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The S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is an (n + 1)-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations S−1 L (s, T )s − TS−1 L (s, T ) = I, and sS−1 R (s, T )−S−1 R (s, T )T = I, respectively, where I denotes the identity operator. These equations allows to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation for the S-functional calculus which is the analogue of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Journal of Geometric Analysis, volume 25, issue 3, in 2015 following peer review. The final publication is available at Springer via DOI: 10.1007/s12220-014-9499-9

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