Kenneth R. Davidson - Polynomially bounded operators

An old problem in operator theory which dates back to the late 1940's asks for a simple criterion to determine if an operator acting on a Hilbert space is similar to a contraction (an operator with norm at most one). In the usual fashion of mathematics, Sz. Nagy proposed a simple criterion as a conjecture. This first attempt was proven wrong, to be replaced by another more sophisticated one popularized by Halmos. This problem was the seed for many deep and important developments in operator theory involving the notion of complete boundedness, which was introduced by Arveson in the late 1960's. This led to the first reasonable condition equivalent to similarity to a contraction by Paulsen. However only very recently has the Halmos conjecture been put to rest by a remarkable counterexample due to Pisier. The purpose of this talk is to survey the history of this problem.