Milan Journal of Mathematics

, Volume 86, Issue 2, pp 225–303 | Cite as

An Application of the S-Functional Calculus to Fractional Diffusion Processes

  • Fabrizio ColomboEmail author
  • Jonathan Gantner


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 law for the propagation of the heat in non homogeneous materials is a vector operator of the form
$$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$
where \({e_{\ell}, {\ell} = 1, 2, 3}\) are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables \({x = (x_{1}, x_{2}, x_{3})}\) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version \({T^{\alpha}, {\rm for} \alpha \in (0, 1)}\), of the Fourier law defined by T. Our new mathematical tools open the way to a large class of 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 on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

Mathematics Subject Classification (2010)

Primary 47A10 47A60 


H functional calculus for quaternionic operators fractional powers of vector operators S-spectrum fractional diffusion and fractional evolution processes 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Politecnico di Milano Dipartimento di MatematicaMilanoItaly

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