arXiv subjects

Qinghai Huo

Publications and source records attributed to Qinghai Huo.

13 recordsLinked to original sources

arXiv2026

Octonionic Riesz-Dunford functional calculus

The Riesz-Dunford functional calculus over the algebra of octonions, denoted by $\mathbb{O}$, has long been an open problem due to the nonassociativity of octonions. Two core obstacles hinder its development: first, the generalization of the resolvent operator series identity produces unexpected associator terms that invalidate standard expansions; second, the nonassociativity spoils the analyticity of the resolvent operator, a key property for defining a functional calculus via Cauchy integrals. In this paper, we initiate the study of the Riesz-Dunford functional calculus for bounded power-associative para-linear operators in Banach octonionic bimodules. To address the above issues, we introduce several pivotal concepts: power-associative operators (to eliminate the unwanted associator terms and recover valid resolvent series expansions), the notions of regular inverse of $R_s-T$ for $s\in \O$ (which serve as the octonionic versions of the resolvent operator), $\mathbb{C}_J$-extendable power-associative operators, and $\mathbb{C}_J$-liftable power-associative operators (to characterize the slice regularity of the resolvent operators). Based on these notions, we define two types of octonionic spectra: the pull-back spectrum $\sigma^*(T)$ and the push-forward spectrum $\sigma_*(T)$. These give rise to the left and right slice regular functional calculi of bounded power-associative para-linear operators, respectively. This theory unifies the Riesz-Dunford functional calculus over division algebras ($ \mathbb{C}, \mathbb{H}, \mathbb{O}$) and fills the six-decade-long gap in octonionic (nonassociative) functional analysis.

math.FA↗

arXiv2026

Octonionic isometric isomorphisms and partial isometry

Very recently, two new notions of para-linear mappings and weak associative orthonormal bases were introduced in octonionic functional analysis, which have been proved to be powerful in formulating the basic theory, such as the Riesz representation theorem and the Parseval theorem. In this article, we continue exploring more properties of these two concepts and initiate the study of octonionic para-linear isometric operators. Surprisingly, it is proven that the condition of the para-linear operator on a Hilbert octonionic bimodule being an isometric isomorphism is equivalent to it mapping any associative orthonormal basis to a weak associative orthonormal basis, which implies also that an octonionic matrix is an isometry if and only if the system of its row vectors is a weak associative orthonormal basis. Furthermore, we introduce the concept of para-linear partial isometric operators and establish the aforementioned analogue in this new setting. Based on these facts, we can provide naturally a new viewpoint of James questions by modifying the definition of octonionic Stiefel space.

math.FA↗

arXiv2026

On the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions over alternative $\ast$-algebras

Recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced and studied over Clifford algebras and octonions, respectively. In this paper, we further develop the theory of generalized partial-slice monogenic functions defined on hypercomplex subspaces and with values in a real alternative $\ast$-algebra and we concentrate on the Fueter-Sce theorem, three types of Cauchy-Kovalevskaya extensions, and their various internal relationships. The paper proposes more bridges between the theories of monogenicity, harmonicity, and generalized partial-slice monogenicity.

math.CV↗

arXiv2025

Octonionic Para-linear Self-Adjoint Operators and Spectral Decomposition

This paper presents a groundbreaking advancement in the theory of operators defined on octonionic Hilbert spaces, successfully resolving a fundamental challenge that has persisted for over six decades. Due to the intrinsic non-associative nature of octonions, conventional linear operator theory encounters profound structural difficulties. We make use of an original conceptual framework termed para-linearity, an innovative generalization of linearity that naturally accommodates the octonionic algebraic structure. Within this newly established paradigm, we systematically develop an appropriate algebraic setting by defining a carefully designed operator algebra and an adjoint operation which, together, recapture essential analytic properties previously inaccessible in this context. We identify a geometric structure, the slice cone, as the fundamental object encoding spectral properties typically derived through sesquilinear forms. We obtain a rigorous characterization of self-adjointness which indicates how to introduce a new notion of strong eigenvalues. For every compact, para-linear, self-adjoint operator with strong eigenvalues, we can establish the spectral decomposition theorem and functional calculi.

math.FA↗

arXiv2025

Monogenic functions over real alternative *-algebras: fundamental results and applications

The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. This paper explores the fundamental properties of these monogenic functions, focusing on the Cauchy-Pompeiu integral formula and Taylor series expansion in hypercomplex subspaces, among which the non-commutativity and especially non-associativity of multiplications demand full considerations. The theory presented herein provides a robust framework for understanding monogenic functions in the context of real alternative $\ast$-algebras, shedding light on the interplay between algebraic structures and hypercomplex analysis.

math.CV↗

arXiv2024

Almansi-type decomposition and Fueter-Sce theorem for generalized partial-slice regular functions

Very recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced to unify the theory of monogenic functions and of slice monogenic functions over Clifford algebras. Inspired by the work of A. Perotti, in this paper we provide two analogous versions of the Almansi decomposition in this new setting. Additionally, two enhancements of the Fueter-Sce theorem have been obtained for generalized partial-slice regular functions.

math.CV↗

arXiv2023

Schwarz lemma for harmonic functions in the unit ball

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics \cite{Markovic}. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples \cite{Melentijevic-P}. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis \cite{Zhang14,Zhang16} and octonionic analysis \cite{Wang-Bian-Liu} in a unified approach.

math.CV↗

arXiv2023

Power-bounded quaternionic operators

Recently, the conception of slice regular functions was allowed to introduce a new quaternionic functional calculus, among which the theory of semigroups of linear operators was developed into the quaternionic setting, even in a more general case of real alternative $*$-algebras. In this paper, we initiate to study the discrete case and introduce the notion of power-bounded quaternionic operators. In particular, by the spherical Yosida approximation, we establish a discrete Hille-Yosida-Phillips theorem to give an equivalent characterization of quaternionic linear operators being power-bounded. A sufficient condition of the power-boundedness for quaternionic linear operators is also given. In addition, a non-commutative version of the Katznelson-Tzafriri theorem (J. Funct. Anal. 68: 313-328, 1986) for power-bounded quaternionic operators is formulated in terms of the $S$-spectrum.

math.SP↗

arXiv2023

Non-associative Categories of Octonionic Bimodules

Category is put to work in the non-associative realm in the article. We focus on a typical example of non-associative category. Its objects are octonionic bimodules, morphisms are octonionic para-linear maps, and compositions are non-associative in general. The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem. An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule Re (f (px)-pf (x))= 0. The composition should be modified so that it preserves the octonionic para-linearity. In this non-associative category, we introduce the Hom and Tensor functors which constitute an adjoint pair. We establish the Yoneda lemma in terms of the new notion of weak functor. To define the exactness in a non-associative category, we introduce the notion of the enveloping category via a universal property. This allows us to establish the exactness of the Hom functor and Tensor functor.

math.CT↗

arXiv2021

Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilbert space. This completely removes the obstacles caused by the failure of the octonionic Parseval equality. As another application, we provide a new approach to studythe Cayley-Dickson algebras, which turn out to be specific examples of octonionic Hilbert spaces. An explicit weak associative orthonormal basis is constructed in each Cayley-Dickson algebra.

math.FA↗

arXiv2021

Para-linearity as the nonassociative counterpart of linearity

In an octonionic Hilbert space $H$, the octonionic linearity is taken to fail for the maps induced by the octonionic inner products, and it should be replaced with the octonionic para-linearity. However, to introduce the notion of the octonionic para-linearity we encounter an insurmountable obstacle. That is, the axiom $$\left\langle pu ,u\right\rangle=p\left\langle u ,u\right\rangle$$ for any octonion $p$ and element $u\in H$ introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces with one being the dual of the other. The dual space consists of continuous left \almost linear functionals and it becomes a right $Ø$-module under the multiplication defined in terms of the second associators which measures the failure of $Ø$-linearity. This right multiplication has an alternative expression $${(f\odot p)(x)}=pf(p^{-1}x)p,$$ which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e., $$\fsh{f\odot p}=\fsh{f}\abs{p}.$$ Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.

math.FA↗

arXiv2020

Octonionic bimodule

The structure of octonionic bimodules is formulated in this paper. It turns out that every octonionic bimodule is a tensor product, the category of octonionic bimodules is isomorphic to the category of real vector spaces. We show that there is also a real part structure on octonionic bimodules similar to the quaternion case. Different from the quaternion setting , the octonionic bimodule sturcture is uniquely determined by its left module structure and hence the real part can be obtained only by left multiplication. The structure of octonionic submodules generated by one element is more involved, which leads to many obstacles to further development of the octonionic functional analysis. We introduce a notion of cyclic decomposition to deal with this difficulty. Using this concept, we give a complete description of the submodule generated by one element in octonionic bimodules. This paper clears the barrier of the structure of $Ø$-modules for the later study of octonionic functional analysis.

math.RA↗

arXiv2019

Classification of left octonion modules

It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element $m$ may be the whole module and may be not in the form $Øm$. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions.

math.RA↗