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EMBEDDING INTO THE SPACE OF BOUNDED OPERATORS ON CERTAIN BANACH SPACES
 

Summary: EMBEDDING INTO THE SPACE OF BOUNDED OPERATORS ON
CERTAIN BANACH SPACES
G. ANDROULAKIS, K. BEANLAND
, S.J. DILWORTH, F. SANACORY
Abstract. Sufficient conditions are given on a Banach space X which ensure that em-
beds in L (X), the space of all bounded linear operators on X. A basic sequence (en) is said
to be quasisubsymmetric if for any two increasing sequences (kn) and ( n) of positive integers
with kn n for all n, (ekn
) dominates (e n
). If a Banach space X has a seminormalized
quasisubsymmetric basis then embeds in L (X).
1. Introduction
The famous open problem of whether there exists an infinite-dimensional Banach space on
which every (bounded linear) operator is a compact perturbation of a multiple of the identity,
is attributed to S. Banach (see related papers [G], [GM], and [S]). One of the reasons that
this problem has attracted a lot of attention is that if such a space X exists then by the
results of [AS] or [L], X provides a positive solution to the Invariant Subspace Problem for
Banach spaces, namely every operator on X has a non-trivial (i.e. different from zero and
the whole space) invariant subspace. Notice that if a space X satisfies the assumptions of
the above problem of Banach and if in addition X has the Approximation Property (AP)

  

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

 

Collections: Mathematics