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SOME MORE WEAK HILBERT SPACES GEORGE ANDROULAKIS, PETER G. CASAZZA, AND DENKA N. KUTZAROVA
 

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 non-trivial 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
2-saturated. That is, every subspace of the space contains a further subspace isomorphic to

  

Source: Androulakis, George - Department of Mathematics, University of South Carolina

 

Collections: Mathematics