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In this paper we introduce fractional powers of quaternionic operators. Theirdefinition is based on the theory of slice-hyperholomorphic functions and onthe $S$-resolvent operators of the quaternionic functional calculus. Theintegral representation formulas of the fractional powers and the quaternionicversion of Kato's formula are based on the notion of $S$-spectrum of aquaternionic operator.  The proofs of several properties of the fractional powers of quaternionicoperators rely on the $S$-resolvent equation. This equation, which is veryimportant and of independent interest, has already been introduced in the caseof bounded quaternionic operators, but for the case of unbounded operators someadditional considerations have to be taken into account. Moreover, we introducea new series expansion for the pseudo-resolvent, which is of independentinterest and allows to investigate the behavior of the $S$-resolvents close tothe $S$-spectrum.  The paper is addressed to researchers working in operator theory and incomplex analysis.

Fractional powers of quaternionic operators and Kato's formula using slice hyperholomorphicity