Opuscula Math. 37, no. 1 (2017), 81-107

Opuscula Mathematica

Seminormal systems of operators in Clifford environments

Mircea Martin

Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex \(C^*\)-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a \(C^*\)-algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

Keywords: multidimensional operator theory, joint seminormality, Riesz transforms, Putnam inequality.

Mathematics Subject Classification: 47B20, 47A13, 47A63, 44A15.

Full text (pdf)

  1. A. Athavale, On joint hyponormal operators, Proc. Amer. Math. Soc. 103 (1988), 417-423.
  2. N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, [in:] Grundlehren der mathematischen Wissenschaften, vol. 298, Springer Verlag, Berlin, Heidelberg, New York, 1992.
  3. S. Bochner, Curvature and Betti numbers. I, II, Ann. of Math. 49, 50 (1948, 1949), 379-390, 79-93.
  4. S. Bochner, K. Yano, Curvature and Betti Numbers, Ann. of Math. Studies 32, Princeton University Press, Princeton, 1953.
  5. F. Brachx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman Research Notes in Mathematics Series, 76, 1982.
  6. J. Bunce, The joint spectrum of commuting non-normal operators, Proc. Amer. Math. Soc. 29 (1971), 499-505.
  7. M. Cho, R.E. Curto, T. Huruya, W. Zelazko, Cartesian form of Putnam's inequality for doubly commuting \(n\)-tuples, Indiana Univ. Math. J. 49 (2000), 1437-1448.
  8. K.F. Clancey, Seminormal Operators, Lecture Notes in Math. 742, Springer Verlag, Berlin, Heidelberg, New York, 1979.
  9. J.B. Conway, Subnormal Operators, Pitman, Boston, 1981.
  10. R.E. Curto, Applications of several complex variables to multiparameter spectral theory, [in:] J.B. Conway, B.B. Morrel (eds), Surveys of Recent Results in Operator Theory, vol. II, Longman Publishing Co., London, 278 (1988), 25-90.
  11. R.E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, [in:] W.B. Arveson, R.G. Douglas (eds), Operator Theory, Operator Algebras and Applications, Part II, Proc. Sympos. Pure Math. 51 (1990), 69-91.
  12. R.E. Curto, R. Jian, A matricial identity involving the self-commutator of a commuting \(n\)-tuple, Proc. Amer. Math. Soc. 121 (1994), 461-464.
  13. R.E. Curto, P.A. Muhly, J. Xia, Hyponormal pairs of commuting operators, [in:] Operator Theory: Advances and Applications, vol. 35, Birkhäuser Verlag, 1988, 1-22.
  14. R.G. Douglas, V. Paulsen, K. Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc. 20 (1988), 67-71.
  15. J.E. Gilbert, M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics 26, Cambridge University Press, Cambridge, 1991.
  16. R. Goldberg, Curvature and Homology, Academic Press, New York, London, 1962.
  17. T. Kato, Smooth operators and commutators, Studia Math. 31 (1968), 531-546.
  18. H.B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, 1989.
  19. M. Martin, Joint seminormality and Dirac operators, Integral Equations Operator Theory 30 (1998), 101-121.
  20. M. Martin, Higher-dimensional Ahlfors-Beurling inequalities in Clifford analysis, Proc. Amer. Math. Soc. 126 (1998), 2863-2871.
  21. M. Martin, Convolution and maximal operator inequalities in Clifford analysis, [in:] Clifford Algebras and Their Applications in Mathematical Physics, vol. II, Clifford Analysis, Progress in Physics 9, Birkhäuser Verlag, Basel, 2000, 95-113.
  22. M. Martin, Self-commutator inequalities in higher dimension, Proc. Amer. Math. Soc. 130 (2002), 2971-2983.
  23. M. Martin, Spin geometry, Clifford analysis, and joint seminormality, [in:] Advances in Analysis and Geometry, Vol. I, Trends in Mathematics Series, Birkhäuser Verlag, Basel, 2004, 227-255.
  24. M. Martin, Uniform approximation by solutions of elliptic equations and seminormality in higher dimensions, Operator Theory: Advances and Applications, vol. 149, Birkhäuser Verlag, Basel, 2004, 387-406.
  25. M. Martin, Deconstructing Dirac operators. I: Quantitative Hartogs-Rosenthal theorems, Proceedings of the 5th International Society for Analysis, Its Applications and Computation Congress, ISAAC 2005, Catania, Italy, 2005, More Progress in Analysis, World Scientific, 2009, 1065-1074.
  26. M. Martin, Deconstructing Dirac operators. III: Dirac and semi-Dirac pairs of differential operators, Operator Theory: Advances and Applications, vol. 203, Birkhäuser Verlag, Basel, 2009, 347-362.
  27. M. Martin, Deconstructing Dirac operators. II: Integral representation formulas, Proceedings of the 7th International Society for Analysis, Its Applications and Computation Congress, ISAAC 2009, London, United Kingdom, 2009, Trends in Mathematics: Hypercomplex Analysis and Applications, Springer Verlag, Berlin, Heidelberg, New York, 2011, 195-211.
  28. M. Martin, M. Putinar, Lectures on Hyponormal Operators, Operator Theory: Advances and Applications, vol. 39, Birkhäuser Verlag, Basel, 1989.
  29. M. Martin, N. Salinas, Weitzenböck type formulas and joint seminormality, [in:] Contemporary Math., Amer. Math. Soc. 212 (1998), 157-167.
  30. M. Martin, P. Szeptycki, Sharp inequalities for convolution operators with homogeneous kernels and applications, Indiana Univ. Math. J. 46 (1997), 975-988.
  31. S. McCullough, V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195.
  32. P.S. Muhly, A note on commutators and singular integrals, Proc. Amer. Math. Soc. 54 (1976), 117-121.
  33. J.D. Pincus, Commutators and systems of singular integral equations, Acta Math. 121 (1968), 219-249.
  34. J.D. Pincus, D. Xia, The analytic model of a hyponormal operator with rank one self-commutator, Integral Equations Operator Theory 4 (1981), 134-150.
  35. J.D. Pincus, D. Xia, J. Xia, The analytic model of a hyponormal operator with rank one self-commutator, Integral Equations Operator Theory 7 (1984), 516-535.
  36. C.R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer Verlag, Berlin, Heidelberg, New York, 1967.
  37. C.R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323-330.
  38. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.
  39. J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172-191.
  40. F.-H. Vasilescu, A characterization of the joint spectrum in Hilbert spaces, Rev. Roumaine Math. Pures Appl. 22 (1977), 1003-1009.
  41. F.-H. Vasilescu, A multidimensional spectral theory in \(C^*\)-algebras, Banach Center Publ. 8 (1982), 471-491.
  42. D. Xia, On non-normal operators. I, Chinese J. Math. 3 (1963), 232-246; II, Acta Math. Sinica 21 (1987), 103-108.
  43. D. Xia, Spectral Theory of Hyponormal Operators, Birkhäuser Verlag, Basel, Boston, Stuttgart, 1983.
  44. D. Xia, On some classes of hyponormal tuples of commuting operators, Operator Theory: Advances and Applications, vol. 48, Birkhäuser Verlag, 1990, 423-448.
  • Mircea Martin
  • Baker University, Department of Mathematics, Baldwin City, KS 66006, USA
  • Communicated by P.A. Cojuhari.
  • Received: 2016-10-02.
  • Accepted: 2016-11-02.
  • Published online: 2016-12-14.
Opuscula Mathematica - cover

Cite this article as:
Mircea Martin, Seminormal systems of operators in Clifford environments, Opuscula Math. 37, no. 1 (2017), 81-107, http://dx.doi.org/10.7494/OpMath.2017.37.1.81

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.