Opuscula Math. 44, no. 3 (2024), 391-407

Opuscula Mathematica

On the Möbius invariant principal functions of Pincus

Sagar Ghosh
Gadadhar Misra

Abstract. In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator \(T\) with \(\operatorname{rank} (T^*T-TT^*) =1\), modulo unitary equivalence, is the unilateral shift.

Keywords: hyponormal operator, multiplicity, trace formula, homogeneous operators, principal function.

Mathematics Subject Classification: 47B20.

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  1. B. Bagchi, G. Misra, Constant characteristic functions and homogeneous operators, J. Operator Theory 37 (1997), 51-65.
  2. B. Bagchi, G. Misra, Homogeneous operators and projective representations of the Möbius group: a survey, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 415-437. https://doi.org/10.1007/BF02829616
  3. B. Bagchi, G. Misra, The homogeneous shifts, J. Funct. Anal. 204 (2003), 293-319. https://doi.org/10.1016/S0022-1236(02)00088-5
  4. B. Bagchi, S. Hazra, G. Misra, A product formula for homogeneous characteristic functions, Integral Equations Operator Theory 95 (2023), Article no. 8. https://doi.org/10.1007/s00020-023-02730-x
  5. C. Berger, B.I. Shaw, Selfcommutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc. 79 (1973), 1193-119. https://doi.org/10.1090/S0002-9904-1973-13375-0
  6. R.W. Carey, J.D. Pincus, An invariant for certain operator algebras, Proc. Natl. Acad. Sci. USA 71 (1974), 1952-1956. https://doi.org/10.1073/pnas.71.5.1952
  7. R.W. Carey, J.D. Pincus, Construction of seminormal operators with prescribed mosaic, Indiana Univ. Math. J. 23 (1974), 1155-1165. https://doi.org/10.1512/iumj.1974.23.23092
  8. A. Chattopadhyay, K.B. Sinha, On the Carey-Helton-Howe-Pincus trace formula, J. Funct. Anal. 274 (2018), 2265-2290. https://doi.org/10.1016/j.jfa.2017.08.006
  9. K.F. Clancey, Seminormal Operators, Lecture Notes in Mathematics, vol. 742, Springer, Berlin, 1979.
  10. K.F. Clancey, B.L. Wadhwa, Local spectra of seminormal operators, Trans. Amer. Math. Soc. 280 (1983), 415-428. https://doi.org/10.2307/1999622
  11. D.N. Clark, G. Misra, On homogeneous contractions and unitary representations of \(SU(1,1)\), J. Operator Theory 30 (1993), 109-122.
  12. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators, Translations of Mathematical Monographs, vol. 18, Providence, RI: AMS, 1969.
  13. B. Gustafsson, M. Putinar, Hyponormal Quantization of Planar Domains: Exponential Transform in Dimension Two, Lecture Notes in Mathematics, vol. 2199, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-65810-0
  14. J. Helton, R. Howe, Integral operators: traces, index, and homology, Proc. Conf. Operator Theory, Dalhousie Univ., Halifax 1973, Lect. Notes Math., vol. 345, Springer, Berlin, 1973, 141-209.
  15. A. Korányi, G. Misra, A classification of homogeneous operators in the Cowen-Douglas class, Adv. Math. 226 (2011), 5338-5360. https://doi.org/10.1016/j.aim.2011.01.012
  16. M. Martin, M. Putinar, Lectures on Hyponormal Operators, Operator Theory: Advances and Applications, vol. 39, Birkhäuser Verlag, Basel, 1989.
  17. G. Misra, Curvature and the backward shift operator, Proc. Amer. Math. Soc. 91 (1984), 105-107. https://doi.org/10.2307/2045279
  18. J.D. Pincus, Commutators and systems of singular integral equations, I, Acta Math. 121 (1968), 219-249. https://doi.org/10.1007/BF02391914
  19. J.D. Pincus, The spectrum of seminormal operators, Proc. Natl. Acad. Sci. USA 68 (1971), 1684-1685. https://doi.org/10.1073/pnas.68.8.1684
  20. J.D. Pincus, The determining function method in the treatment of commutator systems, Hilbert Space Operators Operator Algebras, Colloquia Math. Soc. Janos Bolyai 5 (1972), 443-477.
  21. M. Putinar, Extensions scalaires et noyaux distribution des opérateurs hyponormaux, C.R. Acad. Sci. Paris, Sér. I Math. 301 (1985), 739-741.
  22. M. Putinar, Extreme hyponormal operators, Special Classes of Linear Operators and Other Topics, Operator Theory: Advances and Applications, 28, Birkhäuser, Basel, 1988, 249-265.
  23. C.R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323-330. https://doi.org/10.1007/BF01111839
  24. J. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469-476. https://doi.org/10.1090/S0002-9947-1965-0173161-3
  • Sagar Ghosh
  • Indian Statistical Institute, Theoretical Statistics and Mathematics Unit, Banaglore 560 059, India
  • Gadadhar Misra (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-8096-2039
  • Indian Statistical Institute, Theoretical Statistics and Mathematics Unit, Banaglore 560 059, India
  • Indian Institute of Technology Gandhinagar, Palaj, Gujarat 382 055, India
  • Communicated by Alexander Gomilko.
  • Received: 2023-04-17.
  • Revised: 2023-08-22.
  • Accepted: 2023-08-23.
  • Published online: 2024-02-15.
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Cite this article as:
Sagar Ghosh, Gadadhar Misra, On the Möbius invariant principal functions of Pincus, Opuscula Math. 44, no. 3 (2024), 391-407, https://doi.org/10.7494/OpMath.2024.44.3.391

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