Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.08420

Corona theorem for the quaternionic Hardy space

Zhaopeng Lin·Yufeng Lu·Chao Zu
View original source ↗Read source document ↗Export metadata ↓

Abstract

The finite-generator corona theorem for bounded slice regular functions on the quaternionic unit ball was recently established by Colombo, Pozzi, Sabadini, and Wick. In the present paper, we extend the quaternionic \(H^\infty\)-corona theorem to countably many generators and obtain quantitative estimates that are independent of the cardinality of the generating family. We also establish the corresponding \(H^p\)-corona theorem for the full range \(1\leq p<\infty\), with quantitative norm estimates for both finite and countable families of generators. In the Hilbert-space setting, we prove a quaternionic Leech factorization theorem for Hardy-space multipliers and derive, as a consequence, a Toeplitz corona characterization. Our approach is based on a fixed-slice \(2\times2\) complex matrix realization of the slice regular product, together with operator-valued corona and factorization techniques.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhaopeng Lin, Yufeng Lu, Chao Zu. 2026-08-14. Corona theorem for the quaternionic Hardy space. https://arxiv.org/abs/2608.08420

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗
arXiv2026

Compactness of Toeplitz Operators on the Bergman Space

Let $φ\in L^\infty(\D)$. We study compactness criteria for \(T_φ\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $φ$, however, its Berezin transform $\tildeφ$ does not readily reveal the intrinsic properties of $φ$. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_φ$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.

math.FA↗
arXiv2026

Order automorphisms of partial isometries in $M_n(\mathbb C)$

We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms considered herein admit no such conventional matrix representations. Instead, they are essentially governed by matrices such that $I-(A+A^*)$ is either positive or negative invertible. The results reveal that the structural features of order automorphisms for partial isometries are substantially more intricate than those of classical subspace automorphisms.

math.FA↗