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ISSN 1079-6762



Möbius transformations and
monogenic functional calculus

Author: Vladimir V. Kisil
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 26-33
MSC (1991): Primary 46H30, 47A13; Secondary 30G35, 47A10, 47A60, 47B15, 81Q10
MathSciNet review: 1405966
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Abstract: A new way of doing functional calculi is presented. A functional calculus $\Phi : f(x)\rightarrow f(T)$ is not an algebra homomorphism of a functional algebra into an operator algebra, but an intertwining operator between two representations of a group acting on the two algebras (as linear spaces). This scheme is shown on the newly developed monogenic functional calculus for an arbitrary set of non-commuting self-adjoint operators. The corresponding spectrum and spectral mapping theorem are included.

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  • 1. L. V. Ahlfors, Möbius transformations in ${ {{\mathbb R}^{n}} }$ expressed through $2\times 2$ matrices of Clifford numbers, Complex Variables Theory Appl. 5 (1986), no. 2, 215--224. MR 88a:15052
  • 2. R. F. V. Anderson, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240--267. MR 58:30405
  • 3. F. Brackx, R. Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman Advanced Publishing Program, Boston, 1982. MR 85j:30103
  • 4. J. Cnops, Hurwitz pairs and applications of Möbius transformations, Habilitation dissertation, Universiteit Gent, Faculteit van de Wetenschappen, 1994.
  • 5. R. E. Curto and F.-H. Vasilescu, Standard operator models in several variables, preprint.
  • 6. ------, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), 65--78. MR 94i:47013
  • 7. R. Delanghe, F. Sommen, and V. Sou\v{c}ek, Clifford algebra and spinor-valued functions, Kluwer Academic Publishers, Dordrecht, 1992. MR 94d:30084
  • 8. A. Y. Helemskii, Banach and locally convex algebras, Clarendon Press, Oxford, 1993. MR 94f:46001
  • 9. A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, New York, 1976. MR 54:447
  • 10. V. V. Kisil, Construction of integral representations in spaces of analytical functions, Dokl. Akad. Nauk SSSR, to appear.
  • 11. ------, Do we need that observables form an algebra?, in preparation.
  • 12. ------, Integral representation and coherent states, Bull. Soc. Math. Belg. Sér. A 2 (1995), 529--540. CMP 96:08
  • 13. ------, Spectrum of operator, functional calculi and group representations, in preparation.
  • 14. V. V. Kisil and E. Ramírez de Arellano, The Riesz-Clifford functional calculus for several non-commuting operators and quantum field theory, Math. Methods Appl. Sci., to appear.
  • 15. A. McIntosh and A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421--439. MR 88i:47007
  • 16. M. Reed and B. Simon, Functional analysis, Methods of Modern Mathematical Physics, vol. 1, Academic Press, Orlando, second ed., 1980. MR 85e:46002
  • 17. F. Riesz and B. Sz-Nagy, Functional analysis, Ungar, New York, 1955. MR 17:175i
  • 18. G.-C. Rota and W. G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Indag. Math. 22 (1960), 379--381. MR 26:5434
  • 19. J. Ryan, Some application of conformal covariance in Clifford analysis, Clifford Algebras in Analysis and Related Topics (J. Ryan, ed.), CRC Press, Boca Raton, 1996, pp. 128--155. CMP 96:10
  • 20. M. Takesaki, Structure of factors and automorphism groups, Regional Conference Series in Mathematics, vol. 51, American Mathematical Society, Providence, Rhode Island, 1983. MR 84k:46043
  • 21. J. L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1--38. MR 42:6622
  • 22. ------, A general framework for a multioperator functional calculus, Adv. Math. 9 (1972), 183--252. MR 48:6967
  • 23. ------, Functions of several noncommuting variables, Bulletin of the American Mathematical Society 79 (1973), no. 1, 1--34. MR 47:3995
  • 24. M. E. Taylor, Noncommutative harmonic analysis, Math. Surv. and Monographs, vol. 22, American Mathematical Society, Providence, Rhode Island, 1986. MR 88a:22021
  • 25. F.-H. Vasilescu, Analytic functional calculus and spectral decomposition, Mathematics and Its Applications, vol. 1, D. Reidel Publ. Comp., Dordrecht, Holland, 1982. MR 85b:47016

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Additional Information

Vladimir V. Kisil
Affiliation: Institute of Mathematics, Economics and Mechanics, Odessa State University, ul. Petra Velikogo, 2, Odessa-57, 270057, Ukraine

Keywords: Functional calculus, joint spectrum, group representation, intertwining operator, Clifford analysis, quantization
Received by editor(s): October 6, 1995
Received by editor(s) in revised form: March 9, 1996
Additional Notes: This work was partially supported by the INTAS grant 93-0322. It was finished while the author enjoyed the hospitality of Universiteit Gent, Vakgroep Wiskundige Analyse, Belgium.
Communicated by: Alexandre Kirillov
Article copyright: © Copyright 1996 American Mathematical Society

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