Room P3.10, Mathematics Building

Michael Stessin, State University of New York, Albany
Subalgebras dense in Hardy spaces

Let A be a subalgebra of H . For each p>0 the Hardy space in the unit disk Hp (D) has a structure of A-module. The problem of describing the lattice of closed submodules of this module gives an example of the lattice of closed subspaces invariant under the action of a family of commuting operators. The investigation of the module structure of the disk-algebra was initiated by Wermer in 1950s. It leads to Beurling type results and includes the classical theorem of Beurling as a special case corresponding to the subalgebra of all polynomials. In particular, if the algebra A is dense in all Hp , p<, every closed submodule is z-invariant in a weak sense, and if A is weak- * dense in H , then every closed A-submodule of Hp is z-invariant. In the talk we will discuss some recent results in this area.