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


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.


Hilbert Space Hardy Space Carleson Measure Invertible Operator Hankel Operator 
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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

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