The backward shit on Hp

William T. Ross, University of Richmond


In this semi-expository paper, we examine the backward shift operator, Bf := f-f(0)/z on the classical Hardy space Hp. Though there are many aspects of this operator worth of study [20], we will focus on the description of its invariant subspaces by which we mean the closed linear manifolds E ⊂ Hp for which BE ⊂ E. When 1 < p < ∞, a seminal paper of Douglas, Shapiro, and Shields [8] describes these invariant supspaces by using the important concept of the pseudocontinuation developed earlier by Shapiro [26]. When p = 1, the description is the same [1] except that in the proof, one must be mindful of some technical considerations involving the functions of bounded mean oscillation.