共查询到20条相似文献,搜索用时 31 毫秒
1.
Paula Cerejeiras Fabrizio Colombo Uwe Käehler Irene Sabadini 《Mathematical Methods in the Applied Sciences》2023,46(2):2093-2116
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. 相似文献
2.
Stefan Hartmann 《Advances in Applied Clifford Algebras》2016,26(1):137-149
Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case. 相似文献
3.
The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as
a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains
natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations
and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular
functions. This result holds in a formulation that reflects the peculiarities of the quaternionic setting and the structure
of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like
the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions. 相似文献
4.
《Linear and Multilinear Algebra》2008,56(1):179-184
The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool. 相似文献
5.
In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge–Ampère operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge–Ampère measures \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampère operator is discussed. 相似文献
6.
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern–Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-Sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes. 相似文献
7.
8.
In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on
a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula
which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new
version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to
functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional
calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e
tA
when A is a linear quaternionic operator.
相似文献
9.
Daniel Alpay Maria-Elena Luna-Elizarrarás Michael Shapiro 《Comptes Rendus Mathematique》2005,340(9):639-643
We consider bounded linear operators defined on real normed spaces, and with range in quaternionic spaces. We study the norms of the quaternionic extensions of such operators. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005). 相似文献
10.
11.
本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系. 相似文献
12.
Jing Wen LUAN Fu Liu ZHU 《数学学报(英文版)》2005,21(6):1295-1308
In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field. 相似文献
13.
Bogdan Alexandrov 《Annals of Global Analysis and Geometry》2002,22(1):75-98
We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space. 相似文献
14.
In this paper, we derive the explicit formulas for computing the zeros of certain cubic quaternionic polynomial. From these, we obtain a necessary and sufficient condition to quaternionic cubic polynomial have a spherical zero, and some examples are also provided. Moreover, we will discuss some applications of the cubic quaternionic formulas. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
15.
Wei Wang 《Bulletin des Sciences Mathématiques》2011,(8):988-1010
Gaveau?s optimal control method for real and complex Monge–Ampere operators is generalized to that for quaternionic Monge–Ampere operator. It is also applied to investigate quaternionic regular functions: the characterization of the Silov boundary of a smooth quaternionic pseudoconvex domain. 相似文献
16.
María Elena Luna‐Elizarrarás Marco Antonio Pérez‐de la Rosa Ramón M. Rodríguez‐Dagnino Michael Shapiro 《Mathematical Methods in the Applied Sciences》2013,36(9):1080-1094
It has been found recently that there exists a theory of functions with quaternionic values and in two real variables, which is determined by a Cauchy–Riemann‐type operator with quaternionic variable coefficients, and that is intimately related to the so‐called Mathieu equations. In this work, it is all explained as well as some basic facts of the arising quaternionic function theory. We establish analogues of the basic integral formulas of complex analysis such as Borel–Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. This theory turns out to be in the same relation with the Schrödinger operator with special potential as the usual holomorphic functions in one complex variable, or quaternionic hyperholomorphic functions, or functions of Clifford analysis, are with the corresponding Laplace operator. Moreover, it is similar to that of α‐hyperholomorphic functions and the Helmholtz operator. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
17.
L. C. de Andrés M. Fernández S. Ivanov J. A. Santisteban L. Ugarte D. Vassilev 《Annali di Matematica Pura ed Applicata》2014,193(1):261-290
We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kähler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kähler with either a closed fundamental four form or fundamental two forms defining a differential ideal. 相似文献
18.
Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.Partially supported by the National Science Foundation grant DMS-9296168. 相似文献
19.
Oleg Chterental 《Linear algebra and its applications》2008,428(4):1178-1201
We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size n?4. 相似文献
20.
In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces. 相似文献