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Formulations of the -functional calculus and some consequences

Part of: General theory of linear operators

Published online by Cambridge University Press:  05 May 2016

Fabrizio Colombo  and
Jonathan Gantner
Fabrizio Colombo
Affiliation:
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi 9, 20133 Milano, Italy (fabrizio.colombo@polimi.it; jonathan.gantner@polimi.it)
Jonathan Gantner
Affiliation:
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi 9, 20133 Milano, Italy (fabrizio.colombo@polimi.it; jonathan.gantner@polimi.it)
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In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

Keywords

Fueter–Sce mapping theorem in integral form-spectrumformulations of the -functional calculus for n-tuples of operatorsprojectorsquaternionic version of the -functional calculusquaternionic -resolvent equation

MSC classification

Secondary: 47A10: Spectrum, resolvent 47A60: Functional calculus
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 3 , June 2016 , pp. 509 - 545
Copyright
Copyright © Royal Society of Edinburgh 2016