Analytic semigroups in the noncommutative framework

Functional analysis in Banach spaces over the skew field H
of quaternions is receiving increasing attention and relies
upon the theory of slice regular functions, the quaternionic
counterpart of analytic functions.
In this talk we introduce the class of sectorial operators
on these spaces and we prove that they generate a one
quaternionic parameter operator family which is slice regular
and satisfies a new semigroup law induced by the noncommutative
structure of H.
Moreover these semigroups can be represented by a suitable
Cauchy integral formula.
These results can be generalized to spaces over any finite
dimensional real involutive algebra, Clifford algebras included.