Shift operators and their adjoints in several contexts

I will give a very broad overview discussing various uses and generalizations of the shift operator (and its adjoint). In the classical case we consider the Hardy space of analytic functions on the complex disk with square summable Taylor coefficients. The shift operator is simply multiplication by z and this "shifts" the coefficients of the function. The backward shift does the opposite, and in the case of the Hardy space, it's actually the adjoint of the shift. (This doesn't happen in every function space!) There are many classical results about subspaces that are invariant under the shift or its adjoint and connecting these to functions and operators. I'll discuss some of the generalizations of the shift operators and some of my recent and current projects and how they connect to the classical theory.