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Spin Geometry, Clifford Analysis, and Joint Seminormality

  • Mircea Martin
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

The first part of this article studies the integral and maximal operators associated with fundamental solutions of Dirac operators on Clifford bundles. The main goal is to obtain explicit estimates for integral transforms of this kind in terms of the corresponding maximal functions. As direct cones-quences of such estimates one derives several quantitative Hartogs-Rosenthal type theorems concerning monogenic approximation on compact sets.

The second part illustrates a Clifford analysis approach to the theory of seminormal systems of Hilbert space operators. The four existing concepts of joint seminormality are reevaluated by assuming that the remainders in some Bochner-Kodaira identities are semidefinite, and a new concept is introduced based on a Bochner-Weitzenböck identity. A rather general singular integral model of jointly seminormal pairs of systems of self-adjoint operators that involves Riesz transforms is presented, and a Putnam type commutator inequality for that model is proved.

Keywords

Clifford algebras Dirac operators hyponormal operators Putnam’s inequality Riesz transforms 

Mathematics Subject Classification (2000)

Primary: 42B20, 47B20 Secondary: 44A35, 47A13, 47A30 

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References

  1. [AB]
    Ahlfors, L. and Beurling, A., Conformal invariants and function theoretic null sets, Acta Math 83 (1950), 101–129.MathSciNetzbMATHGoogle Scholar
  2. [All]
    Alexander, H., Projections of polynomial hulls, J. Funct. Anal. 13 (1973), 13–19.CrossRefzbMATHGoogle Scholar
  3. [Al2]
    Alexander, H., On the area of the spectrum of an element of a uniform algebra. In Complex Approximation Proceedings, Quebec, July 3–8, 1978, Birkhäuser, 1980, 3–12.Google Scholar
  4. [Ar]
    Aronszajn, N., A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. 36 (1957), 235–249.MathSciNetzbMATHGoogle Scholar
  5. [At]
    Athavale, A., On joint hyponormal operators, Proc. Amer. Math. Soc. 103 (1988), 417–423.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [AS]
    Axler, S. and Shapiro, J. H., Putnam’s theorem, Alexander’s spectral area estimate, and VMO, Math. Ann. 271 (1985), 161–183.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BGV]
    Berline, H., Getzler, E., and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren der Mathematischen Wissenschaften, 298, Springer Verlag, New York, 1992.Google Scholar
  8. [Bo]
    Bochner, S., Curvature and Betti numbers. I; II, Ann. of Math. 49 (1948), 379–390; 50 (1949), 79–93.Google Scholar
  9. [BoY]
    Bochner, S. and Yano, K. Curvature and Betti Numbers, Ann. of Math. Studies 32, Princeton University Press, Princeton, 1953.Google Scholar
  10. [BDS]
    Brackx, F., Delanghe, R., and Sommen, F., Clifford Analysis, Pitman Research Notes in Mathematics Series 76, 1982.zbMATHGoogle Scholar
  11. [Ca]
    Calderbank, D. M. J., Dirac operators and Clifford analysis on manifolds with boundary, Odense Universitet, Preprint 53, 1997.Google Scholar
  12. [Cl]
    Clancey, K. F., Seminormal Operators, Lecture Notes in Math. 742, Springer Verlag, 1979.zbMATHGoogle Scholar
  13. [CCHZ]
    Cho, M., Curto, R. E., Huruya, T., and Zelazko, W., Cartesian form of Put­nam’s inequality for doubly commuting n-tuples, Indiana Univ. Math. J. 49 (2000),1437–1448.MathSciNetzbMATHGoogle Scholar
  14. [Cn]
    Cnops, J., An Introduction to Dirac Operators on Manifolds, Progress in Mathematical Physiscs, Birkhäuser, Basel, 2002.CrossRefGoogle Scholar
  15. [Co]
    Conway, J. B., The Theory of Subnormal Operators, Mathematical Surveys and Monographs, 36, American Mathematical SocietyProvidence, RI, 1991.Google Scholar
  16. [Cu]
    Curto, R. E. Joint hyponormality: a bridge between hyponormality and subnor-mality, in Operator Theory, Operator Algebras and Applications, W. B. Arveson and R. G. Douglas (eds.); Part 2, Proc. Sympos. Pure Math. 51 (1990), 69–91.Google Scholar
  17. [CJ]
    Curto, R. E. and Jian, R. A matricial identity involving the self-commutator of a commuting n-tuple, Proc. Amer. Math. Soc. 121 (1994), 461–464.MathSciNetzbMATHGoogle Scholar
  18. [CMX]
    Curto, R. E., Muhly, P. A., and Xia, J., Hyponormal pairs of commuting oper­ators, in Operator Theory: Advances and Applications 35, Springer Basel AG, 1988, 1–22.Google Scholar
  19. [DSS]
    Delanghe, R., Sommen, F., and Soucek, V., Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publishers, 1992.CrossRefzbMATHGoogle Scholar
  20. [DPY]
    Douglas, R. G., Paulsen, V., and Yan, K., Operator theory and algebraic geometry, Bull. Amer. Math. Soc. 20 (1988), 67–71.MathSciNetCrossRefGoogle Scholar
  21. [F]
    Friedrich, T., Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics 25, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  22. [GM]
    Gilbert, J. E. and Murray, M. A. M., Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics 26, Cam­bridge University Press, 1991.CrossRefGoogle Scholar
  23. [Gi]
    Gilkey, P. B., The spectral geometry of a Riemann manifold, J. Diff. Geom. 10 (1975), 601–608.MathSciNetzbMATHGoogle Scholar
  24. [GLP]
    Gilkey, P. B., Leahy, J. V., and Park, JH., Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture, CRC Press, Boca Raton, FL, 1999.zbMATHGoogle Scholar
  25. [G]
    Goldberg, R., Curvature and Homology, Academic Press, New York, London, 1962.zbMATHGoogle Scholar
  26. [GS1]
    Gürlebeck, K. and Sprössig, W., Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser, Basel, 1990.CrossRefzbMATHGoogle Scholar
  27. [GS2]
    Gürlebeck, K. and Sprössig, W.Quaternionic and Clifford Calculus for Physicists cists and Engineers, John Wiley & Sons, New York, 1997.zbMATHGoogle Scholar
  28. [He]
    Hedberg, L., On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.MathSciNetCrossRefGoogle Scholar
  29. [Ka]
    Kato, T., Smooth operators and commutators, Studia Math. 31 (1968), 531–546.MathSciNetGoogle Scholar
  30. [LM]
    Lawson, H. B. and Michelson, M.-L., Spin Geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, 1989.Google Scholar
  31. [L]
    Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math. 118 (1983) 349–379.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Mnl]
    Martin, M., Joint seminormality and Dirac operators, Integral Equations Operator Theory 30 (1998), 101–121.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Mn2]
    Martin, M., Higher-dimensional Ahlfors-Beurling inequalities in Clifford anal­ysis, Proc. Amer. Math. Soc. 126 (1998), 2863–2871.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Mn3]
    Martin, M., Convolution and maximal operator inequalities in Clifford analysis, in Clifford Algebras and Their Applications in Mathematical Physics, Vol. 2: Clifford Analysis, Progress in Physics 9, Springer Basel AG, Basel, 2000, 95–113.Google Scholar
  35. [Mn4]
    Martin, M., Self-commutator inequalities in higher dimension, Proc. Amer. Math. Soc. 130 (2002), 2971–2983.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [Mn5]
    Martin, M., Uniform approximation by closed forms in several complex vari­ables, preprint, 2002.Google Scholar
  37. [MnPr]
    Martin M. and Putinar, M., Lectures on Hyponormal Operators, Operator Theory: Advances and Applications 39, Springer Basel AG, Basel, 1989.Google Scholar
  38. [MnSa]
    Martin, M. and Salinas, N., Weitzenböck type formulas and joint seminormality, Contemporary Math., Amer. Math. Soc. 212 (1998), 157–167.MathSciNetGoogle Scholar
  39. [MnSz]
    Martin, M. and Szeptycki, P., Sharp inequalities for convolution operators with homogeneous kernels and applications, Indiana Univ. Math. J. 46 (1997), 975–988.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [McCP]
    McCullough, S. and Paulsen, V., A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187–195.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Mcl]
    McIntosh, A., Operators which have a H functional calculus, in Miniconference on Operator Theory and Partial Differential Equations, 1986, Proceedings of the Centre for Mathematical Analysis, ANU, Canberra 14 (1986), 210–23MathSciNetGoogle Scholar
  42. [Mc2]
    McIntosh, A., Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, in Clifford Algebras in Analysis and Related Topics, J. Ryan (Ed.), CRC Press, Boca Raton, FL, 1995.Google Scholar
  43. [McP]
    McIntosh, A. and Pryde, A., A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421–439.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [McPR]
    McIntosh, A., Pryde, A., and Ricker, W., Comparison of joint spectra for certain classes of commuting operators, Studia Math. 88 (1988), 23–36.MathSciNetzbMATHGoogle Scholar
  45. [Mi]
    Mitrea, M., Singular Integrals, Hardy Spaces, and Clifford Wavelets,Lecture Notes in Mathematics, 1575, Springer-Verlag, Heidelberg, 1994.Google Scholar
  46. [My]
    Muhly, P. S., A note on commutators and singular integrals, Proc. Amer. Math. Soc. 54 (1976), 117–121.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [Ps]
    Pincus, J. D., Commutators and systems of singular integral equations, Acta Math. 121 (1968), 219–249.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [PX]
    Pincus, J. D. and Xia, D., The analytic model of a hyponormal operator with rank one self-commutator, Integral Equations Operator Theory 4 (1981), 134–150.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [PXX]
    Pincus, J. D., Xia, D., and Xia, J., The analytic model of a hyponormal operator with rank one self-commutator, Integral Equations Operator Theory 7 (1984), 516–535.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [P]
    Putinar, M., Extreme hyponormal operators, Operator Theory: Advances and Applications 28 (1988), 249–265.MathSciNetGoogle Scholar
  51. [Pml]
    Putnam, C. R., Commutation Properties of Hilbert Space Operators and Related Topics, Springer-Verlag, Berlin, Heidelberg, New York, 1967.CrossRefGoogle Scholar
  52. [Pm2]
    Putnam, C. R., An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [Ro]
    Roe, J., Elliptic Operators, Topology, and Asymptotic Methods, 2nd ed., Pitman Research Notes, 395, Longman, 1998.Google Scholar
  54. [Ryl]
    Ryan, J. (Ed.), Clifford Algebras in Analysis and Related Topics, CRC Press, Boca Raton, FL, 1995.Google Scholar
  55. [Ry2]
    Ryan, J., Dirac operators, conformal transformations and aspects of classical harmonic analysis, Journal of Lie Theory 8 (1998), 67–82.MathSciNetzbMATHGoogle Scholar
  56. [RyS]
    Ryan, J. and Sprößig, W. (Eds.), Clifford Algebras and Their Applications in Mathematical Physics, Volume 2: Clifford Analysis, Progress in Physics 19, Birkhäuser, Basel, 2000.Google Scholar
  57. [S1]
    Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ„ 1970.zbMATHGoogle Scholar
  58. [S2]
    Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.Google Scholar
  59. [T]
    Tarkhanov, N. N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995.zbMATHGoogle Scholar
  60. [Xl]
    D. Xia, On non-normal operators. I, Chinese J. Math. 3(1963), 232–246; II, Acta Math. Sinica 21 (1987), 103–108.Google Scholar
  61. [X2]
    D. Xia, Spectral Theory of Hyponormal Operators, Springer Basel AG, Basel Boston-Stuttgart, 1983.zbMATHGoogle Scholar
  62. [X3]
    D. Xia, On some classes of hyponormal tuples of commuting operators, in Operator Theory: Advances and Applications 48, Springer Basel AG, 1990, 423–448.Google Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Mircea Martin
    • 1
  1. 1.Department of MathematicsBaker UniversityBaldwin CityUSA

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