Some Remarks on the Operator of Foias and Williams

In this paper we study the Foias-Williams operator T (Hg) = S∗ Hg 0 S where g ∈ L∞, and Hg is a Hankel operator with symbol g. We exhibit a relationship between the similarity of T (Hg) to a contraction and the rate of decay of {|gn|}n=0, the absolute values of the Fourier coefficients of the symbol g. Let H denote a complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. Recall that an operator T in L(H) is said to be polynomially bounded if there exists an M ≥ 1 such that ‖p(T )‖ ≤M sup{|p(ζ)| : |ζ| = 1} for all polynomials p. We denote the class of all polynomially bounded operators by (PB). Also, an operator T is similar to a contraction (notation: T ∈(SC)) if there exists a bounded invertible operator L such that ‖LTL−1‖ ≤ 1. Halmos [4] raised the question whether every polynomially bounded operator is similar to a contraction; this question is still open. While there is a number of results dealing with sufficient conditions for a polynomially bounded operator to be similar to a contraction (cf. [9], [6], [5]), there are very few publications dedicated to the search for a counterexample. In [3] (see also [2]) Foias and Williams have studied the operators of the form T (X) = ( S∗ X 0 S ) acting on H2⊕H2, the direct orthogonal sum of two copies of the Hardy space H2 (to be defined below). Here S is a forward unilateral shift on H2 and X ∈ L(H2). In particular, they conjectured that there exists a Hankel operator Hg with symbol g (to be defined below) such that the operator T (Hg) ∈ (PB) \ (SC). In this paper we continue this line of investigation and we show that the membership in the aforementioned classes depends on the rate at which the sequence {gn}n∈N0 of the Fourier coefficients of g tends to 0. 1991 Mathematics Subject Classification. primary 47A, secondary 47B.