Applications of Hilbert Module Approach to Multivariable Operator Theory

Sarkar, Jaydeb

doi:10.1007/978-3-0348-0692-3_69-1

Cited by 5 publications

(5 citation statements)

References 62 publications

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“…[XCGu03,P94,R88]) that no proper submodule is unitarily equivalent to the Bergman module itself. These issues are thoroughly discussed in [Sa14a]. (3) In a sense, the existence of a Hilbert module with unitarily equivalent submodules is a rare phenomenon.…”

Section: Further Results and Commentsmentioning

confidence: 99%

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An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Sarkar

2015

Operator Theory

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Section: Further Results and Commentsmentioning

confidence: 99%

“…a (D) and S 1 ∼ = S 2 , then S 1 = S 2 (see [R88] or Corollary 8.5 in [Sa14a]). Therefore, on the one hand every non-zero submodule is isometrically isomorphic to the module itself while on the other hand, no proper submodule is.…”

Section: Unitarily Equivalent Submodulesmentioning

confidence: 99%

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Sarkar

2015

Operator Theory

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“…. , M z d acting on H ca (T ) (refer to [23,Section 2] for the general theory of Hilbert modules over the algebra of polynomials). We refer to H ca (T ) as the Drury-Arveson-type Hilbert module associated with T .…”

Section: A Classification Problemmentioning

confidence: 99%

Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

Chavan¹,

Pradhan²,

Trivedi³

2017

Preprint

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Given a directed Cartesian product T of locally finite, leafless, rooted directed trees T1, . . . , T d of finite joint branching index, one may associate with T the Drury-Arveson-type C[z1, . . . , z d ]-Hilbert module Hc a (T ) of vector-valued holomorphic functions on the open unit ball B d in C d , where a > 0. In case all directed trees under consideration are without branching vertices, Hc a (T ) turns out to be the classical Drury-Arveson-type Hilbert module Ha associated with the reproducing kernel 1 (1− z,w ) a defined on B d . Unlike the case of d = 1, the above association does not yield a reproducing kernel Hilbert module if we relax the assumption that T has finite joint branching index. The main result of this paper classifies all directed Cartesian product T for which the Hilbert modules Hc a (T ) are isomorphic in case a is a positive integer. One of the essential tools used to establish this isomorphism is an operator-valued representing measure arising from Hc a (T ). Further, a careful analysis of these Hilbert modules allows us to prove that the cardinality of the k th generation (k = 0, 1, . . .) of T1, . . . , T d are complete invariants for Hc a (•) provided ad = 1. Failure of this result in case ad = 1 may be attributed to the von Neumann-Wold decomposition for isometries. Along the way, we identify the joint cokernel E of the multiplication d-tuple Mz on Hc a (T ) with orthogonal direct sum of tensor products of certain hyperplanes. In case T has finite joint branching index, this readily yields a neat formula for the dimension of E.

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“…(2) If H = L 2 a (D) and S 1 ∼ = S 2 , then S 1 = S 2 (see [R88] or Corollary 8.5 in [Sa14a]). Therefore, on the one hand every non-zero submodule is isometrically isomorphic to the module itself while on the other hand, no proper submodule is.…”

Section: Unitarily Equivalent Submodulesmentioning

confidence: 99%

“…[XCGu03, P94, R88]) that no proper submodule is unitarily equivalent to the Bergman module itself. These issues are thoroughly discussed in [Sa14a].…”

Section: Hilbert Modules Over A(d)mentioning

confidence: 99%

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Sarkar¹

2013

Preprint

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Let {T 1 , . . . , T n } be a set of n commuting bounded linear operators on a Hilbert space H. Then the n-tuple (T 1 , . . . , T n ) turns H into a module over C[z 1 , . . . , z n ] in the following sense: C[z 1 , . . . , z n ] × H → H, (p, h) → p(T 1 , . . . , T n )h, where p ∈ C[z 1 , . . . , z n ] and h ∈ H. The above module is usually called the Hilbert module over C[z 1 , . . . , z n ]. Hilbert modules over C[z 1 , . . . , z n ] (or natural function algebras) were first introduced by R. G. Douglas and C. Foias in 1976.The two main driving forces were the algebraic and complex geometric views to multivariable operator theory. This article gives an introduction of Hilbert modules over function algebras and surveys some recent developments. Here the theory of Hilbert modules is presented as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces and its applications to the theory of n-tuples (n ≥ 1) of commuting operators. The topics which are studied include: model theory from Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, curvature and Fredholm Hilbert modules. More developments in the study of Hilbert module approach to operator theory can be found in a companion paper, "Applications of Hilbert Module Approach to Multivariable Operator Theory".

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Applications of Hilbert Module Approach to Multivariable Operator Theory

Abstract: Abstract. A commuting n-tuple (T 1 , . . . , T n ) of bounded linear operators on a Hilbert space H associate a Hilbert module H over C[z 1 , . . . , z n ] in the following sense:

Cited by 5 publications

References 62 publications

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

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