  We investigate algebraic and topological transitivity and, more generally,
k-transitivity for linear spaces of operators. In finite dimensions, we
determine minimal dimensions of k-transitive spaces for every k, and find
relations between the degree of transitivity of a product or tensor product on
the one hand and those of the factors on the other. We present counterexamples
to some natural conjectures. Some infinite dimensional analogues are discussed.
A simple proof is given of Arveson's result on the weak-operator density of
transitive spaces that are masa bimodules.
