Abstract
A classification of weakly compact multiplication operators on
L(L_p)
,
1
, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of
\ell_p
-strictly singular operators, and we also investigate the structure of general
\ell_p
-strictly singular operators on
L_p
. The main result is that if an operator
T
on
L_p
,
1
, is
\ell_p
-strictly singular and
T_{|X}
is an isomorphism for some subspace
X
of
L_p
, then
X
embeds into
L_r
for all
r
, but
X
need not be isomorphic to a Hilbert space. It is also shown that if
T
is convolution by a biased coin on
L_p
of the Cantor group,
1\le p
, and
T_{|X}
is an isomorphism for some reflexive subspace
X
of
L_p
, then
X
is isomorphic to a Hilbert space. The case
p=1
answers a question asked by Rosenthal in 1976.