On the similarity problem for polynomially bounded operators on hilbert space

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 ² ⊕l 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.