Georgia Institute of Technology College of Sciences

In the first part of the talk we will give a brief survey of significant results going from S. Brown pioneering work showing the existence of invariant subspaces for subnormal operators (1978) to Ambrozie-Muller breakthrough asserting the same conclusion for the adjoint of a polynomially bounded operator (on any Banach space) whose spectrum contains the unit circle (2003). The second part will try to give some insight of the different techniques involved in this series of results, culminating with a brilliant use of Carleson interpolation theory for the last one. In the last part of the talk we will discuss additional open questions which might be investigated by these techniques.