Professor Richter's project will investigate a class of operators on Hilbert space called two-isometries, which satisfy a weaker, order two version of the norm-preserving condition that defines isometries. An important example of a two-isometry is the operator of multiplication by z on the space of functions that map the disc analytically to a region with finite area, i.e. the classical Dirichlet shift on the classical Dirichlet space. The essence of the projected research is to use operator-theoretic results to study functions in spaces of Dirichlet type, and conversely to use function theory to construct models for abstract two-isometries. A central problem is that of characterizing the invariant subspace lattice of the Dirichlet shift and its generalizations. The mathematical research envisioned here has to do with analytic functions on the disc. These may be described geometrically as the continuous mappings of the disc into the plane that preserve angles, except perhaps at scattered singular points. It is a reflection of the important role analytic functions have played in mathematics for well over a century that they can be described in numerous other ways as well. One method of studying analytic functions is to group them into linear spaces, most fruitfully Hilbert spaces defined by an appropriate finiteness condition, and consider the behavior of certain operators that arise naturally on these spaces. One such space is the Dirichlet space consisting of all analytic functions on the disc which map the disc to a planar region with finite area. The operator of multiplication of functions in this space (and its generalizations) by the independent variable will be studied and characterized by Professor Richter.
