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Strictly singular embeddings. (English. Russian original) Zbl 1033.46026
Funct. Anal. Appl. 36, No. 1, 71-73 (2002); translation from Funkts. Anal. Prilozh. 36, No. 1, 85-87 (2002).
Let \(E\) and \(F\) be rearrangement invariant (r.i.) Banach spaces on \((0,1)\) and \(E\subset F\). The present paper deals with conditions under which the natural embedding \(E\subset F\) is strictly singular. That means that for any infinite dimensional subspace \(B\subset E\), the norms of \(E\) and \(F\) are nonequivalent on \(B\). It is known that for every r.i. space \(E\neq L_{\infty}\) the embedding \(L_{\infty}\subset E\) is strictly singular, cf. S. Ya. Novikov [Math. Notes 62, 457–468 (1997; Zbl 0914.46029)], and that this property characterizes \(L_{\infty}\), cf. A. Garcia del Amo, F. L. Hernandez, V. M. Sanchez and E. M. Semenov [J. Lond. Math. Soc. 62, 239–252 (2000; Zbl 0956.46021)].
In the paper under review, the embedding \(E\subset L_1\) is considered. Namely, this embedding is not strictly singular if and only if \(E\supset G\) and this property characterizes \(L_1\). Here \(G\) is the closure of \(L_{\infty}\) in the Orlicz space \(L_M\) with \(M(u)=e^{u^2}-1\). Another result of the paper is as follows. Let \(E\) be separable. The embedding \(E\subset F\) is strictly singular if and only if it is disjointly strictly singular and the norms of \(E\) and \(F\) are not equivalent on the subspace generated by the Rademacher system. (The embedding \(F\subset F\) is called disjointly strictly singular if its restriction to any subspace generated by a sequence of disjoint functions is not an isomorphism.
The paper does not contain the proofs which have been published in [Positivity 7, 119–124 (2003; Zbl 1041.46022)].

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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