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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)].

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Citations:
Zbl 0914.46029; Zbl 0956.46021; Zbl 1041.46022
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