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Transactions of the American Mathematical Society

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Slice regular semigroups


Authors: Riccardo Ghiloni and Vincenzo Recupero
Journal: Trans. Amer. Math. Soc. 370 (2018), 4993-5032
MSC (2010): Primary 30G35, 47D03, 47A60, 47A10
DOI: https://doi.org/10.1090/tran/7354
Published electronically: March 20, 2018
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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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Additional Information

Riccardo Ghiloni
Affiliation: Dipartimento di Matematica Università di Trento Via Sommarive 14 38123 Trento Italy
Email: ghiloni@science.unitn.it

Vincenzo Recupero
Affiliation: Dipartimento di Scienze Matematiche Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy
Email: vincenzo.recupero@polito.it

DOI: https://doi.org/10.1090/tran/7354
Keywords: Slice regular semigroups, analytic semigroups, functions of hypercomplex variables, quaternions, functional calculus, spectrum, resolvent
Received by editor(s): November 4, 2016
Published electronically: March 20, 2018
Additional Notes: The first author was partially supported by INFN-TIFPA and by GNSAGA of INdAM
The second author was a member of GNAMPA of INdAM
Article copyright: © Copyright 2018 American Mathematical Society

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