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\begin{document}$$S$$\end{document}-functional calculus is a functional calculus for (n+1)\documentclass[12pt]{minimal}
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\begin{document}$$(n+1)$$\end{document}-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\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-functional calculus there are two resolvent operators: the left SL-1(s,T)\documentclass[12pt]{minimal}
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\begin{document}$$S_L^{-1}(s,T)$$\end{document} and the right one SR-1(s,T)\documentclass[12pt]{minimal}
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\begin{document}$$S_R^{-1}(s,T)$$\end{document}, where s=(s0,s1,…,sn)∈Rn+1\documentclass[12pt]{minimal}
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\begin{document}$$s=(s_0,s_1,\ldots ,s_n)\in \mathbb {R}^{n+1}$$\end{document} and T=(T0,T1,…,Tn)\documentclass[12pt]{minimal}
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\begin{document}$$T=(T_0,T_1,\ldots ,T_n)$$\end{document} is an (n+1)\documentclass[12pt]{minimal}
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\begin{document}$$(n+1)$$\end{document}-tuple of noncommuting operators. The two S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-resolvent operators satisfy the S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-resolvent equations SL-1(s,T)s-TSL-1(s,T)=I\documentclass[12pt]{minimal}
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\begin{document}$$S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal {I}$$\end{document}, and sSR-1(s,T)-SR-1(s,T)T=I\documentclass[12pt]{minimal}
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\begin{document}$$sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal {I}$$\end{document}, respectively, where I\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I}$$\end{document} denotes the identity operator. These equations allow us to prove some properties of the S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-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\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-resolvent operators simultaneously.
