author: S. C. Arora
published in: Osaka Journal of Mathematics
publication date: 2008-11-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1221656645
review text: Let \((\Omega,{\mathcal A},\mu)\) be a \(\sigma\)-finite measure space. Let \(X\) be a Banach space and \(L_{p\,q}(\Omega,X)\) denote the Lorentz--Bochner space, \({\mathcal B}(X)\) the class of all bounded operators on \(X\). For a strongly measurable function \(u:\Omega\to{\mathcal B}(X)\), the multiplication transform \(M_u:L_{p\,q}(\Omega,\, X) \to L(\Omega,X)\) is defined as \((M_uf)(\omega)=u(\omega)(f(\omega))\) for all \(\omega\in\Omega\), where \(L(\Omega,X)\) is the space of all strongly measurable functions. For a non-singular measurable transformation \(T:\Omega\to\Omega\), the composition transformation \(C_T:L_{pq} (\Omega,X)\to L(\Omega,X)\) is given by \((C_Tf)(\omega)=f(T(\omega))\) for all \(\omega\in\Omega\). In this paper, the authors study the multiplication and composition operators and discuss some of their properties, such as invertibility, range, compactness and spectrum.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5363224
zbMATH Keywords: multiplication operator
zbMATH Keywords: composition operator
zbMATH Keywords: Lorentz-Bochner space
zbMATH Keywords: closedness
zbMATH Keywords: compactness
zbMATH Keywords: spectrum
reviewed by: Dachun Yang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5363224