Let \(M\) be a non compact Hermitian symmetric space with the Kähler form \(\omega_0\) and \(M \subset V\) the canonical Koecher imbedding of \(M\) into the corresponding positive Jordan triple system \(V\). Let \(V \subset M^*\) be the Borel imbedding of \(V\) as an open dense submanifold of the dual compact Hermitian symmetric space \(M^*\) with the Kähler form \(\omega_{FS}\). Using the Bergman operator \(B(z,w) \in \text{End}(V)\), \(z,w \in V\) associated with the triple product of \(V\), the authors construct a real analytic diffeomorphism \[ \Psi : M \rightarrow V, \qquad z \to B(z,z)^{-1/4}z \] and proves that it satisfies some remarkable properties. For example, it transforms the symplectic forms \((\omega_{FS}, \omega_0)\) of \(V\) into the symplectic forms \((\omega_0, \omega_{\text{hyp}})\), respectively, where \(\omega_0\) denotes the standard constant form on \(V\) and its restriction to \(M\). Also, \(\Psi\) commutes with the action of the isotropy group \(H\) of the origin \(0 \in M \subset V\), it maps totally geodesic submanifolds of \(M\) through the origin \(0\) into linear subspaces of \(V\) and it satisfies some heredity property.