  We examine densely defined (but possibly unbounded) multiplication operators
in Hilbert function spaces possessing a complete Nevanlinna-Pick (CNP) kernel.
For such a densely defined operator $T$, the domains of $T$ and $T^*$ are
reproducing kernel Hilbert spaces contractively contained in the ambient space.
We study several aspects of these spaces, especially the domain of $T^*$, which
can be viewed as analogs of the classical deBranges-Rovnyak spaces in the unit
disk.
