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Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators

Authors: Riccardo Ghiloni and Vincenzo Recupero
Journal: Trans. Amer. Math. Soc. 368 (2016), 2645-2678
MSC (2010): Primary 30G35, 47D03, 47A60, 47A10
Published electronically: April 14, 2015
MathSciNet review: 3449252
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Abstract: The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy integral formula. It follows that such a semigroup is analytic in time.

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Additional Information

Riccardo Ghiloni
Affiliation: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo-Trento (TN), Italy

Vincenzo Recupero
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli, Abruzzi 24, 10129 Torino, Italy

Keywords: Functions of hypercomplex variables, functional calculus, semigroups of linear operators, spectrum, resolvent
Received by editor(s): December 3, 2013
Received by editor(s) in revised form: January 26, 2014
Published electronically: April 14, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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