Related papers: On quaternionic Monge-Ampere operator, closed posi…
We introduce and analyze an explicit formulation of fractional powers of the Lam\'e-Navier system of partial differential operators. We show that this fractional Lam\'e-Navier operator is a nonlocal integro-differential operator that…
Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $\tau = - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point…
We analyze the parabolic Dirac operator $D \pm i\partial_t$ in a biquaternionic setting, characterizing its kernel via generalized div-curl systems and Cauchy-Riemann-type relations between the real and imaginary parts. Using the machinery…
The suggested operator manifold formalism enables to develop an approach to the unification of the geometry and the field theory. We also elaborate the formalism of operator multimanifold yielding the multiworld geometry involving the…
We present a construction of fermionic operators in 3+1 dimensions in terms of bosonic fields in the framework of $QED_4$. The basic bosonic variables are the electric fields $E_i$ and their conjugate momenta $A_i$. Our construction…
In this paper, general quaternionic structure are developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Two examples for the Helmholtz equation with local fractional…
In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy…
This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…
A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After…
This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we…
We prove a comparison principle for the pluripotential complex Monge-Amp\`ere flows for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded plurisubharmonic function. As a…
We introduce Wirtinger operators for functions of several quaternionic variables. These operators are real linear partial differential operators which behave well on quaternionic polynomials, with properties analogous to the ones satisfied…
We present new polynomial-based methods for discrete-time quaternionic systems, highlighting how noncommutative multiplication modifies classical control approaches. Defining quaternionic polynomials via a backward-shift operator, we…
Through the study of the degenerate complex Monge-Amp\`ere equation, we establish the optimal regularity of the extremal function associated to intrinsic norms of Chern-Levine-Nirenberg and Bedford-Taylor. We prove a conjecture of…
Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n…
We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n) &= Au(n+2) + f(n,u(n)), \quad n…
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…
It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using…
There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial…
This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the…