 
Summary: BANACH SPACES WITH THE 2SUMMING PROPERTY
A. Arias, T. Figiel, W. B. Johnson and G. Schechtman
Abstract. A Banach space X has the 2summing property if the norm of every
linear operator from X to a Hilbert space is equal to the 2summing norm of the
operator. Up to a point, the theoryof spaces which havethis propertyis independent
of the scalar eld: the propertyis selfdualand any space with the propertyis a nite
dimensional space of maximal distance to the Hilbert space of the same dimension.
In the case of real scalars only the real line and real `2
1 have the 2summingproperty.
In the complex case there are more examples e.g., all subspaces of complex `3
1 and
their duals.
0. Introduction:
Some important classical Banach spaces in particular, C(K) spaces, L1 spaces,
the disk algebra as well as some other spaces (such as quotients of L1 spaces by
re exive subspaces K], Pi]), have the property that every (bounded, linear) oper
ator from the space into a Hilbert space is 2summing. (Later we review equivalent
formulations of the de nition of 2summing operator. Here we mention only that
an operator T : X ! `2 is 2summing provided that for all operators u : `2 ! X
the composition Tu is a HilbertSchmidt operator moreover, the 2summing norm
