Advertisement

Israel Journal of Mathematics

, Volume 54, Issue 2, pp 227–241 | Cite as

On the similarity problem for polynomially bounded operators on hilbert space

  • J. Bourgain
Article

Abstract

Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S −1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining
$$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$
onl 2l 2, whereS is the shift and Γ f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate
$$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$
whereSTS −1 is a contraction and assuming\(\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty \) wheneverp is an analytic polynomial on the disc.

Keywords

Hilbert Space Hardy Space Carleson Measure Invertible Operator Hankel Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W. B. Arveson,Subalgebras of C*-algebras, Acta Math.123 (1969), 141–224.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Bourgain,New Banach space properties of the disc algebra and H , Acta Math.152 (1984), 1–48.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Bourgain,Vector valued singular integrals and the H 1-BMOduality, Case Western Probability Consortium Proceedings, Marcel Dekker, to appear.Google Scholar
  4. 4.
    J. Garnett,Bounded Analytic Functions, Academic Press, 1981.Google Scholar
  5. 5.
    P. R. Halmos,Ten problems in Hilbert space, Bull. Amer. Math. Soc.76 (1970), 887–933.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Holbrook,Spectral dilations and polynomially bounded operators, Indiana Univ. Math. J.20 (1971), 1030–1034.CrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Springer-Verlag, 1977.Google Scholar
  8. 8.
    V. I. Paulsen,Completely bounded maps on C*-algebras and invariant operator ranges, Proc. Amer. Math. Soc.96 (1982), 91–96.CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. I. Paulsen,Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal.55 (1984), 1–17.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. I. Paulsen,Completely bounded homomorphisms of operator algebras, preprint.Google Scholar
  11. 11.
    V. V. Peller,Estimates of functions of power bounded operators on Hilbert space, J. Operator Theory7 (1982), 341–372.zbMATHMathSciNetGoogle Scholar
  12. 12.
    V. V. Peller,Estimates of functions of Hilbert space operators, similarity to a contraction and related function algebras, Lecture Notes in Math.1043, Springer-Verlag, 1984, pp. 199–204.Google Scholar
  13. 13.
    R. Rochberg,A Hankel type operator arising in deformation theory, Proc. Sympos. Pure Math.35 (1979), 457–458.Google Scholar
  14. 14.
    G. C. Rota,On models for linear operators, Comm. Pure Appl. Math.13 (1960), 468–472.CrossRefMathSciNetGoogle Scholar
  15. 15.
    B. Sz.-Nagy,On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. (Szeged)II (1947), 152–157.Google Scholar
  16. 16.
    W. F. Stinespring,Positive functions on C*-algebras, Proc. Amer. Math. Soc.6 (1955), 211–216.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Von Neumann,Eine spectral theorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr.4 (1951), 258–281.zbMATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • J. Bourgain
    • 1
    • 2
  1. 1.University of BrusselsBrusselsBelgium
  2. 2.I.H.E.S.Bures sur YvetteFrance

Personalised recommendations