Quasi-wandering Subspaces in the Bergman Space

Izuchi, Keiji; Izuchi, Kou Hei; Izuchi, Yuko

doi:10.1007/s00020-010-1744-4

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…Their proof was quite difficult, and famously involved biharmonic functions. Since then, considerably simpler proofs have been given by Shimorin, and later Izuchi, and Sun and Zheng ( [34], [35], [20], [37]). Shimorin has extended the ARS theorem to weighted A 2 spaces, A 2 (C, D 1 (0, 1), kdµ) of the unit disk, albeit with restrictions on the weights k such as circular symmetry and a subharmonicity condition ( [34], [35]).…”

Section: The Abstract Beurling's Theoremmentioning

confidence: 99%

Beurling's Theorem for Valuation Hilbert Modules and Several Complex Variables

Neville¹

2022

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We develop a theory of Valuation Hilbert Modules and prove a version of Beurling's theorem for these. Then we apply our version of Beurling's theorem to obtain complete descriptions of the closed invariant subspaces of a number of Hilbert spaces of analytic functions in several complex variables, including H 2 of the polydisk the ball, and bounded symmetric domains, and weighted A 2 spaces on complex analytic manifolds.

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Section: The Abstract Beurling's Theoremmentioning

confidence: 99%

Beurling's Theorem for Valuation Hilbert Modules and Several Complex Variables

Neville¹

2022

Preprint

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“…Their proof was quite difficult, and famously involved biharmonic functions. Since then, considerably simpler proofs have been given by Shimorin, and later Izuchi, and Sun and Zheng ( [22], [23], [13], [24]). Shimorin has extended the ARS theorem to weighted A 2 spaces, A 2 (C, D 1 (0, 1), kdµ) of the unit disk, albeit with restrictions on the weights k such as circular symmetry and a subharmonicity condition ( [22], [23]).…”

Section: A 2 Spacesmentioning

confidence: 99%

An Abstract Beurlings Theorem for Several Complex Variables II

Neville

2021

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In a previous paper, we presented an Abstract Beurling's Theorem for valuation Hilbert modules over valuation algebras. In this paper, we shall apply this theorem to obtain complete descriptions of the closed invariant subspaces of a number of Hilbert spaces of analytic functions in several complex variables.2020 Mathematics Subject Classification. 32A36 (42B30 47A15). Dedicated to the memory of the late Lee Rubel. Lee, this one's for you. 1 2 CHARLES W. NEVILLE but they will be easy to distinguish from the corresponding quantities on H, which will be unsubscripted.If X is a subset of a larger topological space Y , we shall denote the closure of X by cl X = cl Y X, the interior of X by int X = int Y X, and the topological boundary of X by ∂X = ∂ Y X. Finally, card(X) will denote the cardinality of the set X.If p ∈ C n and 0 < r < ∞, D n (p, r) will denote the n-dimensional polydisk of radius r centered at p, D n (p, r)r) will denote the n-dimensional polydisk centered at p of polyradius r. Valuation algebras and valuation Hilbert modules.We also must review the definitions and basic properties of valuation algebras and valuation Hilbert modules from paper I. Again, we quote directly from that paper: Definition 2.1. Let R be a (possibly non-commutative) algebra. An algebra valuation on R is a function ord: R → Z + ∪ {∞} such that for all r and s ∈ R,(2.1.1) ord(r) = 0 if r is a left or right unit of R.(2.1.2) ord(r) = ∞ if and only if r = 0.(2.1.3) ord(rs) ≥ ord(r) + ord(s).(2.1.4) ord(λr) = ord(r) for λ ∈ C, λ = 0 (2.1.5) ord(r + s) ≥ min(ord(r), ord(s))Of course, condition (2.1.1) is satisfied vacuously if R does not have a two sided identity element.Here and throughout, we follow the usual conventions with respect to ∞: m < ∞ for all m ∈ Z, and for such m, mDefinition 2.2. A valuation algebra is an ordered pair (R, ord), where R is a (possibly non-commutative) algebra and ord is an algebra valuation on R.Definition 2.3. Let (R, ord R ) be a valuation algebra, and let H be a Hilbert space which is a left R Hilbert module. A Hilbert module valuation on H (with respect to (R, ord R )) is a function ord : H → Z + ∪ {∞} such that for all h, h 1 and h 2 ∈ H, and for all r ∈ R,(2.3.5) The ord function is upper semi-continuous on H.Note that each set H m = {h ∈ H : ord(h) ≥ m} is a subspace by properties (2.3.3) and (2.3.4). The subspaces H m are closed for m = 0, 1, 2, . . . because ord is upper semi-continuous. The subspace H 0 is automatically a closed subspace because H 0 = H. Fiinaly, as part of the definition of an R Hilbert module, the multiplication map m r : h → rh is continuous for each r ∈ R.Definition 2.4. Let (R, ord R ) be a valuation algebra. A valuation Hilbert module over (R, ord R ) is an ordered pair (H, ord), where H is a left Hilbert module over R and ord is a Hilbert module valuation on H with respect ot (R, ord R ).

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“…Nonorthogonal representations of a closed shift-invariant subspace M of A n,Y based on the wandering subspace E = M ⊖ S n M appears in [6] (for n = 2) and [97] (for n = 3). Another version of a non-orthogonal representation of M is that of Izuchi-Izuchi-Izuchi [63] (see also [39]) is based on the notion of quasi-wandering subspace Q = P M S n | M ⊥ . Fleshing out of this approach for the ω-setting is the topic of Section 6.1.…”

Section: Non-orthogonal Beurling-lax Representations Based On Wanderi...mentioning

confidence: 99%

“…as sets. The following result is the quasi-wandering version of the Beurling-Lax theorem; the proof is adapted from the commutative versions [63,39].…”

Section: Beurling-lax Representations Based On Quasi-wandering Subspacesmentioning

confidence: 99%

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Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting

Ball,

Bolotnikov

2019

Preprint

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Noncommutative formal power series and reproducing kernel Hilbert spaces, contractive and inner multipliers, shift-invariant subspace, hypercontraction, Bergman-inner function and Bergman-inner family, functional model Contents Chapter 1. Introduction 1.1. Overview 1.2. Standard weighted Bergman spaces 1.3. The Fock space setting 1.4. Weighted Bergman-Fock spaces Chapter 2. Formal Reproducing Kernel Hilbert Spaces 2.1. Basic definitions 2.2. Weighted Hardy-Fock spaces Chapter 3. Contractive multipliers 3.1. Contractive multipliers in general 3.2. Contractive multipliers between Fock spaces 3.3. A noncommutative Leech theorem 3.4. Contractive multipliers from H 2 U (F + d ) to H 2 ω,Y (F + d ) for admissible ω 3.5. H 2 ω,Y (F + d )-Bergman-inner multipliers Chapter 4. Stein relations and observability-operator range spaces 4.1. Observability operators and gramians 4.2. Shifted ω-gramians 4.3. The model shift-operator tuple on H 2 ω,Y (F + d ) 4.4. A characterization of the model shift-operator tuple on H 2 ω,Y (F + d ) 4.5. Observability-operator range spaces Chapter 5. Beurling-Lax theorems based on contractive multipliers 5.1. Beurling-Lax theorem via McCT-inner multipliers and more general contractive multipliers 5.2. Representations with model space of the form n j=1 A j,Uj (F + d ) Chapter 6. Non-orthogonal Beurling-Lax representations based on wandering subspaces 6.1. Beurling-Lax representations based on quasi-wandering subspaces 6.2. Non-orthogonal Beurling-Lax representations based on wandering subspaces Chapter 7. Orthogonal Beurling-Lax representations based on wandering subspaces 7.1. Transfer functions Θ ω,U β and metric constraints 7.2. Beurling-Lax representations based on Bergman-inner families 7.3. Expansive multiplier property 7.4. Bergman-inner multipliers as extremal solutions of interpolation problems Chapter 8. Model theory for ω-hypercontractive operator d-tuples 8.1. Model theory based on contractive-multiplier/McCT-inner multiplier as characteristic function 8.2. Model theory for * -ω strongly stable hypercontractions via characteristic Bergman-inner families 8.3. Model theory for n-hypercontractions Chapter 9. Hardy-Fock spaces built from a regular formal power series 9.1. Introduction 9.2. Contractive multipliers from H 2 p,U (F + d ) to H 2 ωp,n,Y (F + d ) 9.3. Output stability, Stein equations and inequalities 9.4. The ω p,n -shift model operator tuple S ω p,n ,R 9.5. Observability operator range spaces in H 2 ωp,n,Y (F + d ) 9.6. Beurling-Lax theorems: (p, n)-versions 9.7. H 2 ωp,n,Y (F + d )-Bergman-inner families 9.8. Operator model theory for c.n.c. * -(p, n)-hypercontractive operator tuples T Bibliography CHAPTER 1 1.2. STANDARD WEIGHTED BERGMAN SPACES 3.5. H 2 ω,Y (F + d )-BERGMAN-INNER MULTIPLIERS 49

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Quasi-wandering Subspaces in the Bergman Space

Cited by 9 publications

References 13 publications

Beurling's Theorem for Valuation Hilbert Modules and Several Complex Variables

Beurling's Theorem for Valuation Hilbert Modules and Several Complex Variables

An Abstract Beurlings Theorem for Several Complex Variables II

Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting

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