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Compact operators on Musielak-Orlicz spaces. (English) Zbl 0676.46024
Summary: Let \(L^{\phi}(\nu)\) be a Musielak-Orlicz space over a non-atomic \(\sigma\)-finite measure space (S,\(\Sigma\),\(\nu)\), determined by a Musielak-Orlicz function \(\phi\) : \({\mathbb{R}}_+\times S\to {\mathbb{R}}_+\), and let \(L^{\phi}_ a(\nu)\) be its subspace consisting of \(\nu\)- continuous elements. It is shown that every compact linear operator from \(L^{\phi}_ a(\nu)\) into any complete topological vector space factors through the inclusion map \(L^{\phi}_ a(\nu)\hookrightarrow L^{{\hat \phi}}_ a(\nu)\) where \({\hat \phi}\) is the convex minorant of \(\phi\). It follows that a non-zero compact operator exists on \(L^{\phi}_ a(\nu)\) if and only if \[ \liminf_{r\to \infty}r^{-1} \phi (r,s)>0 \] on a set of positive measure. Also, the Mackey topology of \(L^{\phi}_ a(\nu)\) is the topology induced from \(L^{{\hat \phi}}_ a(\nu)\). This extends some earlier results of N. J. Kalton concerning ordinary Orlicz function spaces.

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators