This article is cited in 5 papers

On symmetric spaces with convergence in measure on reflexive subspaces

S. V. Astashkin, S. I. Strakhov

Samara national research university, 34 Moskovskoe highway, Samara, 443086 Russia

Abstract: A closed subspace $H$ of a symmetric space $X$ on $[0,1]$ is said to be strongly embedded in $X$ if in $H$ a convergence in $X$-norm is equivalent to the convergence in Lebesgue measure. We study symmetric spaces $X$ with the property that all their reflexive subspaces are strongly embedded in $X$. We prove that it is the case for all spaces, which satisfy an analog of the classical Dunford–Pettis theorem of relatively weakly compact subsets in $L_1$. At the same time the converse assertion fails for a wide class of separable Marcinkiewicz spaces.

Keywords: symmetric spaces, reflexive subspace, Marcinkiewicz space, equicontinuity of norms.

UDC: 517.982

Received: 23.06.2017

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2018, 62:8, 1–8

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