The Journal of Geometric Analysis

, Volume 25, Issue 3, pp 1939–1968 | Cite as

A New Resolvent Equation for the \(S\)-Functional Calculus

  • Daniel Alpay
  • Fabrizio Colombo
  • Jonathan Gantner
  • Irene Sabadini


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.


\(n\)-tuples of noncommuting operators Quaternionic operators \(S\)-spectrum Right \(S\)-resolvent operator Left \(S\)-resolvent operator Resolvent equation Projectors 

Mathematics Subject Classification

47S10 30G35 


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Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  • Daniel Alpay
    • 1
  • Fabrizio Colombo
    • 2
  • Jonathan Gantner
    • 3
  • Irene Sabadini
    • 2
  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  3. 3.Vienna University of TechnologyInstitute for Analysis and Scientific ComputingWienAustria

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