Multiplication and composition operators on Lorentz–Bochner spaces. (English) Zbl 1165.47023

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.


47B38 Linear operators on function spaces (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B33 Linear composition operators
Full Text: Euclid


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