Francesco Mantovani (Politecnico di Milano, Italy)
In this talk we develop a functional calculus for Clifford operators, that contains vector operators as a particular case, and we introduce a theory of fractional powers for bisectorial operators. Special emphasis is placed on the gradient operator with nonconstant coefficients, with the aim of deriving the corresponding nonlocal formulation of Fourier's law.
The construction is based on the spectral theory of the $S$-spectrum and addresses a fundamental difficulty that distinguishes bisectorial operators from the classical sectorial case: the fractional power function is not analytic on the negative real axis and therefore cannot be directly treated within the standard functional calculus framework. To overcome this obstacle, we propose a new definition of fractional powers tailored to the bisectorial setting. We then apply the resulting functional calculus for vector operators to gradient operators, establishing a rigorous framework for defining and studying their fractional powers.
Poster