Slice regular semigroups

Ghiloni, Riccardo; Recupero, Vincenzo

doi:10.1090/tran/7354

Slice regular semigroups
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by Riccardo Ghiloni and Vincenzo Recupero PDF

Trans. Amer. Math. Soc. 370 (2018), 4993-5032 Request permission

Abstract:

In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory and characterizes those semigroups which can be represented by a noncommutative Cauchy integral formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford algebras $\mathbb {R}_n$ included.

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