• Nakazi, Takahiko (Hokkaido University)
  • 투고 : 2017.04.07
  • 심사 : 2017.12.29
  • 발행 : 2018.07.31


For $1{\leq}p{\leq}{\infty}$, let $H^p$ be the usual Hardy space on the unit circle. When ${\alpha}$ and ${\beta}$ are bounded functions, a singular integral operator $S_{{\alpha},{\beta}}$ is defined as the following: $S_{{\alpha},{\beta}}(f+{\bar{g}})={\alpha}f+{\beta}{\bar{g}}(f{\in}H^p,\;g{\in}zH^p)$. When p = 2, we study the hyponormality of $S_{{\alpha},{\beta}}$ when ${\alpha}$ and ${\beta}$ are some special functions.


singular integral operator;Toeplitz operator;Hardy space;hyponormal operator


  1. M. B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976), no. 3, 597-604.
  2. A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102.
  3. C. C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), no. 3, 809-812.
  4. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415.
  5. I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations. Vol. II, translated from the 1979 German translation by S. Roch and revised by the authors, Operator Theory: Advances and Applications, 54, Birkhauser Verlag, Basel, 1992.
  6. C. X. Gu, A generalization of Cowen's characterization of hyponormal Toeplitz operators, J. Funct. Anal. 124 (1994), no. 1, 135-148.
  7. I. S. Hwang, I. H. Kim, and W. Y. Lee, Hyponormality of Toeplitz operators with polynomial symbols, Math. Ann. 313 (1999), no. 2, 247-261.
  8. I. S. Hwang, Hyponormality of Toeplitz operators with polynomial symbols: an extremal case, Math. Nachr. 231 (2001), 25-38.<25::AID-MANA25>3.0.CO;2-X
  9. I. S. Hwang and W. Y. Lee, Hyponormality of Toeplitz operators with polynomial and symmetric-type symbols, Integral Equations Operator Theory 50 (2004), no. 3, 363-373.
  10. T. Nakazi, Hyponormal Toeplitz operators and zeros of polynomials, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2425-2428.
  11. T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 753-767.
  12. T. Nakazi and T. Yamamoto, Normal singular integral operators with Cauchy kernel on $L^2$, Integral Equations Operator Theory 78 (2014), no. 2, 233-248.
  13. D. Sarason, Generalized interpolation in $H^{\infty}$, Trans. Amer. Math. Soc. 127 (1967), 179-203.
  14. K. H. Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equations Operator Theory 21 (1995), no. 3, 376-381.