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.
D. Alpay, F. Colombo, J. Gantner and I. Sabadini. A new resolvent equation for the S-functional calculus. Journal of Geometric Analysis, 25 (2015), no. 3, 1939-1968.