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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, 84105 , Beer-Sheva, Israel

    Daniel Alpay

  2. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 , Milano, Italy

    Fabrizio Colombo & Irene Sabadini

  3. Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, 1040 , Wien, Austria

    Jonathan Gantner

Authors
  1. Daniel Alpay
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  2. Fabrizio Colombo
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  3. Jonathan Gantner
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  4. Irene Sabadini
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Correspondence to Irene Sabadini.

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Communicated by Der-Chen Edward Chang.

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Alpay, D., Colombo, F., Gantner, J. et al. A New Resolvent Equation for the \(S\)-Functional Calculus. J Geom Anal 25, 1939–1968 (2015). https://doi.org/10.1007/s12220-014-9499-9

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