A New Resolvent Equation for the S-Functional Calculus

The S -functional calculus is a functional calculus for ( n + 1 ) -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 S L - 1 ( s , T ) and the right one S R - 1 ( s , T ) , where s = ( s 0 , s 1 , … , s n ) ∈ R n + 1 and T = ( T 0 , T 1 , … , T n ) is an ( n + 1 ) -tuple of noncommuting operators. The two S -resolvent operators satisfy the S -resolvent equations S L - 1 ( s , T ) s - T S L - 1 ( s , T ) = I , and s S R - 1 ( s , T ) - S R - 1 ( s , T ) T = I , respectively, where I 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.