A New Resolvent Equation for the S-Functional Calculus

The $$S$$S-functional calculus is a functional calculus for (Formula presented.)-tuples of not 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 (Formula presented.) and the right one (Formula presented.), where (Formula presented.) and (Formula presented.) is an (Formula presented.)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations (Formula presented.), and (Formula presented.), respectively, where (Formula presented.) denotes the identity operator. These equations allow us to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.