A New Resolvent Equation for the $$S$$ S -Functional Calculus

Abstract

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 $$s=(s_0,s_1,\ldots ,s_n)\in \mathbb {R}^{n+1}$$ and $$T=(T_0,T_1,\ldots ,T_n)$$ is an $$(n+1)$$ -tuple of noncommuting operators. The two $$S$$ -resolvent operators satisfy the $$S$$ -resolvent equations $$S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal {I}$$ , and $$sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal {I}$$ , respectively, where $$\mathcal {I}$$ 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.