Integral Equations and Operator Theory

, Volume 37, Issue 4, pp 437–448

Aluthge transforms of operators

  • Il Bong Jung
  • Eungil Ko
  • Carl Pearcy


Associated with every operatorT on Hilbert space is its Aluthge transform\(\tilde T\) (defined below). In this note we study various connections betweenT and\(\tilde T\), including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if\(\tilde T\) has a nontrivial invariant subspace, then so doesT, and we give various applications of our results.

1991 Mathematics Subject Classification

47A15 47B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Aluthge,On p-hyponormal operators for 0<p<1, Integral Equations and Operator Theory13 (1990), 307–315.Google Scholar
  2. [2]
    —,Some generalized theorems on p-hyponormal operators, Integral Equations and Operator Theory24 (1996), 497–501.Google Scholar
  3. [3]
    C. Berger,Sufficiently high powers of hyponormal operators have rationally invariant subspaces, Integral Equations and Operator Theory1 (1978), 444–447.Google Scholar
  4. [4]
    A. Brown,On a class of operators, Proc. Amer. Math. Soc.4 (1953), 723–728.Google Scholar
  5. [5]
    S. Brown,Hyponormal operators with thick spectrum have invariant subspaces, Ann. of Math.125 (1987), 93–103.Google Scholar
  6. [6]
    R. Curto, P. Muhly and D. Xia,A trace estimate for p-hyponormal operators, Integral Equations and Operator Theory6 (1983), 507–514.Google Scholar
  7. [7]
    B. Duggal,p-hyponormal operators and invariant subspaces, Acta Sci. Math. (Szeged)64 (1998), 249–257.Google Scholar
  8. [8]
    M. Fujii and Y. Nakatsu,On subclasses of hyponormal operators, Proc. Japan Acad.51 (1975). 243–246.Google Scholar
  9. [9]
    T. Hoover,Quasi-similarity of operators. Illinois J. Math.16 (1972), 678–686.Google Scholar
  10. [10]
    T. Huruya,A note on p-hyponormal operators, Proc. Amer. Math. Soc.125 (1997), 3617–3624.Google Scholar
  11. [11]
    K. Löwner,Uber monotone matrix functionen, Math. Z.38 (1983), 507–514.Google Scholar
  12. [12]
    M. Martin and M. Putinar,Lectures on hyponormal operators, Birkhäuser-Verlag, Basel-Boston, 1989.Google Scholar
  13. [13]
    Sz.-Nagy and C. Foias,Harmonic analysis of operators on Hilbert space, North-Holland, New York, 1970Google Scholar
  14. [14]
    K. Tanahashi,On log-hyponormal operators, Integral Equations and Operator Theory34 (1999), 364–372.Google Scholar

Copyright information

© Birkhäuser Verlag 2000

Authors and Affiliations

  • Il Bong Jung
    • 1
  • Eungil Ko
    • 2
  • Carl Pearcy
    • 3
  1. 1.Department of MathematicsKyungpook National UniversityTaeguKorea
  2. 2.Department of MathematicsEwha Women's UniversitySeoulKorea
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations