Linear operators on 𝐿_{𝑝} for 0<𝑝<1

Kalton, N. J.

doi:10.1090/S0002-9947-1980-0567084-3

Linear operators on $L_{p}$ for $0<p<1$
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by N. J. Kalton

Trans. Amer. Math. Soc. 259 (1980), 319-355

DOI: https://doi.org/10.1090/S0002-9947-1980-0567084-3

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Abstract:

If $0 < p < 1$ we classify completely the linear operators $T: {L_p} \to X$ where X is a p-convex symmetric quasi-Banach function space. We also show that if $T: {L_p} \to {L_0}$ is a nonzero linear operator, then for $p < q \leqslant 2$ there is a subspace Z of ${L_p}$, isomorphic to ${L_q}$, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if $p < q < \infty$, the Lorentz space $L(p, q)$ is a quotient of ${L_p}$ which contains no copy of ${l_p}$.

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