Linear operators on 𝐿_{𝑝} for 0

Abstract

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