Extremal functions and invariant subspaces in Dirichlet spaces
Let μ be a positive finite Borel measure on the unit circle of the complex plane, and let D(μ) be the associated Dirichlet space. The Beurling-Deny capacity associated with D(μ) is denoted by cμ. Brown-Shields conjecture for D(μ) says that a function f∈D(μ) is cyclic for D(μ), meaning that {pf:p is a polynomial} is dense in D(μ), if and only if f is outer and the boundary zeros set of f is of cμ− capacity zero. In this talk, we give a class of measures for which Brown-Shields conjecture holds. We also give an explicit description of all invariant subspaces for the shift operator for these measures. Our method is based on a study of the behavior of extremal functions for Dirichlet spaces.
This is joint work with Y. Elmadani and I. Labghail