Quaternionic Closed Operators Fractional Powers and Fractional Diffusion Processes This book presents a new theory for evolution operators and a new method for defining fractional powers of vector operators. This new approach allows to define new classes of fractional diffusion and evolution problems. These innovative methods and techniques based on the concept of S spectrum can inspire researchers from various areas of operator theory and PDEs to explore new research directions in their fields. This monograph is the natural continuation of the book: Spectral Theory on the S Spectrum for Quaternionic Operators by Fabrizio Colombo Jonathan Gantner and David P. Kimsey (Operator Theory: Advances and Applications Vol. 270) Introduction Preliminary results The direct approach to the S functional calculus The Quaternionic Evolution Operator Perturbations of the generator of a group The Phillips functional calculus The H Infinity Functional Calculus Fractional powers of quaternionic linear operators The fractional heat equation using quaternionic techniques. Applications to fractional diffusion Historical notes and References Appendix: Principles of functional Analysis