Operator-stable distributions and stable marginals

Sharpe has shown that full operator-stable distributions μ on Rⁿ are infinitely divisible and for a suitable automorphism B depending on μ satisfy the relation $\text{μ}{}^{\mathrm{t}}\text{= μt}{}^{\mathrm{?B}}\text{1 δ(b(t))}$ for all t > 0. B is called an exponent for μ. It is proved here that if an operator-stable distribution on Rⁿ has n linearly independent univariate stable marginals, then its exponents are semi-simple operators. In addition necessary and sufficient conditions are given for such a distribution on R² to have univariate stable marginals. The proofs use a hitherto unpublished result of Sharpe's that all full operator-stable distributions are absolutely continuous. His proof is provided here.