Unbounded multipliers of complete Pick spaces

Abstract

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 $T^{\ast }$ are reproducing kernel Hilbert spaces contractively contained in the ambient space. We study several aspects of these spaces, especially the domain of $T^{\ast }$, which can be viewed as analogs of the classical deBranges-Rovnyak spaces in the unit disk.