Frame of Operators in quaternionic Hilbert spaces

四元数Hilbert空间中K-框架对偶的刻画

Characterizations of dual of K-frames in quaternionic Hilbert spaces

本文研究了右四元数Hilbert空间中K-框架的对偶，在K值域非必闭合时证明了基于规范K-对偶的刻画，并定义与研究了近似K-对偶及其存在性结果。

四元数Hilbert空间上的框架与算子

Frames and Operators on Quaternionic Hilbert spaces

本文研究四元数Hilbert空间中的框架理论，通过算子刻画框架，并分析由有界算子作用在框架上得到的框架形式。

实与四元数 Hilbert 空间中的正规算子

Normal operators in real and quaternionic Hilbert spaces

本文提出直接由复情形推导实 Hilbert 空间正规算子谱表示的新方法，并将四元数正规算子纳入实正规算子框架，使其谱与相关测度可在复平面子集上经典定义。

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.

Fractional Riesz-Hilbert transforms and fractional monogenic signals

The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fracti…

关于四元数希尔伯特空间上的$K$-框架

On $K$-frames for Quaternionic Hilbert Spaces

本文研究四元数希尔伯特空间上的$K$-框架，推广Douglas定理，并探讨直和空间上的$K_1\oplus K_2$-框架及其对偶与各分量框架的关系。

四元数希尔伯特空间上自伴算子的对数凸函数的若干不等式

Some inequalities for log-convex functions of selfadjoint operators on quaternionic Hilbert spaces

本文针对四元数希尔伯特空间上的自伴算子，利用对数凸函数证明了若干Jensen型不等式，并通过特定对数凸函数得到算子不等式的特殊情况，丰富了该领域的不等式研究。

四元数（超）Hilbert 空间上连续 $K$-框架的结果

Results on Continuous $K$-frames for Quaternionic (Super) Hilbert Spaces

本文探讨了四元数 Hilbert 空间中的连续 $K$-框架，研究了单个空间及超空间的 $K_1 \oplus K_2$-框架，并分析了它们与各子空间框架间的关系与对偶性。

On the Equivalence of Complex and Quaternionic Quantum Mechanics

Due to the existence of incompatible observables, the propositional calculus of a quantum system does not form a Boolean algebra but an orthomodular lattice. Such lattice can be realised as a lattice of subspaces on a real, complex or quaternionic Hilbert space, which motivated the formulation of real and quaternionic quantum mechanics in addition to the usual complex formulation. It was argued that any real quantum system admits a complex structure that turns it into a complex quantum system and hence real quantum mechanics was soon discarded. Several authors however developed a quaternionic …

Perturbation of Continuous Frames on Quaternionic Hilbert Spaces

In this note, following the theory of discrete frame perturbations in a complex Hilbert space, we examine perturbation of rank $n$ continuous frame, rank $n$ continuous Bessel family and rank $n$ continuous Riesz family in a non-commutative setting, namely in a right quaternionic Hilbert space.

On the polar decomposition of right linear operators in quaternionic Hilbert spaces

In this article we prove the existence of the polar decomposition for densely defined closed right linear operators in quaternionic Hilbert spaces: If $T$ is a densely defined closed right linear operator in a quaternionic Hilbert space $H$, then there exists a partial isometry $U_{0}$ such that $T = U_{0}|T|$. In fact $U_{0}$ is unique if $N(U_{0}) = N(T)$. In particular, if $H$ is separable and $U$ is a partial isometry with $T = U|T|$, then we prove that $U = U_{0}$ if and only if either $N(T) = \{0\}$ or $R(T)^{\bot} = \{0\}$.

Frame Multipliers for discrete frames on Quaternionic Hilbert Spaces

In this note, following the complex theory, we examine discrete controlled frames, discrete weighted frames and frame multipliers in a non-commutative setting, namely in a left quaternionic Hilbert space. In particular, we show that the controlled frames are equivalent to usual frames under certain conditions. We also study connection between frame multipliers and weighted frames in the same Hilbert space.

Kato S-spectrum in the quaternionic setting

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local S-spectrum, at a point of a right quaternionic Hilbert space, and the local spectral subsets are introduced and studied. The S-surjectivity spectrum and its connections to the Kato S-spectrum, approximate S-point spectrum and local S-spectrum are investigated. The g…

The Spectral Theorem for Quaternionic Normal Operators

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a bounded normal right quaternionic linear operator on $\mathcal{H}$. In this paper, we prove that there exists a unique spectral measure $E$ in $\mathcal{H}$ such that $$T=\int_{σ_S(T)}λdE_λ,$$ where $σ_S(T)$ denotes the spherical spectrum of $T$.

Fredholm operators and essential S-spectrum in the quaternionic setting

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate $S$-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact operator perturbations. Finally, in association with Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the $S$-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel we study left and r…

Spectral Theorem for quaternionic normal operators: Multiplication form

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis $\mathcal{N}_{m}$ of $\mathcal{H}$, a measure space $(Ω, μ)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(Ω; \mathbb{H}; μ)$ and a $μ$ - measurable function $ϕ\colon Ω\to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{α+ m β; \;α, β\in \mathbb{R}\}$) such that \[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_ϕ$ is the multiplication operator…

On quaternionic functional analysis

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion $B^*$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^*$-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend two results of Kulkarni (namely, we will give the full versions of the Gelfand-Naimark theorem and the Gelfand theorem for quaternion $B^*$-algebras). On our way to …

Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $δ>0$ there exist a compact operator $K$ with $\|K\|< δ$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not …

四元数量子q-振子与量子经济学中的无界次正规算子

Quaternionic Quantum q-Oscillator And Unbounded Subnormal Operators In Quantum Economics

本研究确定四元数希尔伯特空间上的无界次正规算子类对应四元数量子谐振q-振子的解，并借助无界次正规算子的模型理论，探讨了该算子、q-振子与四元数量子经济学的关联。

Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} Λ_{\varepsilon}^{S}(T) := σ_S (T) \bigcup \left \{ q \in \mathbb{H}\setminus σ_S(T):\; \|Δ_{q}(T)^{-1}\| \geq \frac{1}{\varepsilon} \right\}, \end{align*} where $\mathbb{H}$ denotes the division ring of quaternions, $σ_S(T)$ is the $S$-spectrum of $T$ and $Δ_q(T)= T^2-2 \text{Re}(q)T+|q|^2I$. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate severa…

Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

We extend some results of group representation theory and von Neumann algebras to the quaternionic Hilbert space case, proving the double commutant theorem (whose quaternionic proof requires a different procedure) and extend to the quaternionic case a result concerning the classification of irreducible von Neumann algebras. Secondly, we consider elementary relativistic systems in Wigner's view defined as a locally-faithful irreducible strongly-continuous unitary representation of Poincaré group in quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the syst…

Continuous slice functional calculus in quaternionic Hilbert spaces

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable.…

四元数Hilbert空间上的Cowen-Douglas算子

Cowen-Douglas operators on quaternionic Hilbert spaces

本文将Cowen-Douglas算子推广到四元数Hilbert空间，记为$B_n^s(Ω_q)$，建立了相关向量丛刚性定理与酉等价分类，指出曲率不再是其完全酉不变量，并给出了$B_1^{s}(Ω_q)$的复表示等价刻画。

Continuous Functional Calculus for Quaternionic Bounded Normal Operators

In this article we give an approach to define continuous functional calculus for bounded quaternionic normal operators defined on a right quaternionic Hilbert space.

Perturbation of normal quaternionic operators

The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two $S$-resolvent operators and the $S$-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a quatern…

The $H^\infty$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators

In this paper we extend the $H^\infty$ functional calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called $S$-functional calculus. The $S$-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties. The $S$-functional calculus is based on the notion of $S$-spectru…

四元数Hardy空间的Corona定理

Corona theorem for the quaternionic Hardy space

该研究针对四元数Hardy空间，将有界切片正则函数的有限生成元Corona定理推广至可数多个生成元，同时建立对应的Toeplitz Corona定理，解决了相关文献提出的问题。

在四元数福克空间上的Toeplitz算子

Toeplitz Operators on Quaternionic Fock Spaces

本文研究了四元数福克空间上具有正测度符号和切片函数符号的Toeplitz算子的有界性和紧性，通过归一化再生核和对称盒子平均提出判据，同时引入全局四元数福克空间定义具有实值测度符号的Toeplitz算子。

Spectral properties of compact normal quaternionic operators

General, especially spectral, features of compact normal operators in quaternionic Hilbert spaces are studied and some results are established which generalize well-known properties of compact normal operators in complex Hilbert spaces. More precisely, it is proved that the norm of such an operator always coincides with the maximum of the set of absolute values of the eigenvalues (exploiting the notion of spherical eigenvalue). Moreover the structure of the spectral decomposition of a generic compact normal operator $T$ is discussed also proving a spectral characterization theorem for compact …

四元数 Hansen--Pedersen 不等式及其在 $S_r$-变换中的应用

A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms

本文在右四元数 Hilbert 空间上建立四元数 Hansen--Pedersen 不等式，并应用于 $S_r$-变换，证明其保持 $p$-亚正规性及对数亚正规性。

Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks

We present an approach to the spectrum and analytic functional calculus for quaternionic linear operators, following the corresponding results concerning the real linear operators. In fact, the construction of the analytic functional calculus for real linear operators can be refined to get a similar construction for quaternionic linear ones, in a classical manner, using a Riesz-Dunford-Gelfand type kernel, and considering spectra in the complex plane. A quaternionic joint spectrum for pairs of operators is also discussed, and an analytic functional calculus is constructed, via a Martinelli typ…

Decomposable operators, local S-spectrum and S-spectrum in the quaternionic setting

In a right quaternionic Hilbert space, following the complex formalism, decomposable operators, the so-called Bishop's property and the single valued extension property are defined and the connections between them are studied to certain extent. In particular, for a decomposable operator, it is shown that the S-spectrum, approximate S-point spectrum, surjectivity S-spectrum and the union of all local S-spectra coincide. Using continuous right slice-regular functions we have also studied certain properties of local S-spectrum and local S-spectral subspaces.

Weighted quaternionic Cauchy singular integral

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the $k$-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô--Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman p…

四元数算子的核算性与Grothendieck-Lidskii公式

Nuclearity and Grothendieck-Lidskii formula for quaternionic operators

本文在四元数算子中借助伴随矩阵引入迹概念，建立四元数Fredholm行列式并证明Grothendieck-Lidskii型公式，进而推广到四元数局部凸空间中的$\ rac{2}{3}$-核算算子。

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 Hilber…

Beurling-Lax type theorems in the complex and quaternionic setting: the half-space case

We give a generalization of the Beurling-Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.

On the relation between S-spectrum and right spectrum

We use the $\mathbb{R}$-linearity of $Iλ-T$ to define $σ(T)$, the right spectrum of a right $\mathbb{H}$-linear operator $T$ in a right quaternionic Hilbert space. We show that $σ(T)$ coincides with the $S$-spectrum $σ_S(T)$.

Some results on the lattice parameters of quaternionic Gabor frames

Gabor frames play a vital role not only 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…

Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures

The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has been started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigrou…

Admissibility and Frame Homotopy for Quaternionic Frames

We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exist, is it always possible to interpolate between any two while fixing their spectra and norms? In other words, the first question is the admissibility question for quaternionic frames and the second is a generalization of the frame homotopy conjecture. We give complete answers to both questions. For the first question, the existence criterion is exactly the same as in the real and complex cases. For the second, the non-empty spaces of quaterni…

Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators

Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defi…

Cullen-regular quaternionic functions in a Fueter operator framework

We show characterizations of the class of Cullen-regular functions in the sense of Gentili-Struppa for any domain $Ω$ in terms of the Fueter operator. We then state a Integral Theorem and discuss how it can be used to define a more general version of Cullen-regularity, that does not require the function to be of class $C^{1}$.

Fredholm theory in quaternionic Banach algebras

Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectr…

关于有界四元数算子的Browder S-谱的一个注记

A note on the Browder S-spectrum of a bounded quaternionic operator

本文建立了有界四元数算子Browder S-谱的紧扰动交集刻画，并证明了有限秩限制及逐点稳定性结果。

Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in partic…

四元数Banach模上拟李括号的局部刚性及其在非线性PDEs中的应用

Local Rigidity of Quasi--Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs

本文在四元数Banach右模上建立拟李括号的局部刚性定理，通过构造显式双线性校正恢复精确李性质，并结合同伦算子与Neumann级数反演，为非线性PDEs提供局部适定性与延拓准则等应用。

四元数预解式方程与$\u0026mathcal;{S}$-预解算子的级数展开

Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator

本文在四元数框架下证明了$\u0026mathcal;{S}$-预解算子的预解式方程，推导其在$\u0026mathcal;{S}$-预解集内任意四元数开邻域的级数展开及相关几何推论，证明中用到多变量向量值实解析函数概念。

四元数拟李结构的算子范数转移与上同调刚性及其在滑模$\beta$-指数稳定性中的应用

Operator-Norm Transfer and Cohomological Rigidity for Quaternionic Quasi-Lie Structures with Application to Sliding Mode $β$-Exponential Stability

针对四元数拟李结构，开发了算子理论和上同调工具，建立了算子范数精确转移和上同调刚性结果，并应用于滑模控制实现$\beta$-指数稳定性。