Applications of Hilbert Module Approach to Multivariable Operator Theory

TL;DR

This survey explores the algebraic and geometric aspects of Hilbert modules and their applications in multivariable operator theory, covering topics like dilations, submodules, and normality.

Contribution

It provides a comprehensive overview of the Hilbert module approach to multivariable operator theory, emphasizing new algebraic and geometric insights and applications.

Findings

Analysis of generalized canonical models and Cowen-Douglas class

Development of dilation and factorization techniques for reproducing kernel Hilbert spaces

Study of submodules, quotient modules, and normality phenomena in Hilbert modules

Abstract

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and geometric aspects of Hilbert module approach to operator theory and to survey several applications of the theory of Hilbert modules in multivariable operator theory. The topics which are studied include: generalized canonical models and Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert spaces, a class of simple submodules and quotient modules of the Hardy modules over polydisc, commutant lifting theorem, similarity and free Hilbert modules, left invertible multipliers, inner resolutions, essentially normal Hilbert modules, localizations of free resolutions and rigidity phenomenon. This article is a companion paper to "An Introduction to Hilbert Module Approach to Multivariable Operator Theory".