Applications of Hilbert Module Approach to Multivariable Operator Theory

Jaydeb Sarkar ()
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Jaydeb Sarkar: Statistics and Mathematics Unit, Indian Statistical Institute

Chapter 39 in Operator Theory, 2015, pp 1035-1091 from Springer

Abstract: Abstract A commuting n-tuple ( T 1 , … , T n ) $$(T_{1},\ldots,T_{n})$$ of bounded linear operators on a Hilbert space ℋ $$\mathcal{H}$$ associates a Hilbert module ℋ $$\mathcal{H}$$ over ℂ [ z 1 , … , z n ] $$\mathbb{C}[z_{1},\ldots,z_{n}]$$ in the following sense: ℂ [ z 1 , … , z n ] × ℋ → ℋ , ( p , h ) ↦ p ( T 1 , … , T n ) h , $$\displaystyle{\mathbb{C}[z_{1},\ldots,z_{n}] \times \mathcal{H}\rightarrow \mathcal{H},\quad \quad (p,h)\mapsto p(T_{1},\ldots,T_{n})h,}$$ where p ∈ ℂ [ z 1 , … , z n ] $$p \in \mathbb{C}[z_{1},\ldots,z_{n}]$$ and h ∈ ℋ $$h \in \mathcal{H}$$ . 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”.

Keywords: Hilbert Modules; Essential Normality; Commutant Lifting Theorem; Cowen-Douglas Class; Quotient Module (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_69

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DOI: 10.1007/978-3-0348-0667-1_69

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