# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2016mar24.2092

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

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