In this paper, we will examine the backward shift operator Lƒ = (ƒ – ƒ(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 ∏.
Alexandru Aleman, Stefan Richter, and William T. Ross. Pseudocontinuations and the Backward Shift. Technical paper (TR-97-01). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, August, 1997.