On the semigroup of Q-monotone symplectic operators

It is known that the semigroup of Q-monotone symplectic operators has a polar decomposition H exp W and a triple decomposition Γ⁺ HΓ^-. The aim of the present paper is to give a complete representation which decomposes exp W into the triple decomposition Γ⁺ HΓ^-. By the semigroup property of Γ⁺ HΓ^-, we obtain a binary operation on the cone of positive operators whose automorphism group is the orthogonal group O(n).