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STRICTLY SINGULAR, NON-COMPACT OPERATORS EXIST ON THE SPACE OF GOWERS AND MAUREY
 

Summary: STRICTLY SINGULAR, NON-COMPACT OPERATORS EXIST ON THE
SPACE OF GOWERS AND MAUREY
G. ANDROULAKIS AND TH. SCHLUMPRECHT
Abstract We construct a strictly singular non-compact operator on Gowers' and Maurey's
space GM.
1. Introduction
In 1993 W.T. Gowers and B. Maurey [GM] solved the famous "unconditional basic se-
quence problem" by constructing the first known example of a space that does not contain
any unconditional basic sequence. In the present paper this space is denoted by GM. Fur-
thermore it was shown in [GM] that the space GM is hereditarily indecomposable (HI), i.e.
no infinite dimensional subspace of GM can be decomposed into a direct sum of two further
infinite dimensional closed subspaces. As shown in [GM] every bounded operator on a com-
plex HI space can be written as a sum of a multiple of the identity and a strictly compact
operator. Recall that an operator T is strictly singular if no restriction of T to an infinite
dimensional subspace is an isomorphism. Actually Lemma 22 of [GM] implies immediately
that the real version of GM has also the property that every operator on a subspace of it
is a strictly singular perturbation of a multiple of the identity. In [GM] it is asked whether
or not every operator on GM can be written as a compact perturbation of a multiple of the
identity. If the answer to this question were positive then by [AS] the space GM would be
the first known example of an infinite dimensional Banach space such that every operator on

  

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

 

Collections: Mathematics