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 Hβ 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=e1βa(x)βx1ββ+e2βb(x)βx2ββ+e3βc(x)βx3ββ, where eββ, eββ=1,2,3 are orthogonal unit vectors, a, b, c
are suitable real valued function that depend on the space variables
x=(x1β,x2β,x3β) 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
TΞ±, for Ξ±β(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