Geodesic
On a Theorem of Kadets and Pe{\l}czy{\'n}ski
Matematičeskie zametki, Tome 106 (2019) no. 2, pp. 174-187

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Necessary and sufficient conditions are found under which a symmetric space $X$ on $[0,1]$ of type $2$ has the following property, which was first proved for the spaces $L_p$, $p>2$, by Kadets and Pełcziński: if $\{u_n\}_{n=1}^\infty$ is an unconditional basic sequence in $X$ such that $$ \|u_n\|_X\asymp\|u_n\|_{L_1},\qquad n\in\mathbb N, $$ then the norms of the spaces $X$ and $L_1$ are equivalent on the closed linear span $[u_n]$ in $X$. For sequences of martingale differences, this implication holds in any symmetric space of type $2$.
Keywords: Kadets–Pełczyński alternative, symmetric space, Rademacher type, Boyd indices, (disjointly) strictly singular inclusion. 
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     author = {S. V. Astashkin},
     title = {On a {Theorem} of {Kadets} and {Pe{\l}czy{\'n}ski}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {174--187},
     publisher = {mathdoc},
     volume = {106},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_2_a1/}
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S. V. Astashkin. On a Theorem of Kadets and Pe{\l}czy{\'n}ski. Matematičeskie zametki, Tome 106 (2019) no. 2, pp. 174-187. http://geodesic.mathdoc.fr/item/MZM_2019_106_2_a1/