Advertisement

Polynomially Hyponormal Operators

  • Raúl Curto
  • Mihai Putinar
Part of the Operator Theory Advances and Applications book series (OT, volume 207)

Abstract

A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponorrnal operator which is not subnormal. This is achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides an optimal framework for studying k-hyponormality. Non-trivial links with the theory of Toeplitz operators on Hardy space are also exposed in detail. A good selection of intriguing open problems, with precise references to prior works and partial solutions, is offered.

Mathematics Subject Classification (2000)

Primary 47B20 Secondary 47B35 47B37 46A55 30E05 

Keywords

Hyponorrnal operator weighted shift moment problem convex cone Toeplitz operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Dvke Math. J. 43 (1976), 597–604.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    J. Agler, Hypercontractions and subnormality, J. Operator Th. 13 (1985), no. 2, 203–217.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Agier, An abstract approach to model theory, in Surveys of Recent Results in Operator Theory, vol. II (J.B. Conway and B.B. Morrel, eds.), Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 1–23.Google Scholar
  4. [4]
    J.Y. Bae, G. Exner and I.B. Jung, Criteria for positively quadratically hyponormal weighted shifts, Proc. Amer. Math. Soc. 130 (2002), 3287–3294CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    G. Cassier, Problème des moments sur un compact de ℝn et décomposition de polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), no. 3, 254–266.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Y.B. Choi, A propagation of quadratically hyponormal weighted shifts. Bull. Korean Math. Soc. 37 (2000), 347–352.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Y.B. Choi, J.K. Han and W.Y. Lee, One-step extension of the Bergman shift, Proc. Amer. Math. Soc. 128(2000), 3639–3646.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Mathematics and its Applications, vol. 9, Gordon and Breach, Science Publishers, New York-London-Paris, 1968.Google Scholar
  9. [9]
    J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991.Google Scholar
  10. [10]
    C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367(1986), 215–219.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    C. Cowen, Hyponormal and subnormal Toeplitz operators, in Surveys of Some Recent Results in Operator Theory, Vol. I (J.B. Conway and B.B. Morrel, eds.), Pitman Res. Notes in Math., vol. 171, Longman Publ. Co. 1988, pp. 155–167.Google Scholar
  12. [12]
    C. Cowen and J. Long, Some subnormal Toeplitz operators, J. Reine Angein. Math. 351(1984), 216–220.zbMATHMathSciNetGoogle Scholar
  13. [13]
    R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symposia Pure Math. 51(1990), 69–91.MathSciNetGoogle Scholar
  14. [14]
    R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13(1990), 49–66.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    R. Curto, Polynomially hyponormal operators on Hilbert space, in Proceedings of EL AM VII, Revista Unión Mat. Arg. 37(1991), 29–56.zbMATHMathSciNetGoogle Scholar
  16. [16]
    R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17(1993), 202–246.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations Operator Theory, 18(1994), 369–426.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    R. Curto, LS. Hwang and W.Y. Lee, Weak subnormality of operators, Arch. Math. 79(2002), 360–371.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    R. Curto and I.B. Jung, Quadratically hyponormal shifts with two equal weights, Integral Equations Operator Theory 37(2000), 208–231.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    R. Curto, LB. Jung and W.Y. Lee, Extensions and extremality of recursively generated weighted shifts, Proc. Amer. Math. Soc. 130(2002), 565–576.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    R. Curto, I.B. Jung and S.S. Park, A characterization of k-hyponormality via weak subnormality, J. Math. Anal. Appl. 279(2003), 556–568.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    R. Curto and S.H. Lee, Quartically hyponormal weighted shifts need not be 3-hyponormal, J. Math. Anal. Appl. 314 (2006), 455–463.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    R. Curto, S. H. Lee and W.Y. Lee, Subnormality and 2-hyponormality for Toeplitz operators, Integral Equations Operator Theory 44(2002), 138–148.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    R. Curto, S.H. Lee and W.Y. Lee, A new criterion for k-hyponormality via weak subnormality, Proc. Amer. Math. Soc. 133(2005), 1805–1816.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    R. Curto and W.Y. Lee, Joint hyponormality of Toeplitz pairs, Memoirs Amer. Math. Soc. 150, no. 712, Amer. Math. Soc, Providence, 2001.Google Scholar
  26. [26]
    R. Curto and W.Y. Lee, Towards a model theory for 2-hyponormal operators, Integral Equations Operator Theory 44(2002), 290–315.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    R. Curto and W.Y. Lee, Subnormality and k-hyponormality of Toeplitz operators: A brief survey and open questions, in Operator Theory and Banach Algebras, The Theta Foundation, Bucharest, 2003; pp. 73–81.Google Scholar
  28. [28]
    R. Curto and W.Y. Lee, Solution of the quadratically hyponormal completion problem, Proc. Amer. Math. Soc. 131(2003), 2479–2489.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    R. Curto and W.Y. Lee, k-hyponormality of finite rank perturbations of unilateral weighted shifts, Trans. Amer. Math. Soc. 357(2005), 4719-4737.Google Scholar
  30. [30]
    R. Curto and S.S. Park, k-hyponormality of powers of weighted shifts via Schur products, Proc. Amer. Math. Soc. 131(2003), 2761–2769.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    R. Curto, Y.T. Poon and J. Yoon, Subnormality of Bergman-like weighted shifts, J. Math. Anal. Appl. 308(2005), 334–342.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. 115 (1993), no. 2, 480–497.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    M.A. Dritschel and S. McCullough, Model theory for hyponormal contractions, Integral Equations Operator Theory 36(2000), 182–192.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    G. Exner, I.B. Jung and D. Park, Some quadratically hyponormal weighted shifts, Integral Equations Operator Theory 60 (2008), 13–36.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    G. Exner, I.B. Jung and S.S. Park, Weakly n-hyponormal weighted shifts and their examples, Integral Equations Operator Theory 54 (2006), 215–233.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    P.R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2, (1950). 125–134.MathSciNetGoogle Scholar
  37. [37]
    P.R. Halmos, A Hilbert Space Problem Book, Second edition, Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982.Google Scholar
  38. [38]
    P.E. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), 887–933.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    P.R. Halmos, Ten years in Hilbert space, Integral Equations Operator Theory 2(1979), 529–564.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    J.W. Helton and M. Putinar, Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization, in Operator Theory, Structured Matrices, and Dilations, pp. 229–306, Theta Ser. Adv. Math. 7, Theta, Bucharest, 2007.Google Scholar
  41. [41]
    T. Hoover, I.B. Jung, and A. Lambert, Moment sequences and backward extensions of subnormal weighted shifts, J. Austral. Math. Soc. 73 (2002), 27–36.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [42]
    I. Jung and C. Li, A formula for k-hyponormality of backstep extensions of subnormal weighted shifts, Proc. Amer. Math. Soc. 129 (2001), 2343–2351.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    I. Jung and S. Park, Quadratically hyponormal weighted shifts and their examples, Integral Equations Operator Theory 36(2000), 480–498.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    I. Jung and S. Park, Cubically hyponormal weighted shifts and their examples, J. Math. Anal. Appl. 247 (2000), 557–569.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [45]
    S.H. Lee and W.Y. Lee, Quadratic hyponormality and 2-hyponormality for Toeplitz operators, Integral Equations Operator Theory 54 (2006), 597–602.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    S.H. Lee and W.Y. Lee, Hyponormal operators with rank-two self-commutators, J. Math. Anal. Appl. 351(2009), 616–626.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [47]
    S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107(1989), 187–195.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [48]
    T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), 753–767.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), 969–984.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover, New York, 1990.zbMATHGoogle Scholar
  51. [51]
    J. Stampfli, Which weighted shifts are subnormal? Pacific J. Math. 17(1966), 367–379.zbMATHMathSciNetGoogle Scholar
  52. [52]
    S. Sun, Bergman shift is not unitarily equivalent to a Toeplitz operator, Kexue Tongbao (English Ed.) 28 (1983), 1027–1030.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Raúl Curto
    • 1
  • Mihai Putinar
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations