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Applications of Hilbert Module Approach to Multivariable Operator Theory

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Operator Theory

Abstract

A commuting n-tuple \((T_{1},\ldots,T_{n})\) of bounded linear operators on a Hilbert space \(\mathcal{H}\) associates a Hilbert module \(\mathcal{H}\) over \(\mathbb{C}[z_{1},\ldots,z_{n}]\) in the following sense:

$$\displaystyle{\mathbb{C}[z_{1},\ldots,z_{n}] \times \mathcal{H}\rightarrow \mathcal{H},\quad \quad (p,h)\mapsto p(T_{1},\ldots,T_{n})h,}$$

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 polydisk, 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”.

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  1. Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, India

    Jaydeb Sarkar

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  1. Jaydeb Sarkar
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  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

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Sarkar, J. (2015). Applications of Hilbert Module Approach to Multivariable Operator Theory. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_69

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