Abstract
We refine some earlier results by Flores, Hernández, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator acting in X is compact. We show that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with the non-trivial Boyd indices has compact square, and that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space X whenever \(X\supset G\), where G is the closure of \(L_\infty \) in the Orlicz space, generated by the function \(e^{u^2}-1\). Moreover, a partial converse to the latter result is given under the assumption that \(X\subset L\log ^{1/2}L\). As an application we find rather sharp conditions, under which a Lorentz space \(\Lambda (2,\psi )\) possesses the Kato property. In particular, \(\Lambda (2,\log ^{-\alpha }(e/u))\), with \(0<\alpha \le 1\), is a 2-DH and 2-convex rearrangement invariant space, which does not have the Kato property. This gives a negative answer to the question posed by Hernández, Semenov, and Tradacete.
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Notes
Sometimes such a set is termed also as L-weakly compact or equi-integrable.
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The author has been supported by the Ministry of Science and Higher Education of the Russian Federation (Project 1.470.2016/1.4) and by the RFBR Grant 18–01–00414a.
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Astashkin, S.V. Compact and strictly singular operators in rearrangement invariant spaces and Rademacher functions. Positivity 25, 159–175 (2021). https://doi.org/10.1007/s11117-020-00755-9
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DOI: https://doi.org/10.1007/s11117-020-00755-9
Keywords
- Banach lattice
- Rearrangement invariant space
- Orlicz space
- Lorentz space
- Disjointly homogeneous lattice
- p-disjointly homogeneous lattice
- Compact operator
- Strictly singular operator
- Disjointly strictly singular operator