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We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by \(\Phi\) are isotonic if and only if \(\Phi\) is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.

We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by Φ are isotonic if and only if Φ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012. [ABSTRACT FROM AUTHOR]

We discuss the latticial structure of projection sets and antiprojection sets in Orlicz-Musielak spaces L Φ, with the distance given by modular ρ Φ and classical F-norms generated by ρ Φ. In particular, we show that the case of Amemiya F-norm is essentially different from others. This extends results given for sets of best approximants by Landers and Rogge, Kilmer and Kozlowski, and Mazzone. [ABSTRACT FROM AUTHOR]