Resumen de Reducibility and triangularizability of semitransitive spaces of operators

Janez Bernik, Roman Drnovsek, Damjana Kokol Bukovsek, Tomaz Kosir, Matjaz Omladic

• A linear space L of operators on a vector space X is called semitransitive if, given two nonzero vectors x, y in X, there exists an element A in L such that either y=Ax or x=Ay. In this paper we consider semitransitive spaces of operators on a finite dimensional vector space X over an algebraically closed field. In particular, we are interested in the existence of nontrivial invariant subspaces of X for a semitransitive space L. We are able to relate the existence of an invariant subspace for L to the properties of some rank varieties that we associate to L. Using this relation we show that, if the dimension of L is the same as the dimension of X, which is minimal possible, then L is triangularizable. By contrast we show that, from n=3 onwards, there exists a minimal semitransitive space L of dimension n+1 of operators on an n-dimensional vector space X which is also irreducible. We also give a new characterization of semitransitive spaces of operators on finite dimensional vector spaces