The method of minimal vectors was introduced by Ansari and Enflo (1998) in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. Pearcy (2002) used it to prove a version of Lomonosov's theorem. Androulakis (2002) adapted the technique to super-reflexive Banach spaces. The method was independently generalized to reflexive Banach spaces by Chalendar, Partington, and Smith (2002). There has been a hope that this technique could provide a positive solution to the invariant subspace problem for these spaces. In this note we present a version of the method of minimal vectors (based the paper of Androulakis) that works for arbitrary Banach spaces. In particular, it applies in the spaces where there are known examples of operators without invariant subspaces (e.g. by Enflo and Read) This shows that the method of minimal vectors alone cannot solve the invariant subspace problem for "good" spaces.
Precisely, a variant of the method of minimal vectors is used in the paper in order to prove the following result:
Theorem. Let Q be a quasinilpotent operator on a Banach space, and suppose that there exists a closed ball B such that B does not contain the origin and for every sequence (xn) in B there is a subsequence (yi) and a sequence of operators (Ki) in the commutant of Q such that each Ki is of norm at most 1 and the sequence (Kiyi) converges in norm to a non-zero vector. Then Q has a hyperinvariant subspace.