Abstract. Using the technique of minimal vectors, as introduced by Ansari and Enflo and generalized by Troitsky, we provide new results concerning the existence of invariant subspaces for operators satisfying conditions similar to those of Lomonosov's theorems. Most of the operators to which the theory applies are quasinilpotent, but a wide class of weighted shifts, including vectorial shifts, is studied as a further application.

AMS Subject Classification (1991): 41A29, 47A15, 47B37, 46B10

Keyword(s): Minimal vectors, Hyperinvariant subspaces, Weighted shifts, Quasinilpotent operators

Received September 6, 2004, and in final form June 1, 2005. (Registered under 5889/2009.)