Let $mathbb{H}_g$ and $mathbb{D}_g$ be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let $mathbb{C}^{(h,g)}$ be the Euclidean space of all $h{ imes}g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk $mathbb{D}_g{ imes}mathbb{C}^{(h,g)}$ onto the Siegel-Jacobi space $mathbb{H}_g{ imes}mathbb{C}^{(h,g)}$ which gives a partial bounded realization of $mathbb{H}_g{ imes}mathbb{C}^{(h,g)}$ by $mathbb{D}_g{ imes}mathbb{C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on $mathbb{D}_g{ imes}mathbb{C}^{(h,g)}$. and $mathbb{H}_g{ imes}mathbb{C}^{(h,g)}$. are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel Jacobi disk $mathbb{D}_g{ imes}mathbb{C}^{(h,g)}$. invariant under the natural action of the Jacobi group $mathbb{D}_g{ imes}mathbb{C}^{(h,g)}$ explicitly.