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 \(T^*\) 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.