Illinois Journal of Mathematics

Curvature inequalities and extremal operators

Gadadhar Misra and Md. Ramiz Reza

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A curvature inequality is established for contractive commuting tuples of operators T in the Cowen–Douglas class Bn(Ω) of rank n defined on some bounded domain Ω in Cm. Properties of the extremal operators (that is, the operators which achieve equality) are investigated. Specifically, a substantial part of a well-known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.

Article information

Source
Illinois J. Math., Volume 63, Number 2 (2019), 193-217.

Dates
Received: 26 December 2016
Revised: 3 April 2019
First available in Project Euclid: 1 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1564646431

Digital Object Identifier
doi:10.1215/00192082-7768711

Mathematical Reviews number (MathSciNet)
MR3987495

Zentralblatt MATH identifier
07088305

Subjects
Primary: 30C40: Kernel functions and applications
Secondary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.) 47A25: Spectral sets

Citation

Misra, Gadadhar; Reza, Md. Ramiz. Curvature inequalities and extremal operators. Illinois J. Math. 63 (2019), no. 2, 193--217. doi:10.1215/00192082-7768711. https://projecteuclid.org/euclid.ijm/1564646431


Export citation

References

  • [1] M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Adv. Math. 19 (1976), no. 1, 106–148.
  • [2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217.
  • [3] S. R. Bell, “The Cauchy transform, potential theory, and conformal mapping” in Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
  • [4] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187–261.
  • [5] M. J. Cowen and R. G. Douglas, “Operators possessing an open set of eigenvalues,” in Functions, Series, Operators, Vol. I, II (Budapest, 1980), Colloq. Math. Soc. János Bolyai, Vol. 35, North Holland, 1983, 323–341.
  • [6] R. E. Curto and N. Salinas, Generalized Bergman kernels and the Cowen–Douglas theory, Amer. J. Math. 106 (1984), no. 2, 447–488.
  • [7] R. G. Douglas, D. K. Keshari, and A. Xu, Generalized bundle shift with application to multiplication operator on the Bergman space, J. Operator Theory 75 (2016), no. 1, 3–19.
  • [8] S. D. Fisher, “Function theory on planar domains” in Pure and Applied Mathematics, Wiley & Sons, New York, 1983.
  • [9] G. Misra, Curvature and the backward shift operators, Proc. Amer. Math. Soc. 91 (1984), no. 1, 105–107.
  • [10] G. Misra, Curvature inequalities and extremal properties of bundle shifts, J. Operator Theory 11 (1984), no. 2, 305–317.
  • [11] G. Misra and A. Pal, Contractivity, complete contractivity and curvature inequalities, J. Anal. Math. 136, (2018), 31–54.
  • [12] G. Misra and V. Pati, Contractive and completely contractive modules, matricial tangent vectors and distance decreasing metrics, J. Operator Theory 30 (1993), 353–380.
  • [13] G. Misra and N. S. N. Sastry, Bounded modules, extremal problems, and a curvature inequality, J. Funct. Anal. 88 (1990), no. 1, 118–134.
  • [14] Z. Nehari, Conformal Mapping, Dover, New York, 1975.
  • [15] M. R. Reza, Curvature inequalities for operators in the Cowen–Douglas class of a planar domain, Indiana Univ. Math. J. 67 (2018), no. 3, 1255–1279.
  • [16] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147–189.
  • [17] N. Suita, On a metric induced by analytic capacity. II, Kōdai Math. Sem. Rep. 27 (1976), nos. 1–2, 159–162.
  • [18] M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Transactions of the American Mathematical Society 319 (1990), no. 1, 405–415.
  • [19] M. Voichick, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc. 111 (1964), 493–512.
  • [20] K. Wang and G. Zhang, Curvature inequalities for operators of the Cowen–Douglas class, Israel J. Math. 222 (2017), no. 1, 279–296.