## 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.

### 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
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### References:

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