On submultiplicativity of spectral radius and transitivity of semigroups

It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a $k$ such that the spectral radius satisfies $r(AB) \leq k r(A) r(B)$ for all $A$ and $B$ in the semigroup.

It is also shown that the spectral radius is not $k$-submultiplicative on any transitive semigroup of compact operators.