Unbounded multipliers of complete Pick spaces

TL;DR

This paper investigates unbounded multiplication operators in Hilbert function spaces with complete Pick kernels, focusing on their domains and analogs of deBranges-Rovnyak spaces, revealing new structural insights.

Contribution

It introduces a detailed analysis of unbounded multiplication operators in CNP spaces, highlighting the structure of their domains and associated spaces, which was not previously well-understood.

Findings

Domains of operators and adjoints are reproducing kernel Hilbert spaces

Identifies analogs of deBranges-Rovnyak spaces in this context

Provides new structural understanding of unbounded multipliers

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.