In this semi-expository paper, we examine the backward shift operator
Bf := (f-f(0)/z
on the classical Hardy space Hp. Through there are many aspects of this operator worthy of study , we will focus on the description of its invariant subspaces by which we mean the closed linear manifolds Ɛ ⊂ Hp for which BƐ ⊂ Ɛ. When 1 < p < ∞, a seminal paper of Douglas, Shapiro, and Shields  describes these invariant subspaces by using the important concept of a pseudocontinuation developed earlier by Shapiro . When p = 1, the description is the same  except that in the proof, one must be mindful of some technical considerations involving the functions of bounded mean oscillation.
Copyright © 2005 Birkhäuser Verlag Basel.
The definitive version is available at: https://link.springer.com/chapter/10.1007%2F3-7643-7340-7_14
Ross, William T. "The Backward Shift on Hp." Operator Theory: Advances and Applications Selected Topics in Complex Analysis 158 (2005): 191-211. doi:10.1007/3-7643-7340-7_14.
Ross, William T., "The backward shift on Hp" (2005). Math and Computer Science Faculty Publications. 185.