  We prove that independent rectangular random matrices, when embedded in a
space of larger square matrices, are asymptotically free with amalgamation over
a commutative finite dimensional subalgebra $D$ (under an hypothesis of unitary
invariance). Then we consider elements of a finite von Neumann algebra
containing $D$, which have kernel and range projection in $D$. We associate
them a free entropy with the microstates approach, and a free Fisher's
information with the conjugate variables approach. Both give rise to
optimization problems whose solutions involve freeness with amalgamation over
$D$. It could be a first proposition for the study of operators between
different Hilbert spaces with the tools of free probability. As an application,
we prove a result of freeness with amalgamation between the two parts of the
polar decomposition of $R$-diagonal elements with non trivial kernel.
