Semigroups over real alternative \(^\ast\)-algebras: generation theorems and spherical sectorial operators (Q2787984)

The authors have two goals in mind. At first they want to prove generation theorems for semigroups in Banach spaces whose set of scalars belongs besides real and complex numbers, also to quaternions, octonions, and Clifford algebras. They do not need any functional calculus. Instead, they reduce the generalized theorems to the classical commutative case. NEWLINENEWLINENEWLINENEWLINE The second aim is to study a natural extension of the sectorial operator to general real alternative *-algebras. The authors introduce a concept of spherical sectorial operators and prove that such an operator generates a semigroup analytic in time, that can be represented by a Cauchy integral formula. In the quaternion framework, this concept of spherical sectorial operators and the analyticity in time of the corresponding semigroups is new.NEWLINENEWLINEThe authors recall the basic concepts and properties concerning real alternative *-algebras and slice regular functions defined over it. Moreover, they give the definition of a Banach bimodule over such a *-algebra, and they study unbounded right linear operators acting on such a Banach bimodule.NEWLINENEWLINEFinally, the authors provide a new notion of spherical sectorial operators and prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy integral formula.NEWLINENEWLINEThe paper contains an appendix, in which the authors review the theory of line integrals for vector functions with values in a Banach bimodule.