An introduction to fractional powers of quaternionic operators and new fractional diffusion processes

The spectral theory based on the notion of S-spectrum, for quaternionic operators and in particular for vector operators, has been introduced about ten years ago and it is nowadays well developed. It turned out that this theory allows to define new fractional diffusion processes. In this note we announce new results on the quaternionic version of the H ∞ functional calculus and we use them to compute the fractional powers of vector operators. Such fractional powers of generalized gradient operators define a new Fourier’s law for the propagation of the heat. This paper is devoted to researchers in different research fields such as: quaternionic analysis, operator theory and fractional diffusion problems.