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}
.