The monogenic functional calculus

Brian Jefferies; Alan McIntosh; James Picton-Warlow

Studia Mathematica (1999)

  • Volume: 136, Issue: 2, page 99-119
  • ISSN: 0039-3223

Abstract

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A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.

How to cite

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Jefferies, Brian, McIntosh, Alan, and Picton-Warlow, James. "The monogenic functional calculus." Studia Mathematica 136.2 (1999): 99-119. <http://eudml.org/doc/216667>.

@article{Jefferies1999,
abstract = {A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.},
author = {Jefferies, Brian, McIntosh, Alan, Picton-Warlow, James},
journal = {Studia Mathematica},
keywords = {functional calculus; Clifford algebra; monogenic function; monogenic functional calculus},
language = {eng},
number = {2},
pages = {99-119},
title = {The monogenic functional calculus},
url = {http://eudml.org/doc/216667},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Jefferies, Brian
AU - McIntosh, Alan
AU - Picton-Warlow, James
TI - The monogenic functional calculus
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 2
SP - 99
EP - 119
AB - A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.
LA - eng
KW - functional calculus; Clifford algebra; monogenic function; monogenic functional calculus
UR - http://eudml.org/doc/216667
ER -

References

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  1. [1] R. F. V. Anderson, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240-267. Zbl0191.13403
  2. [2] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Res. Notes Math. Ser. 76, Pitman, Boston, 1982. 
  3. [3] B. Jefferies and A. McIntosh, The Weyl calculus and Clifford analysis, Bull. Austral. Math. Soc. 57 (1998), 329-341. Zbl0915.47015
  4. [4] B. Jefferies and B. Straub, Lacunas in the support of the Weyl calculus for two hermitian matrices, submitted. Zbl1059.47016
  5. [5] V. V. Kisil, Möbius transformations and monogenic functional calculus, ERA Amer. Math. Soc. 2 (1996), 26-33. 
  6. [6] V. V. Kisil and E. Ramírez de Arellano, The Riesz-Clifford functional calculus for non-commuting operators and quantum field theory, Math. Methods Appl. Sci. 19 (1996), 593-605. Zbl0853.47012
  7. [7] C. Li, A. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994), 665-721. Zbl0817.42008
  8. [8] A. McIntosh and A. Pryde, The solution of systems of operator equations using Clifford algebras, in: Miniconf. on Linear Analysis and Function Spaces 1984, Centre for Mathematical Analysis, ANU, Canberra, 9 (1985), 212-222. 
  9. [9] A. McIntosh and A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421-439. Zbl0694.47015
  10. [10] A. McIntosh, A. Pryde and W. Ricker, Comparison of joint spectra for certain classes of commuting operators, Studia Math. 88 (1988), 23-36. Zbl0665.47002
  11. [11] J. Ryan, Plemelj formulae and transformations associated to plane wave decompositions in complex Clifford analysis, Proc. London Math. Soc. 64 (1992), 70-94. Zbl0703.30043
  12. [12] F. Sommen, Plane wave decompositions of monogenic functions, Ann. Polon. Math. 49 (1988), 101-114. Zbl0673.30038
  13. [13] J. L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1-38. Zbl0233.47025

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