Reducibility and triangularizability of semitransitive spaces of operators
Bernik, Janez, 1967- ...
A linear space ▫$\mathcal{L}$▫ of operators on a vector space ▫$X$▫ is called semitransitive if,given two nonzero vectors ▫$x,y \in X$▫, there exists an element ▫$A \in \mathcal{L}$▫ 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 ▫$\mathcal{L}$▫. We are able to relate the existence of an invariant subspace for ▫$\mathcal{L}$▫ to the properties of some rank varieties that we associate to ▫$\mathcal{L}$▫. Using this relation we show that, if the dimension of ▫$\mathcal{L}$▫ is the same as the dimension of ▫$X$▫, which is minimal possible, then ▫$\mathcal{L}$▫ 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.
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