This article is cited in 1 paper

Mathematics

On a characteristic of strongly embedded subspaces in symmetric spaces

S. I. Strakhov

Samara National Research University, Samara, Russian Federation

Abstract: It is shown that the presence of a lower $p$-estimate with constant $1$ in the symmetric space $E$ is sufficient for the condition of equivalence of convergence in norm and in measure on the subspace $H$ of the space $E$ to be satisfied if and only if the numerical characteristic $\eta_ {E}(H) <1. $ The last criterion is also valid for symmetric spaces “close” to $L_ {1},$ more precisely, for which an analog of the Dunford–Pettis criterion of weak compactness is valid. In particular, it is shown that spaces “close” to $L_ {1},$ have the binary property: the characteristic $\eta_{E}(H)$ takes only two values, $0$ and $1$. This gives an example of binary Orlicz spaces different from the spaces $L_{p}$.

Keywords: rearrangement invariant space, Orlicz space, Luxemburg norm, Orlicz norm, lower $p$-estimate with constant one, strongly embedded subspace, equivalent norms, convergence in measure.

UDC: 517.982.22

Received: 11.03.2021
Revised: 15.04.2021
Accepted: 28.05.2021

DOI: 10.18287/2541-7525-2021-27-2-25-32