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