 
Summary: SOME EQUIVALENT NORMS ON THE HILBERT SPACE
G. ANDROULAKIS, F. SANACORY
Abstract. We present a family of some new Tsirelsontype norms for the separable Hilbert
space. Our results extend some results of S. Bellenot, J. BernuŽes and I. Deliyanni and provide
candidates for distorted norms on the Hilbert space.
In this note we present a family of some new Tsirelsontype norms for the separable Hilbert
space 2. The motivation for presenting these norms is the following question of E. Odell
and Th. Schlumprecht [1] which is also mentioned by T.W. Gowers [5]:
Question 1. Is it possible, for > 0 to explicitly define an equivalent norm  ·  on 2
such that every infinite dimensional subspace Y of 2 contains two vectors y1 and y2 with
y1 2 = y2 2 = 1 (where · 2 denotes the usual norm of 2) and y1/y2 > ?
An implicitly defined norm with the above property exists by the solution of the famous
distortion problem by Odell and Schlumprecht [1, 2]. The family of norms that we present
gives candidates for the solution of Question 1. Some of the norms of our family were first
presented by S. Bellenot [3] which recently A.M. Pelczar [6] proved that these norms do not
answer Question 1. Another purpose of present note, is to extend some results of Bellenot
[3], J. BernuŽes and I. Deliyanni [4].
In order to define the new norms on 2 we first introduce some notation. For x = (x(i)) 2
and E N we denote by Ex the natural projection of x on E, i.e. Ex = ((Ex)(i)) where
(Ex)(i) = x(i) for all i E and (Ex)(i) = 0 otherwise. Let c00 be the vector space of scalar
