Geodesic
Orthogonality in nonseparable rearrangement-invariant spaces
Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1553-1570

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Let $E$ be a nonseparable rearrangement-invariant space and let $E_0$ be the closure of the space of bounded functions in $E$. Elements of $E$ orthogonal to $E_0$, that is, elements $x\in E$, $x\ne 0$, such that $\|x\|_{E} \le\|x+y\|_{E}$ for each $y\in E_0$, are investigated. The set of orthogonal elements $\mathcal{O}(E)$ is characterized in the case when $E$ is a Marcinkiewicz or an Orlicz space. If an Orlicz space $L_M$ is considered with the Luxemburg norm, then the set $L_M\setminus (L_M)_0$ is the algebraic sum of $\mathcal{O}(L_M)$ and $(L_M)_0$. Each nonseparable rearrangement-invariant space $E$ such that $\mathcal{O}(E)\ne\varnothing$ is shown to contain an asymptotically isometric copy of the space $l_\infty$. Bibliography: 17 titles.
Keywords: rearrangement-invariant space, nonseparable Banach space, Orlicz space
Mots-clés : Marcinkiewicz space, orthogonal element. 
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     author = {S. V. Astashkin and E. M. Semenov},
     title = {Orthogonality in nonseparable rearrangement-invariant spaces},
     journal = {Sbornik. Mathematics},
     pages = {1553--1570},
     publisher = {mathdoc},
     volume = {212},
     number = {11},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_11_a1/}
}
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S. V. Astashkin; E. M. Semenov. Orthogonality in nonseparable rearrangement-invariant spaces. Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1553-1570. http://geodesic.mathdoc.fr/item/SM_2021_212_11_a1/