Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus; but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.
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…
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…
Regarding quaternions as normal matrices, we first characterize the $2\times 2$ matrix-valued functions, defined on subsets of quaternions, whose values are quaternions. Then we investigate the regularity of quaternionic-valued functions, defined by the analytic functional calculus. Constructions of analytic functional calculi for real linear operators, in particular for quaternionic linear ones, are finally discussed, using a Riesz-Dunford-Gelfand type kernel in one variable, or a Martinelly type kernel in two variables.
Denoting by $\mathbb{M}$ the complexification of the quaternionic algebra $\mathbb{H}$, we characterize the family of those $\mathbb{M}$-valued functions, defined on subsets of $\H$, whose values are actually quaternions, using an intrinsic approach. In particular, we show that the slice quaternionic regularity can be characterized via a Cauchy type transform acting on the space of analytic $\mathbb{M}$-valued stem functions.
In this article we give an approach to define continuous functional calculus for bounded quaternionic normal operators defined on a right quaternionic Hilbert space.
In this paper, we first prove that the S-spectrum of a bounded right quaternionic linear operator on a two-sided quaternionic Banach space is a union of the spectrum of some bounded linear operators on a complex Banach space. Furthermore, we show that the S-functional calculus is obtained by the Riesz-Dunford functional calculus for complex linear operators. We also give simple proofs of some already existing results.
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…
The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for $n$-tuples of noncommuting operators. With this paper we show that the spectral theory on the $S$-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. Mor…
In this paper we show how the spectral theory based on the notion of $S$-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $H^\infty$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier laws for the propagation of the heat in non homogeneous materials is a vector operator of the form \[ T=e_1\,a(x)\partial_{x_1} + e_2\,b(x)\partial_{x_2} + e_3\,c(x)\partial_{x_3}, \] where $e_\ell$, $e_\ell=1,2,3$ are orthogonal unit vectors, $a$, $b$, $c$ …
