Beurling's theorem for Hilbert spaces

An expository talk about Beurling’s theorem for Hilbert spaces, which characterizes the closed subspaces of a Hilbert space that are invariant under a shift operator. We explicitly discuss the proof of Beurling’s theorem for the Hardy space, and then state the other generalizations of the theorem.

Further reading: Beurling, “On two problems concerning linear transformations in Hilbert space” (Acta Math. 81, 1949); Hoffman, Banach Spaces of Analytic Functions (Dover); Rosenblum–Rovnyak, Hardy Classes and Operator Theory (Dover).