DOI QR코드

DOI QR Code

TRUNCATED HANKEL OPERATORS AND THEIR MATRICES

  • Received : 2018.02.25
  • Accepted : 2018.06.21
  • Published : 2019.01.31

Abstract

Truncated Hankel operators are compressions of classical Hankel operators to model spaces. In this paper we describe matrix representations of truncated Hankel operators on finite-dimensional model spaces. We then show that the obtained descriptions hold also for some infinite-dimensional cases.

Keywords

References

  1. A. Baranov, I. Chalendar, E. Fricain, J. E. Mashreghi, and D. Timotin Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators, J. Funct. Anal. 259 (2010), no. 10, 2673-2701. https://doi.org/10.1016/j.jfa.2010.05.005
  2. A. Bottcher and B. Silbermann, Analysis of Toeplitz Operators, second edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
  3. J. A. Cima, S. R. Garcia, W. T. Ross, and W. R. Wogen, Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity, Indiana Univ. Math. J. 59 (2010), no. 2, 595-620. https://doi.org/10.1512/iumj.2010.59.4097
  4. J. A. Cima, W. T. Ross, and W. R. Wogen, Truncated Toeplitz operators on finite dimensional spaces, Oper. Matrices 2 (2008), no. 3, 357-369.
  5. D. N. Clark, One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), 169-191. https://doi.org/10.1007/BF02790036
  6. P. L. Duren, Theory of $H^p$ Spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970.
  7. S. R. Garcia, J. Mashreghi, and W. T. Ross, Introduction to Model Spaces and Their Operators, Cambridge Studies in Advanced Mathematics, 148, Cambridge University Press, Cambridge, 2016.
  8. S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285-1315. https://doi.org/10.1090/S0002-9947-05-03742-6
  9. S. R. Garcia and W. T. Ross, A non-linear extremal problem on the Hardy space, Comput. Methods Funct. Theory 9 (2009), no. 2, 485-524. https://doi.org/10.1007/BF03321742
  10. S. R. Garcia and W. T. Ross, The norm of a truncated Toeplitz operator, in Hilbert spaces of analytic functions, 59-64, CRM Proc. Lecture Notes, 51, Amer. Math. Soc., Providence, RI, 2010.
  11. S. R. Garcia and W. T. Ross, Recent progress on truncated Toeplitz operators, in Blaschke products and their applications, 275-319, Fields Inst. Commun., 65, Springer, New York, 2013.
  12. C. Gu, Algebraic properties of truncated Hankel operators, preprint.
  13. D.-O. Kang and H. J. Kim, Products of truncated Hankel operators, J. Math. Anal. Appl. 435 (2016), no. 2, 1804-1811. https://doi.org/10.1016/j.jmaa.2015.11.019
  14. B. Lanucha, Matrix representations of truncated Toeplitz operators, J. Math. Anal. Appl. 413 (2014), no. 1, 430-437. https://doi.org/10.1016/j.jmaa.2013.11.065
  15. R. A. Martinez-Avendano and P. Rosenthal, An Introduction to Operators on the Hardy-Hilbert Space, Graduate Texts in Mathematics, 237, Springer, New York, 2007.
  16. N. K. Nikolski, Treatise on the Shift Operator, translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin, 1986.
  17. V. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
  18. D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491-526. https://doi.org/10.7153/oam-01-29