 
Summary: Contemporary Mathematics
A note on the method of minimal vectors
George Androulakis
Abstract: The methods of "minimal vectors" were introduced by Ansari and
Enflo and strengthened by Pearcy, in order to prove the existence of hyperinvariant
subspaces for certain operators on Hilbert space. In this note we present the method
of minimal vectors for operators on superreflexive Banach spaces and we give a
new sufficient condition for the existence of hyperinvariant subspaces of certain
operators on these spaces..
1. Introduction
The Invariant Subspace Problem (I.S.P.) asks whether there exists a separa
ble infinite dimensional Banach space on which every operator has a nontrivial
invariant subspace. By "operator" we always mean "continuous linear map", by
"subspace" we mean "closed linear manifold", and by "nontrivial" we mean "dif
ferent than zero and the whole space". Several negative solutions to the I.S.P.
are known [4] [5] [13] [14], [15], [16]. It remains unknown whether the separable
Hilbert space is a positive solution to the I.S.P.. There is an extensive literature of
results towards a positive solution of the I.S.P. especially in the case of the infinite
dimensional separable complex Hilbert space 2. We only mention Lomonosov's
result: every operator which is not a multiple of the identity and commutes with
