Pseudocontinuations and the Backward Shift

In this paper, we will examine the backward shift operator Lf = (f − f(0))/z on certain Banach spaces of analytic functions on the open unit disk D. In particular, for a (closed) subspace M for which LM ⊂M, we wish to determine the spectrum, the point spectrum, and the approximate point spectrum of L|M. In order to do this, we will use the concept of “pseudocontinuation” of functions across the unit circle T. We will first discuss the backward shift on a general Banach space of analytic functions and then for the weighted Hardy and Bergman spaces, we will show that σ(L|M) = σap(L|M) and moreover whenever M does not contain all of the polynomials, then σ(L|M) ∩ D = σp(L|M) ∩ D = σap(L|M) ∩ D and is a Blaschke sequence. In fact, for certain measures, we will show that M is contained in the Nevanlinna class and every function in M has a pseudocontinuation across T to a function in the Nevanlinna class of the exterior disk. For the Dirichlet and Besov spaces however, the spectral picture of σ(L|M) is quite different. For example σap(L|M) and σ(L|M) can differ and even when σ(L|M) ∩ D = σp(L|M) ∩ D = σap(L|M) ∩ D and is discrete, it need not be a Blaschke sequence. Moreover,M may contain functions which do not have pseudocontinuations across any set of positive measure in T. As an application of our pseudocontinuation techniques and the so-called “H2-duality”, we will look at the index of the Mz-invariant subspaces of the Bergman spaces and weighted Dirichlet spaces. In particular, whenever f and g belong to 223 224 Alexandru Aleman, Stefan Richter & William T. Ross the unweighted Bergman space La(D) and f/g has finite nontangential limits almost everywhere on a set of positive Lebesgue measure in the circle, then the Mz-invariant subspace generated by f and g has index equal to one. For a large class of weighted Dirichlet spaces, we will show that every non-zero Mz-invariant subspace has index equal to one.