Hankel operators and similarity problems
by
Catalin Badea
Universite de Lille

Let T in B( K) be a coisometry on Hilbert space K, (that is its adjoint T^* is an isometry) and let V in B(H) be an isometry. Given X in B(H, K), let R(X;T, V) be a (general) Foguel operator

R(X;T, V) = é ê ê ê ë T X 0 V ù ú ú ú û in B(K\oplusH).

The operator R(X;T, V) is said to be a Foguel-Hankel operator if X satisfies TX = XV. Well-known examples studied by Foguel (the first power bounded operator not polynomially bounded), Foias-Williams, Pisier (the first polynomially bounded operator not similar to a contraction), Davidson-Paulsen and others are operators of this type.

The aim of my talk is to discuss several recent results concerning the similarity to contractions of general and concrete Foguel and Foguel-Hankel operators. In particular, we give an operator-theoretic equivalent for the following result due to Bonami-Bruna and, independently, to Gasch-Gilbert: The restriction of the operator of triangular truncation to Hankel matrices in B(l₂) is bounded.

Their original proof is based on estimates of Lacey and Thiele on the bilinear Hilbert transform.

Date received: June 14, 2002