Quaternionic Operators Research Articles

Quaternionic spherical sectorial and dissipative operators via S-resolvent kernels

In this paper, we treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute and develop a theory of quaternionic non-negative operator, spherical sectorial operator and dissipative operator via S-resolvent kernels in quaternionic locally convex spaces (short for q.l.c.s.). The notions of quaternionic non-negative operators and quaternionic (m-)dissipative operators are introduced via S-resolvent operators and H-valued inner product on Hilbert H-bimodule. By choosing the suitable spherical sector, the spherical sectorial operator is introduced to establish the relationship with the quaternionic non-negative operator. It is crucial to note that the quaternionic operators we consider do not necessarily commute.

Rank one quaternionic operators and additive preservers

In this paper, we completely describe all additive surjective maps, on the set of all bounded finite rank right linear operators acting on a right quaternionic Banach space, that preserve the set of all bounded rank one (resp. rank one idempotent, rank one nilpotent) right linear operators, in both directions.

The H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty $$\\end{document}-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

In recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, we identify four distinct classes of functions: slice hyperholomorphic functions (leading to the S-functional calculus), axially harmonic functions (leading to the Q-functional calculus), axially polyanalytic functions of order 2 (leading to the P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}-functional calculus), and axially monogenic functions (leading to the F-functional calculus). By applying the respective product rule, we establish the four different H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty $$\\end{document}-versions of these functional calculi.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Dual quaternion operations for rigid body motion and their application to the hand–eye calibration

The dual quaternion plays an advantageous role in the field of robotic kinematics and dynamics due to its compact expression. It is well known that quaternion can be easily implemented as algebraic operations on vectors through matrix operators. Similarly, the dual matrix operators can convert dual quaternion operations to matrix operations. However, it has not received adequate attention, and thus there are very few applications at present. With the dual matrix operators of dual quaternion, this paper re-verifies the equivalency between the conjugate formula of unit dual quaternion and dual Euler–Rodrigues formula. To combine this equivalence with the homomorphic mapping of Lie groups, a theoretical correlation of the current hand-eye calibration methods is established. Motivated by the attempt to establish a more complete scheme for hand-eye calibration, a new simultaneous method based on the conjugate formula of dual quaternion is proposed. The method verifies that the rotation problem can be extended to the rigid-body transformation problem via the dual algebra. Finally, the simulation and experimental validations for the hand–eye calibration method are given.

Almansi decomposition of polynomials of quaternionic Dirac operators

This article presents an innovative extension of the classical Almansi decomposition. Traditionally, the Almansi decomposition is employed to decompose polymonogenic functions into their monogenic counterparts, thereby elucidating the relationship between the Dirac operator and its successive iterations. Diverging from this conventional method, the new extension delves into the kernel of a polynomial related to the quaternionic Dirac operator, as opposed to focusing on the iterated operator. This is achieved by employing a polynomial-based partition of unity. The outcome of this approach is a distinctive decomposition that leverages the kernel of the Dirac–Helmholtz operators.

Properties of Fredholm, Weyl and Jeribi essential S-spectra in a right quaternionic Hilbert space

The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of [Formula: see text] block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

Quaternionic Davis–Wielandt shell in a right quaternionic Hilbert space

UDC 517.5 We derive some results concerning the quaternionic Davis–Wielandt shell for a bounded right linear operator in a right quaternionic Hilbert space. The relations between the geometric properties of the quaternionic Davis–Wielandt shells and the algebraic properties of quaternionic operators are obtained.

Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed.

Adaptive reweighted quaternion sparse learning for data recovery and classification

Sparse representation (SR) methods in quaternion space have been attracting increasing interests recently. However, most existing quaternion SR methods adopt the quaternion ℓ1 norm, which penalizes all the entries of the quaternion sparse vector equally and ignores the differences and significance of different entries. Ideally, the entries with large magnitude should be less penalized while those with small magnitude (such as zero entries) should be more penalized. Therefore, we propose an Adaptive Weighted Quaternion Sparse Representation (AWQSR) method in this paper, which can learn weights for distinct entries of the quaternion sparse entries in an adaptive manner. Due to the noncommutativity of quaternion multiplication, it is difficult to tackle the resulting optimization problem of AWQSR. For this reason, we devise an effective iteratively reweighted optimization algorithm based on quaternion operators. To further improve the classification performance, we also develop a Supervised AWQSR based Classification (SAWQSRC) method by leveraging the label information of training samples to learn discriminative weights. Theoretical analysis of SAWQSRC has also been established to show that SAWQSRC succeeds in classification under appropriate conditions. The experiments on simulated data and real data prove the validity of the proposed methods for quaternion signal recovery and classification.

Characterizations of Unit Darboux Ruled Surface with Quaternions

This paper presents a quaternionic approach to generating and characterizing the ruled surface drawn by the unit Darboux vector. The study derives the Darboux frame of the surface and relates it to the Frenet frame of the base curve. Moreover, it obtains the quaternionic shape operator and its matrix representation using the normal and geodesic curvatures to provide a more detailed analysis. To illustrate the concepts discussed, the paper offers a clear example that will help readers better understand the concepts and showcases the quaternionic shape operator, Gauss curvature, mean curvature, and rotation matrix. Finally, it emphasizes the need for further research on this topic.

Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview

<abstract><p>An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.</p></abstract>

A Universal Quaternion Hypergraph Network for Multimodal Video Question Answering

Fusion and interaction of multimodal features are essential for video question answering. Structural information composed of the relationships between different objects in videos is very complex, which restricts understanding and reasoning. In this paper, we propose a quaternion hypergraph network (QHGN) for multimodal video question answering, to simultaneously involve multimodal features and structural information. Since quaternion operations are suitable for multimodal interactions, four components of the quaternion vectors are applied to represent the multimodal features. Furthermore, we construct a hypergraph based on the visual objects detected in the video. Most importantly, the quaternion hypergraph convolution operator is theoretically derived to realize multimodal and relational reasoning. Question and candidate answers are embedded in quaternion space, and a Q&A reasoning module is creatively designed for selecting the answer accurately. Moreover, the unified framework can be extended to other video-text tasks with different quaternion decoders. Experimental evaluations on the TVQA dataset and DramaQA dataset show that our method achieves state-of-the-art performance.

Orientation and Kinematics of Rotation: Quaternionic and Four-Dimensional Skew-Symmetric Operators, Equations, and Algorithms

Orientation and Kinematics of Rotation: Quaternionic and Four-Dimensional Skew-Symmetric Operators, Equations, and Algorithms

S-spectrum and numerical range of a quaternionic operator

We study the numerical range of bounded linear operators on quaternionic Hilbert spaces and its relation with the S-spectrum. The class of complex operators on quaternionic Hilbert spaces is introduced and the upper bild of normal complex operators is completely characterized in this setting.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Fractional powers of higher-order vector operators on bounded and unbounded domains

AbstractUsing the $H^{\infty }$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^{2}(\Omega,\mathbb {C}\otimes \mathbb {H})$. The operators that we consider are of the type \begin{align*} T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \quad x=(x_1,\, x_2,\, x_3)\in \overline{\Omega}, \end{align*}where $\overline {\Omega }$ is the closure of either a bounded domain $\Omega$ with $C^{1}$ boundary, or an unbounded domain $\Omega$ in $\mathbb {R}^{3}$ with a sufficiently regular boundary, which satisfy the so-called property $(R)$ (see Definition 1.3), $e_1,\, e_2,\, e_3\in \mathbb {H}$ which are pairwise anticommuting imaginary units, $a_1,\,a_2,\, a_3: \overline {\Omega } \subset \mathbb {R}^{3}\to \mathbb {R}$ are the coefficients of $T$. In particular, it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_{\alpha }(T)$ for $\alpha \in (0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial _{x_l}^{m}$, do not commute among themselves. This kind of result is to be understood in the more general setting of the fractional diffusion problems. The method used to construct the fractional power of a quaternionic linear operator is a generalization of the method developed by Balakrishnan.

Pseudo S-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε>0, the pseudo S-spectrum of T is defined asΛεS(T):=σS(T)⋃{q∈H∖σS(T):‖Qq(T)−1‖≥1ε}, where H denotes the division ring of quaternions, σS(T) is the S-spectrum of T and Qq(T)=T2−2Re(q)T+|q|2I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S-spectrum and explicitly compute the pseudo S-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε>0 the pseudo S-spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G1-condition for quaternionic operators and prove some results in this setting.

Riesz Projection and Essential S-spectrum in Quaternionic Setting

This paper is devoted to the investigation of the Weyl and the essential S-spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S-eigenvalue of finite type is both introduced and studied. In particular, we have shown that the Weyl and the essential S-spectra do not contain eigenvalues of finite type. We have also described the boundary of the Weyl S-spectrum and the particular case of the spectral theorem of the essential S-spectrum.

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation.