Dilations, Wandering Subspaces, and Inner Functions

TL;DR

This paper explores wandering subspaces for operator tuples on Hilbert spaces, characterizing them via inner functions in analytic spaces, and refines results on dilations of pure row contractions.

Contribution

It introduces a new description of wandering subspaces using inner functions for a broad class of Hilbert spaces and improves a key dilation uniqueness result.

Findings

Wandering subspaces are characterized by $\\mathcal{H}_K$-inner functions.

$\\mathcal{H}_K$-inner functions are shown to be contractive multipliers.

A refinement of Arveson's dilation uniqueness theorem is proved.

Abstract

The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb C^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1}, \ldots ,M_{z_n})$ can be described in terms of suitable $\mathcal{H}_K$-inner functions. We prove that $\mathcal{H}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way we prove a refinement of a result of Arveson on the uniqueness of minimal dilations of pure row contractions.