Hardy Spaces and Toeplitz operators on the Pentablock
Abstract
The pentablock is a domain in ℂ3 arising in the study of the μ-synthesis problem and is defined by 𝒫 = { (a21,tr(A),det(A)): A=[aij]i,j=12∈ M2(ℂ), |A|<1 }. We introduce the Hardy space H2(𝒫) associated with the canonical measure μ on the distinguished boundary b𝒫. We identify H2(𝒫) with a closed subspace of L2(b𝒫,μ) and show that it is a reproducing kernel Hilbert space. We establish a new description of 𝒫 and its closure and obtain a uniqueness set. In particular, we prove that 𝒫 is the interior of the polynomial convex hull of Ψ(𝔻̅3), where Ψ(ξ,λ1,λ2) = ( (ξ(λ1−λ2))/2, λ1+λ2, λ1λ2 ) is the parametrization map for 𝒫. We also show that H2(𝒫) embeds isometrically as a closed subspace of H2(𝔻3). We then obtain a Brown-Halmos type characterization of Toeplitz operators on H2(𝒫) and a commutant lifting theorem for 𝒫-isometries and their minimal 𝒫-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize 𝒫-unitaries as images of commuting triples of unitaries under Ψ and present a class of 𝒫-contractions from triples having 𝔻̅3 as a spectral set.
Related
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Two-parameter Toeplitz-type operators between Fock spaces
- Toeplitz and Hankel operators on Bergman spaces with doubling weights
- Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces
- Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
BibTeX
@misc{su2026hardy,
title = {{Hardy Spaces and Toeplitz operators on the Pentablock}},
author = {Guicong Su and Nitin Tomar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17729},
eprint = {2610.17729},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17729},
}
RIS
TY - UNPB
TI - Hardy Spaces and Toeplitz operators on the Pentablock
AU - Guicong Su
AU - Nitin Tomar
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17729
UR - https://arxiv.org/abs/2610.17729
AB - The pentablock is a domain in ℂ^3 arising in the study of the μ-synthesis problem and is defined by 𝒫 = { (a_21,tr(A),det(A)): A=[a_ij]_i,j=1^2∈ M_2(ℂ), |A|<1 }. We introduce the Hardy space H^2(𝒫) associated with the canonical measure μ on the distinguished boundary b𝒫. We identify H^2(𝒫) with a closed subspace of L^2(b𝒫,μ) and show that it is a reproducing kernel Hilbert space. We establish a new description of 𝒫 and its closure and obtain a uniqueness set. In particular, we prove that 𝒫 is the interior of the polynomial convex hull of Ψ(𝔻̅^3), where Ψ(ξ,λ_1,λ_2) = ( (ξ(λ_1−λ_2))/2, λ_1+λ_2, λ_1λ_2 ) is the parametrization map for 𝒫. We also show that H^2(𝒫) embeds isometrically as a closed subspace of H^2(𝔻^3). We then obtain a Brown-Halmos type characterization of Toeplitz operators on H^2(𝒫) and a commutant lifting theorem for 𝒫-isometries and their minimal 𝒫-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize 𝒫-unitaries as images of commuting triples of unitaries under Ψ and present a class of 𝒫-contractions from triples having 𝔻̅^3 as a spectral set.
ER -