Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RANGE INCLUSION OF TWO SAME TYPE CONCRETE OPERATORS

  • Received : 2015.12.07
  • Published : 2016.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let H and K be two Hilbert spaces, and let A and B be two bounded linear operators from H to K. We are interested in $RangeB^*{\supseteq}RangeA^*$. It is well known that this is equivalent to the inequality $A^*A{\geq}{\varepsilon}B^*B$ for a positive constant ${\varepsilon}$. We study conditions in terms of symbols when A and B are singular integral operators, Hankel operators or Toeplitz operators, etc.

Keywords

References

  1. M. Cotlar and C. Sadosky, On the Helson-Szego theorem and a related class of modified Toeplitz kernels, Harmonic analysis in Euclidean spaces (Williamstown, MA, 1978), Part 1, pp. 383-407, eds. G. Weiss and S. Wainger, Proc. Symp. Pure Math. 35, Amer. Math. Soc., Providence, 1979.
  2. R. G. Douglas, On majorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-416. https://doi.org/10.1090/S0002-9939-1966-0203464-1
  3. C. Gu, A generalization of Cowen's characterization of hyponormal Toeplitz operators, J. Funct. Anal. 124 (1994), no. 1, 135-148. https://doi.org/10.1006/jfan.1994.1102
  4. E. Hayashi, The kernel of a Toeplitz operator, Integral Equations Operator Theory 9 (1986), no. 4, 589-591.
  5. B. A. Lotto, Range inclusion of Toeplitz and Hankel operators, J. Operator Theory 24 (1990), no. 1, 17-22.
  6. B. Sz.-Nagy and C. Foias, Analyse harmonique des operateurs de l'espace de Hilbert, Budapest, Akadeniai Kaido, 1967.
  7. T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 753-767. https://doi.org/10.1090/S0002-9947-1993-1162103-7
  8. D. Sarason, Generalized interpolation in $H^{\infty}$, Trans. Amer. Math. Soc. 127 (1967), 179-203.
  9. K. Takahashi, Another proof of the $H^{\infty}$-interpolation theorem on the unit circle, Technical Note, 1982.
  10. T. Yamamoto, Majorization of singular integral operators with Cauchy kernel on $L^2$, Ann. Funct. Anal. 5 (2014), no. 1, 101-108. https://doi.org/10.15352/afa/1391614574