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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Linear operators on $ L\sb{p}$ for $ 0<p<1$


Author: N. J. Kalton
Journal: Trans. Amer. Math. Soc. 259 (1980), 319-355
MSC: Primary 47B38; Secondary 46E30
MathSciNet review: 567084
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0567084-3
PII: S 0002-9947(1980)0567084-3
Article copyright: © Copyright 1980 American Mathematical Society