2018/03/28 by Colombo, Fabrizio, Gantner, Jonathan · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1803.10528
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^∞ 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=e1 a(x)∂x1 + e2 b(x)∂x2 + e3 c(x)∂x3, where e_ℓ, e_ℓ=1,2,3 are orthogonal unit vectors, a, b, c are suitable real valued function that depend on the space variables x=(x1,x2,x3) 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 Tα, for α∈ (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 theory based on the S-spectrum for vector operators. This paper is devoted to researchers in different research fields such as: fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.