Compact and strictly singular operators in rearrangement invariant spaces and Rademacher functions
The relation between strictly singular and compact operators is a classical topic with deep consequences in the theory of spaces of integrable functions, since the structure of the spaces determines how these classes of operators are related to each other. In this paper, the author analyses how these classes of operators are related in the context of rearrangement invariant spaces, starting from the work by \textit{J. Flores} et al. [Isr. J. Math. 188, 323--352 (2012; Zbl 1262.46017)]. Regarding the analysis of these operators on function spaces, the main reference is a classical theorem of Milman that establishes that every strictly singular operator in \(L^p(\mu)\) has a compact square [\textit{V. D. Mil'man}, Teor. Funkts. Funkts. Anal. Prilozh. 10, 15--26 (1970; Zbl 0217.17201)]. The above mentioned authors extended Milman's result to other classes of function spaces, getting results like the following one: each strictly singular operator, bounded in a disjointly homogeneous Banach lattice with an unconditional basis and finite Rademacher cotype, has a compact square. Recall that a Banach space has the Kato property if every strictly singular operator acting in the space is compact. These authors also stated that every 2-disjointly homogeneous Banach lattice with finite cotype has the Kato property. This is the starting poing of the present paper that focuses its attention on the identification of rearrangement invariant spaces with the Kato property; that was also analysed in the paper by \textit{F.~L. Hernández} et al. [Funct. Approximatio, Comment. Math. 50, No.~2, 215--232 (2014; Zbl 1311.46026)]. In this direction, the author proves that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with non-trivial Boyd indices has a compact square (Theorems~1 and~2). Moreover, consider the closure \(G\) of \(L_\infty\) in the Orlicz space generated by the function \(e^{u^2}-1\). The author proves that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space \(X\) if \(G \subset X\) (Theorem~3). A sort of converse is given under the assumption that \(X \subseteq L \log ^{1/2} L\) (Theorem~4). Some results on when a Lorentz space \(\Lambda(2 , \psi)\) has the Kato property are also given in Section~5.
scientific article; zbMATH DE number 7345609
scientific article