Hilbert Modules in Analytic Function Spaces

The Invariant Subspace Problem is the most well-known unsolved problem in Operator Theory. For normal operators, the spectral decomposition theorem completely describes the structure of their invariant subspaces. For non-normal operators, a fundamental example is the unilateral shift operator S on a Hilbert space. In 1949, Beurling gave a complete description of its invariant using analytic function theory in the setting of classical Hardy space H2(D): a closed subspace M⊂H2(D) is invariant for S if and only if M = θH2(D) for some inner function θ. This settlement relies on additional structures of analytic function space: multiplication, boundedness and boundary values of analytic functions. These structures are not available in general Hilbert spaces. Beurling’s theorem not only suggested a good approach to Invariant Subspace Problem but also became a foundation for many other developments in Operator Theory.

A great amount of work has been done to search for generalizations of Beurling’s idea. Now the theory has grown much beyond the scope of invariant subspace problem. This workshop aims to bring together experts to report and exchange ideas on recent developments.