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
DOI:
https://doi.org/10.1090/S1079-6762-96-00004-2
MathSciNet review:
1405966
Full-text PDF Free Access
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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.
- Lars V. Ahlfors, Möbius transformations in ${\bf R}^n$ expressed through $2\times 2$ matrices of Clifford numbers, Complex Variables Theory Appl. 5 (1986), no. 2-4, 215–224. MR 846490, DOI https://doi.org/10.1080/17476938608814142
- Robert F. V. Anderson, The Weyl functional calculus, J. Functional Analysis 4 (1969), 240–267. MR 0635128, DOI https://doi.org/10.1016/0022-1236%2869%2990013-5
- W. Arveson, The harmonic analysis of automorphisms groups, American Mathematical Society, Amer. Math. Soc. Summer Institute, 1980.
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
- J. Cnops, Hurwitz pairs and applications of Möbius transformations, Habilitation dissertation, Universiteit Gent, Faculteit van de Wetenschappen, 1994.
- A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.
- R. E. Curto and F.-H. Vasilescu, Standard operator models in several variables, preprint.
- R. E. Curto and F.-H. Vasilescu, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 65–78. MR 1243269
- R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, Mathematics and its Applications, vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992. A function theory for the Dirac operator; Related REDUCE software by F. Brackx and D. Constales; With 1 IBM-PC floppy disk (3.5 inch). MR 1169463
- A. Ya. Helemskii, Banach and locally convex algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. Translated from the Russian by A. West. MR 1231796
- A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt; Grundlehren der Mathematischen Wissenschaften, Band 220. MR 0412321
- V. V. Kisil, Construction of integral representations in spaces of analytical functions, Dokl. Akad. Nauk SSSR, to appear.
- ---, Do we need that observables form an algebra?, in preparation.
- ---, Integral representation and coherent states, Bull. Soc. Math. Belg. Sér. A 2 (1995), 529–540.
- ---, Spectrum of operator, functional calculi and group representations, in preparation.
- 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.
- Alan McIntosh and Alan Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), no. 2, 421–439. MR 891783, DOI https://doi.org/10.1512/iumj.1987.36.36024
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- F. Riesz and B. Sz-Nagy, Functional analysis, Ungar, New York, 1955.
- Gian-Carlo Rota and Gilbert Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 379–381. MR 0147922
- 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.
- Masamichi Takesaki, Structure of factors and automorphism groups, CBMS Regional Conference Series in Mathematics, vol. 51, Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R.I., 1983. MR 703512
- Joseph L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38. MR 271741, DOI https://doi.org/10.1007/BF02392329
- Joseph L. Taylor, A general framework for a multi-operator functional calculus, Advances in Math. 9 (1972), 183–252. MR 328625, DOI https://doi.org/10.1016/0001-8708%2872%2990017-5
- Joseph L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34. MR 315446, DOI https://doi.org/10.1090/S0002-9904-1973-13077-0
- Michael E. Taylor, Noncommutative harmonic analysis, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986. MR 852988
- Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. MR 690957
- 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.
- R. F. V. Anderson, The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240–267.
- W. Arveson, The harmonic analysis of automorphisms groups, American Mathematical Society, Amer. Math. Soc. Summer Institute, 1980.
- F. Brackx, R. Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman Advanced Publishing Program, Boston, 1982.
- J. Cnops, Hurwitz pairs and applications of Möbius transformations, Habilitation dissertation, Universiteit Gent, Faculteit van de Wetenschappen, 1994.
- A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.
- R. E. Curto and F.-H. Vasilescu, Standard operator models in several variables, preprint.
- ---, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), 65–78.
- R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, Kluwer Academic Publishers, Dordrecht, 1992.
- A. Y. Helemskii, Banach and locally convex algebras, Clarendon Press, Oxford, 1993.
- A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, New York, 1976.
- V. V. Kisil, Construction of integral representations in spaces of analytical functions, Dokl. Akad. Nauk SSSR, to appear.
- ---, Do we need that observables form an algebra?, in preparation.
- ---, Integral representation and coherent states, Bull. Soc. Math. Belg. Sér. A 2 (1995), 529–540.
- ---, Spectrum of operator, functional calculi and group representations, in preparation.
- 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.
- A. McIntosh and A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421–439.
- M. Reed and B. Simon, Functional analysis, Methods of Modern Mathematical Physics, vol. 1, Academic Press, Orlando, second ed., 1980.
- F. Riesz and B. Sz-Nagy, Functional analysis, Ungar, New York, 1955.
- G.-C. Rota and W. G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Indag. Math. 22 (1960), 379–381.
- 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.
- M. Takesaki, Structure of factors and automorphism groups, Regional Conference Series in Mathematics, vol. 51, American Mathematical Society, Providence, Rhode Island, 1983.
- J. L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38.
- ---, A general framework for a multioperator functional calculus, Adv. Math. 9 (1972), 183–252.
- ---, Functions of several noncommuting variables, Bulletin of the American Mathematical Society 79 (1973), no. 1, 1–34.
- M. E. Taylor, Noncommutative harmonic analysis, Math. Surv. and Monographs, vol. 22, American Mathematical Society, Providence, Rhode Island, 1986.
- F.-H. Vasilescu, Analytic functional calculus and spectral decomposition, Mathematics and Its Applications, vol. 1, D. Reidel Publ. Comp., Dordrecht, Holland, 1982.
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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
Email:
vk@imem.odessa.ua
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
