Erik Avelin: Invariant Subspaces for Integration and Differentiation Operators on Spaces of Holomorphic Functions

Abstract: Given a continuous linear operator T on a Fréchet space X, a closed subspace M of X is said to be T-invariant if \(T(M) \subset M\). As we know from elementary linear algebra, it is often useful in the study of an operator to determine all invariant subspaces. For the classical integration operator \((Tf)(x) = \int_0^x f\) on \(L^2(0,1)\), this was done by Agmon in 1949. The corresponding problem for the differentiation operator \(Df = f'\) on \(C^\infty(0,1)\) is open, but partial results were obtained by Aleman and Korenblum in 2008.

This talk presents complex-variable analogues of the theorems of Agmon and Aleman-Korenblum. For a conformal map g of the unit disc onto a Jordan domain \(\Omega\) with smooth boundary we consider the operators \((T_g f)(z) = \int_a^z fg'\) (\(|a| \le 1\)) and \(D_g f = f'/g'\) on the Hardy space \(H^2\) and the smooth disc algebra \(A^\infty\), respectively. Complete results are obtained under the assumption that \(\partial\Omega\) has positive curvature.