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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) ≤ kr(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.