Contractive Hilbert modules on quotient domains

2024/09/17 by Shibananda Biswas, Biswas, Shibananda, Gargi Ghosh +4 · 1 citation
Mathematics · #20F55 #47A13 #47A25 #47B32 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2409.11101

openalex publication_date 2024/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let the complex reflection group G(m,p,n) act on the unit polydisc \mathbb Dⁿ in \mathbb Cⁿ. A \boldsymbolΘ_n-contraction is a commuting tuple of operators on a Hilbert space having \boldsymbolΘ_n:=\\boldsymbolθ(z)=(θ₁(z),…,θ_n(z)):z∈\mathbb Dⁿ\ as a spectral set, where \θ_i\_i=1ⁿ is a homogeneous system of parameters associated to G(m,p,n). A plethora of examples of \boldsymbolΘ_n-contractions is exhibited. Under a mild hypothesis, it is shown that these \boldsymbolΘ_n-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of \boldsymbolΘ_n-isometries is obtained.

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• Function theoretic aspects of the symmetrized polydisc and generalization

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