$L_p+L_q$ and $L_p\cap L_q$ are not isomorphic for all $1\le p,q\le \infty$, $p\ne q$

Abstract

We prove that if $1\le p,q\le \infty$, then the spaces $L_p+L_q$ and $L_p\cap L_q$ are isomorphic if and only if $p=q$. In particular, $L_2+L_{\infty }$ and $L_2\cap L_{\infty }$ are not isomorphic which is an answer to a question formulated in the paper S. V. Astashkin and L. Maligranda, \textit{$L_p+L_{\infty }$ and $L_p\cap L_{\infty }$ are not isomorphic for all $1\le p<\infty ,p\ne 2$}, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2181--2194.

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