Multiplication and composition operators on Lorentz--Bochner spaces
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.
scientific article; zbMATH DE number 5363224
scientific article