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 Ϲ 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 б(L│M) = б ap (L│M) and moreover whenever M does not contain all of the polynomials, then
б (L│M)∩D = б p(L│M) ∩D = б ap(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 б (L│M) is quite different. For example б ap (L│M) and б (L│M) can differ and even when
б (L│M)∩D = бp(L│M) ∩D = б ap(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 the unweighted Bergman space Lpa(D) and f/g has finite non-tangential limits almost everywhere on a set of positive Lebesgue measure in the circle, then the Mz-invariant subspace generatedby 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.
Copyright © 1998 Indiana University Mathematics Journal. This article first appeared in Indiana University Mathematics Journal 47:1 (1998), 223-276.
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Ross, William T., Alexandru Aleman, and Stefan Richter. "Pseudocontinuations and the Backward Shift." Indiana University Mathematics Journal 47, no. 1 (1998): 223-276. doi:10.1512/iumj.1998.47.1583.