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Journal of Operator Theory

Volume 77, Issue 2,  Spring  2017  pp. 287-331.

A direct approach to the $S$-functional calculus for closed operators

Authors:  Jonathan Gantner
Author institution: Dipartimento di Matematica, Politecnico di Milano, Milano, 20133, Italy

Summary:  We define the $S$-functional calculus for unbounded closed quaternionic operators and $n$-tuples of operators directly via a Cauchy integral. This allows us to consider also operators, whose $S$-resolvent sets do not contain real points. We show that the main properties of the calculus also hold true with this definition and that it is compatible with intrinsic polynomials, although polynomials are not included in the set of admissible functions. We also prove that the $S$-functional calculus is able to create spectral projections. For this purpose, we remove the assumption that admissible functions are defined on slice domains, which leads to an unexpected phenomenon: the $S$-functional calculi for left and right slice hyperholomorphic functions become inconsistent and give different operators for functions that are both left and right slice hyperholomorphic. Any such function is however the sum of a locally constant and an intrinsic function. For intrinsic functions both functional calculi agree, but for locally constant functions this may in general not be the case.

Keywords:  $S$-functional calculus, $n$-tuples of operators, quaternionic operators

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