For which~$p$ and~$r$ is the equation $\Pi_p(L^r,\cdot)=I_p(L^r,\cdot)$ true?
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 237-242
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It is explained when the classes of $p$-absolutely summing and $p$-integral operators given on the space $L^r(\mu)$ coincide. For a Banach space $X$ there is considered the following subset of the real line:
$$
J_X\stackrel{\mathrm{def}}=\{p\colon1\le p\infty,\ \Pi_p(X,Y)=I_p(X,Y)\ \forall Y\}.
$$
In the case when $X$ is an infinite-dimensional subspace of the space $L^r(\mu)$, it is proved that $J_X=(1,2]$ if $1\le r\le2$, and $J_X=\{2\}$ if $2$ and $X$ is not isomorphic with a Hilbert space.
@article{ZNSL_1981_113_a16,
author = {N. G. Sidorenko},
title = {For which~$p$ and~$r$ is the equation $\Pi_p(L^r,\cdot)=I_p(L^r,\cdot)$ true?},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {237--242},
publisher = {mathdoc},
volume = {113},
year = {1981},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a16/}
}
N. G. Sidorenko. For which~$p$ and~$r$ is the equation $\Pi_p(L^r,\cdot)=I_p(L^r,\cdot)$ true?. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 237-242. http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a16/