 
Summary: SOME MORE WEAK HILBERT SPACES
GEORGE ANDROULAKIS, PETER G. CASAZZA, AND DENKA N. KUTZAROVA
Abstract: We give new examples of weak Hilbert spaces.
1. Introduction
The Banach space properties weak type 2 and weak cotype 2 were introduced and
studied by V. Milman and G. Pisier [MP]. Later, Pisier [P] studied spaces which are both
of weak type 2 and weak cotype 2 and called them weak Hilbert spaces. Weak Hilbert
spaces are stable under passing to subspaces, dual spaces, and quotient spaces. The canonical
example of a weak Hilbert space which is not a Hilbert space is convexified Tsirelson space
T(2)
[CS, J1, J2, P]. Tsirelson's space was introduced by B.S. Tsirelson [T] as the first
example of a Banach space which does not contain an isomorphic copy of c0 or p, 1 p < .
Today, we denote by T the dual space of the original example of Tsirelson since in T we
have an important analytic description of the norm due to Figiel and Johnson [FJ]. In [J1],
Johnson introduced modified Tsirelson space TM . Later, Casazza and Odell [CO] proved
the surprising fact that TM is naturally isomorphic to the original Tsirelson space T. At
this point, all the nontrivial examples of weak Hilbert spaces (i.e. those which are not
Hilbert spaces) had unconditional bases and had subspaces which failed to contain 2. A.
Edgington [E] introduced a class of weak Hilbert spaces with unconditional bases which are
2saturated. That is, every subspace of the space contains a further subspace isomorphic to
