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

Arora, S. C.; Datt, Gopal; Verma, Satish

Osaka J. Math. 45, No. 3, 629-641 (2008).

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.

Reviewer: Yang Dachun (Beijing)

MSC:

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

Keywords:

multiplication operator; composition operator; Lorentz-Bochner space; closedness; compactness; spectrum

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