#### DOI

10.1512/iumj.1998.47.1583

#### Abstract

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 *L**M* 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 = *б _{p}*(

*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 “*H*^{2}-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 *L ^{p}_{a}*(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.

#### Document Type

Article

#### Publication Date

1998

#### Publisher Statement

Copyright © 1998 Indiana University Mathematics Journal. This article first appeared in *Indiana University Mathematics Journal *47:1 (1998), 223-276.

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#### Recommended Citation

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.