A New Resolvent Equation for the S-Functional Calculus

Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene Sabadini

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


The $$S$$S-functional calculus is a functional calculus for (Formula presented.)-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 (Formula presented.) and the right one (Formula presented.), where (Formula presented.) and (Formula presented.) is an (Formula presented.)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations (Formula presented.), and (Formula presented.), respectively, where (Formula presented.) 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.

Original languageEnglish
Pages (from-to)1939-1968
Number of pages30
JournalJournal of Geometric Analysis
Issue number3
StatePublished - 20 Jul 2015


  • Left s-resolvent operator
  • Projectors
  • Quaternionic operators
  • Resolvent equation
  • Right s-resolvent operator
  • n-tuples of noncommuting operators
  • s-spectrum

ASJC Scopus subject areas

  • Geometry and Topology


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