The structure of the fractional powers of the noncommutative Fourier law

Colombo, Fabrizio; Peloso, Marco M.; Pinton, Stefano

doi:10.1002/mma.5719

Cited by 24 publications

(16 citation statements)

References 57 publications

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“…Suppose x ∈ D(I(f )) and letting i → ∞ in (5. 19), we obtain the equality in (5.18). Finally, using the fact that E(M i )y → y as i → ∞ for all y ∈ H n and I(f…”

Section: Thus Limmentioning

confidence: 75%

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The spectral theorem for normal operators on a Clifford module

Colombo¹,

Kimsey²

2021

Preprint

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In this paper, using the recently discovered notion of the Sspectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be −1). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

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“…Suppose x ∈ D(I(f )) and letting i → ∞ in (5. 19), we obtain the equality in (5.18). Finally, using the fact that E(M i )y → y as i → ∞ for all y ∈ H n and I(f…”

Section: Thus Limmentioning

confidence: 75%

“…In the quaternionic setting, fractional Fourier's law, the has been considered in various papers, see for example [15,19], and the references therein.…”

Section: Motivationmentioning

confidence: 99%

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The spectral theorem for normal operators on a Clifford module

Colombo¹,

Kimsey²

2021

Preprint

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“…The theory of slice hyperholomorphic functions was somewhat abandoned until 2006 when G. Gentili and D. C. Struppa (inspired by C. G. Cullen [51]) introduced in [59] the notion of slice regular functions for the quaternions. Further developments of the theory of slice regular functions were discussed also in [28] and the above definition was extended by F. Colombo, I. Sabadini and D.C. Struppa, in [47], (see also [35,49,50]) to the Clifford algebra setting. Slice regular functions as defined in [59] and their generalization to the Clifford algebra as in [47], called slice monogenic functions, possess good properties on specific open sets that are called axially symmetric slice domains.…”

Section: Spectral Theories In the Hyperholomorphic Settingmentioning

confidence: 99%

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

Colombo

Gantner

Pinton

2021

Adv. Appl. Clifford Algebras

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The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

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“…When T is the Fourier law for the heat diffusion problems with the homogeneous Dirichlet boundary condition, there are no further boundary conditions that are necessary to generate the fractional powers of T . In the case is bounded we studied this problem in the papers [14,18,19]. In this paper we consider the case in which is unbounded.…”

Section: Introductionmentioning

confidence: 99%

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Colombo

González

Pinton

2021

Complex Anal. Oper. Theory

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Using the spectral theory on the S-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$\begin{aligned} T=\sum _{\ell =1}^3e_\ell a_\ell (x)\partial _{x_\ell }, \ \ \ x=(x_1,x_2,x_3)\in \overline{\Omega }, \end{aligned}$$ T = ∑ ℓ = 1 3 e ℓ a ℓ ( x ) ∂ x ℓ , x = ( x 1 , x 2 , x 3 ) ∈ Ω ¯ , where, $$\Omega $$ Ω can be either a bounded or an unbounded domain in $$\mathbb {R}^3$$ R 3 whose boundary $$\partial \Omega $$ ∂ Ω is considered suitably regular, $$\overline{\Omega }$$ Ω ¯ is the closure of $$\Omega $$ Ω and $$e_\ell $$ e ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 are the imaginary units of the quaternions $$\mathbb {H}$$ H . The operators $$T_\ell :=a_\ell (x)\partial _{x_\ell }$$ T ℓ : = a ℓ ( x ) ∂ x ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 , are called the components of T and $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R are the coefficients of T. In this paper we study the generation of the fractional powers of T, denoted by $$P_{\alpha }(T)$$ P α ( T ) for $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , when the operators $$T_\ell $$ T ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 do not commute among themselves. To define the fractional powers $$P_{\alpha }(T)$$ P α ( T ) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T. In this paper we consider two different boundary conditions. If $$\Omega $$ Ω is unbounded we consider Dirichlet boundary conditions. If $$\Omega $$ Ω is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator $$\sum _{\ell =1}^3a_\ell ^2(x)n_\ell (x) \partial _{x_\ell }+a(x)I$$ ∑ ℓ = 1 3 a ℓ 2 ( x ) n ℓ ( x ) ∂ x ℓ + a ( x ) I , for $$x\in \partial \Omega $$ x ∈ ∂ Ω , where I is the identity operator, $$a:\partial \Omega \rightarrow \mathbb {R}$$ a : ∂ Ω → R is a given function and $$n=(n_1,n_2,n_3)$$ n = ( n 1 , n 2 , n 3 ) is the outward unit normal vector to $$\partial \Omega $$ ∂ Ω . The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type $$ \sum _{\ell =1}^3a_\ell (x)n_\ell (x) \partial _{x_\ell }+b(x)I$$ ∑ ℓ = 1 3 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ + b ( x ) I , $$x\in \partial \Omega . $$ x ∈ ∂ Ω . For this reason we also discuss the conditions on the coefficients $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R of T and on the coefficient $$b:\partial \Omega \rightarrow \mathbb {R}$$ b : ∂ Ω → R so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

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The structure of the fractional powers of the noncommutative Fourier law

Cited by 24 publications

References 57 publications

The spectral theorem for normal operators on a Clifford module

The spectral theorem for normal operators on a Clifford module

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

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