Invariant subspaces of translation semigroups

In these lectures I shall give an account of some recent results and open problems relating to subspaces of square integrable functions on the real line which are jointly invariant for a pair of semigroups of unitary operators. These semigroups are quite fundamental, namely, translation semigroups, Fourier translation semigroups, and dilation semigroups. A celebrated theorem of Beurling gives a description of the closed subspaces that are simply invariant for translations (or Fourier translations) and as a result these subspaces are in bijective correspondence with the set of all unimodular functions. In contrast, subspaces that are invariant for two of these semigroups turn out to be finitely parametrised by a family of specific unimodular functions. I shall indicate how one goes about the identification of these sets of invariant subspaces and how, with the natural topology, they are identifiable as Euclidean manifolds. Also I shall discuss aspects of the relatively novel nonselfadjoint operator algebras that are associated with them. These algebras are generated by two non-commuting copies of H∞(R). To identify the topology on the set of invariant subspaces it turns out that one needs to establish some essentially function theoretic assertions, in which a limit of a sequence of (projections onto) purely invariant subspaces is a particular reducing subspace (projection). We sketch below how one can obtain such “strange limits”. I would like to thank Alfonso Montes-Rodríguez for the opportunity to present these lectures at the University of Seville to an ideal audience composed of a good mix of old hands and young minds