Document Type

Article

Publication Date

2016

Abstract

In this paper we extend the H∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties.

The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem.

The main purpose of this paper is to construct the H∞ functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H∞ functional calculus for (n+1)-tuples of operators applies, in particular, to the Dirac operator.

Comments

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Functional Analysis, volume 271, in 2016. DOI: 10.1016/j.jfa.2016.06.009

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Elsevier

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