S-Spectrum and the quaternionic Cayley transform of an operator

Muraleetharan, B.; Sabadini, Irene; Thirulogasanthar, K.

doi:10.1016/j.geomphys.2017.12.001

Cited by 9 publications

(7 citation statements)

References 13 publications

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“…From an historical point of view, the research for an appropriate notion of quaternionic spectrum, the S-spectrum, started because in quaternionic quantum mechanics, see [1,10], the notion of quaternionic spectrum was unclear. Recent investigations in quaternionic quantum mechanics based on the current knowledge in quaternionic operator theory can be found for example in [29,38,39]. The basic tool for quaternionic quantum mechanics is the spectral theorem based on the S-spectrum that been developed in [4,5], and in [26] for the case of matrices.…”

Section: Introductionmentioning

confidence: 99%

Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space

Colombo¹,

Gantner²

2018

J. Phys. A: Math. Theor.

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In this paper we give a concrete application of the spectral theory based on the notion of S-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials, that is the heat flow is given by the vector operator:where e ℓ , ℓ = 1, 2, 3 are orthogonal unit vectors in R 3 , a, b, c are given real valued functions that depend on the space variables x = (x1, x2, x3), and possibly also on time. Using the H ∞ -version of the S-functional calculus we have recently defined fractional powers of quaternionic operators, which contain, as a particular case, the vector operator T . Hence, we can define the non-local version T α , for α ∈ (0, 1), of the Fourier law defined by T . We will see in this paper how we have to interpret T α , when we introduce our new approach called: "The S-spectrum approach to fractional diffusion processes". This new method allows us to enlarge the class of fractional diffusion and fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations and non commutative operator theory. Our theory applies not only to the heat diffusion process but also to Fick's law and more in general it allows to compute the fractional powers of vector operators that arise in different fields of science and technology.This can also be done for quaternionic sectorial operators (and in particular for sectorial vector operators) using the quaternionic version of the H ∞ -functional calculus. The study of the fractional powers of quaternionic and of vector operators started with the paper [16]. The H ∞ -functional calculus has been extended to the quaternionic setting in [6], following the original paper of McIntosh [37]. This calculus can be developed with less restrictions on the operator as in [34]. Recently,in [17], we have extended the H ∞ -functional calculus in its full generality following this approach. In the same paper, we also proved some non trivial results like the spectral mapping theorem in this setting and we used this functional calculus to define and study fractional powers of quaternionic linear sectorial operators and in particular vector operators. In our papers we have extended several classical results of fractional powers of operators in [9,33,35,36,37,41,42] to the quaternionic setting.This paper has two main goals. The first one is to define a new approach to fractional diffusion processes that is based on the spectral theory on the S-spectrum, that we will call: The S-spectrum approach to fractional diffusion problems. The second is to give a concrete application to fractional diffusion problems and fractional evolution when the operator T defined in (1.1) has commuting components. Our theory applies more in general to operators T with non commuting components (a(x)∂ x 1 , b(x)∂ x 2 , c(x)∂ x 3 ) but, for the sake o...

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Section: Introductionmentioning

confidence: 99%

Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space

Colombo¹,

Gantner²

2018

J. Phys. A: Math. Theor.

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“…Recent investigations in quaternionic quantum mechanics based on the current knowledge in quaternionic operator theory can be found for example in other studies. () The basic tool for quaternionic quantum mechanics is the spectral theorem based on the S ‐spectrum that been developed previous works() for the case of matrices. Beyond the spectral theorem there are further directions of research.…”

Section: Discussionmentioning

confidence: 99%

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Colombo

Mongodi

Peloso

et al. 2019

Math Methods in App Sciences

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Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in R3 and let normalΩ⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T=a(xtrue_)∂xe1+b(xtrue_)∂ye2+c(xtrue_)∂ze3, into the conservation of energy law, here a, b, c:normalΩ→double-struckR are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c:normalΩ→double-struckR the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.

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“…The H ∞ -functional calculus was further extended following the book [52] in [14]. For more recent developments associated with quaternionic quantum mechanics see [59,60,61,62]. Remark 6.6.…”

Section: Noncommuting Matrix Variables and Some Final Remarksmentioning

confidence: 99%

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

Colombo,

Gantner,

Kimsey

et al. 2021

Preprint

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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. More precisely, we show that the S-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S-functional calculus goes beyond quaternionic and Clifford analysis. Indeed we show that the S-functional calculus has a certain universality property. This fact makes the spectral theory on the S-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

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S-Spectrum and the quaternionic Cayley transform of an operator

Cited by 9 publications

References 13 publications

Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space

Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

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