Basic properties of multiplication and composition operators between distinct Orlicz spaces. (English) Zbl 1423.47009
Summary: First, we present some simple (and easily verifiable) necessary conditions and sufficient conditions for the boundedness of the multiplication operator \(M_u\) and the composition operator \(C_T\) acting from the Orlicz space \(L^{\Phi _1}(\Omega )\) into the Orlicz space \(L^{\Phi _2}(\Omega )\) over an arbitrary complete, \(\sigma \)-finite measure space \((\Omega ,\Sigma ,\mu )\). Next, we investigate the problem of conditions on the generating Young functions, the function \(u\), and/or the function \(h=d(\mu \circ T^{-1})/d\mu \), under which the operators \(M_u\) and \(C_T\) are of closed range or finite rank. Finally, we give necessary and sufficient conditions for the boundedness of the operators \(M_u\) and \(C_T\) in terms of techniques developed within the theory of Musielak-Orlicz spaces.
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 47B33 | Linear composition operators |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
multiplication operator; composition operator; continuous operators; closed-range operators; finite-rank operators; Orlicz space; Musielak-Orlicz space; Lebesgue space; measurable transformation; Radon-Nikodým derivativeReferences:
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