A Structure Formula for Slice Monogenic Functions and Some of its Consequences

  • Fabrizio Colombo
  • Irene Sabadini
Conference paper
Part of the Trends in Mathematics book series (TM)


In this paper we show a structure formula for slice monogenic functions (see Lemma 2.2 and [1] for further details): we will show that this formula is a key tool to prove several results, among which we mention the Cauchy integral formula with slice monogenic kernel. This Cauchy formula allows us to extend the validity of the functional calculus for n-tuples of noncommuting operators introduced in [6]. In this wider setting, most of the properties which hold for the Riesz-Dunford functional calculus of a single operator, such as the Spectral Mapping Theorem and the Spectral Radius Theorem, still hold.


Slice monogenic functions slice monogenic kernel structure formula for slice monogenic functions functional calculus for n-tuples of linear operators 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 47A10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Colombo, I. Sabadini, The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators, preprint, 2008.Google Scholar
  2. [2]
    F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, preprint, 2008.Google Scholar
  3. [3]
    F. Colombo, G. Gentili, I. Sabadini, A Cauchy kernel for slice regular functions, preprint, 2008.Google Scholar
  4. [4]
    F. Colombo, G. Gentili, I. Sabadini, D.C. Struppa, An overview on functional calculus in different settings, this volume.Google Scholar
  5. [5]
    F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, to appear in Israel Journal of Mathematics.Google Scholar
  6. [6]
    F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), 2255–2274.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    B. Jefferies, Spectral properties of noncommuting operators, Lecture Notes in Mathematics, 1843, Springer-Verlag, Berlin, 2004.Google Scholar
  8. [8]
    B. Jefferies, A. McIntosh, The Weyl calculus and Clifford analysis, Bull. Austral. Math. Soc., 57 (1998), 329–341.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    B. Jefferies, A. McIntosh, J. Picton-Warlow, The monogenic functional calculus, Studia Math., 136 (1999), 99–119.zbMATHMathSciNetGoogle Scholar
  10. [10]
    V.V. Kisil, E. Ramirez de Arellano, The Riesz-Clifford functional calculus for noncommuting operators and quantum field theory, Math. Methods Appl. Sci., 19 (1996), 593–605.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A. McIntosh, A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J., 36 (1987), 421–439.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    F. Sommen, Special functions in Clifford analysis and axial symmetry, J. Math. Anal. Appl., 130 (1988), 110–133.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Fabrizio Colombo
    • 1
  • Irene Sabadini
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations