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
Abstract |
References |
Similar Articles |
Additional Information
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.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
46H30,
47A13,
30G35,
47A10,
47A60,
47B15,
81Q10
Retrieve articles in all journals
with MSC (1991):
46H30,
47A13,
30G35,
47A10,
47A60,
47B15,
81Q10
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