Skip to content
DraconiticResearch
Search Sources API
182,460 studies31 sourcesUpdated 32 min ago
PreprintCC0arXiv

Functional Model for Isometric Pairs Satisfying an Automorphic Covariance Relation - I

  • Sourav Ghosh
  • Haripada Sau
Open at sourcePDFDOICite

Abstract

We derive a canonical functional model for a pair of isometric operators (V1, V2) acting on a Hilbert space β„‹ that satisfy the automorphic covariance relation V1 V2 = V2 Ο†(V1), where Ο† is an automorphism of the unit disk 𝔻. This extends the classical Bercovici--Douglas--Foias functional model for commuting isometric pairs to a dynamically twisted setting. When both V1 and V2 are pure isometries (shifts), the model simplifies to (Mz, MΘ WΟ†*) acting on H2(β„±), where WΟ† is a unitary weighted composition operator on the Hardy space H2 and Θ∈ H∞(𝔻, ℬ(β„±)) is an operator-valued inner function such that MΘ is a shift. As an application, we establish a complete unitary classification theorem for automorphically covariant isometric pairs by identifying a complete set of joint unitary invariants. Notably, our results are novel even in the special case of rotation automorphisms (V1 V2 = q V2 V1 with |q| = 1).

Related

  1. Stable base change from unitary groups in three variables and integral relation of automorphic periods
    Tristan Ricoul
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Functional Causal Discovery via Conditional Covariance Ordering
    Keyu Li, Ruoxu Tan
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  3. Mixture-of-Finite-Mixtures Wishart Model for Clustering Covariance Matrices with an Application to Brain Functional Connectivity
    Zongyu Li, Stefano Castruccio, Zhiyong Zhang
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  4. Equivariance, Curvature and Symmetry in Functional Covariance Estimation
    Jocelyn Nembe
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  5. Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
    Jaesung Kwon
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{ghosh2026functional,
  title = {{Functional Model for Isometric Pairs Satisfying an Automorphic Covariance Relation - I}},
  author = {Sourav Ghosh and Haripada Sau},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26773},
  eprint = {2610.26773},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26773},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Functional Model for Isometric Pairs Satisfying an Automorphic Covariance Relation - I
AU  - Sourav Ghosh
AU  - Haripada Sau
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26773
UR  - https://arxiv.org/abs/2610.26773
AB  - We derive a canonical functional model for a pair of isometric operators (V_1, V_2) acting on a Hilbert space β„‹ that satisfy the automorphic covariance relation V_1 V_2 = V_2 Ο†(V_1), where Ο† is an automorphism of the unit disk 𝔻. This extends the classical Bercovici--Douglas--Foias functional model for commuting isometric pairs to a dynamically twisted setting. When both V_1 and V_2 are pure isometries (shifts), the model simplifies to (M_z, M_Θ W_Ο†^*) acting on H^2(β„±), where W_Ο† is a unitary weighted composition operator on the Hardy space H^2 and Θ∈ H^∞(𝔻, ℬ(β„±)) is an operator-valued inner function such that M_Θ is a shift. As an application, we establish a complete unitary classification theorem for automorphically covariant isometric pairs by identifying a complete set of joint unitary invariants. Notably, our results are novel even in the special case of rotation automorphisms (V_1 V_2 = q V_2 V_1 with |q| = 1).
ER  -