The Backward Shift on Hp

Abstract

z on the classical Hardy space H. Though there are many aspects of this operator worthy of study [20], we will focus on the description of its invariant subspaces by which we mean the closed linear manifolds E ⊂ H for which BE ⊂ E . When 1 1 case involves heavy use of duality and especially the Hahn-Banach separation theorem where one gets at E by first looking at E⊥, the annihilator of E , and then returning to E by ⊥(E⊥). On the other hand, when 0 < p < 1, H is no longer locally convex and the Hahn-Banach separation theorem fails [12]. In fact, as we shall see in § 4, there are invariant subspaces E 6= H, 0 < p < 1, for which ⊥(E⊥) = H. Despite these difficulties, an ingenious tour de force approach of Aleksandrov [1] (see also [6]), using such tools as distribution theory and the atomic decomposition theorem, characterizes these invariant subspaces. The first several sections of this paper are a leisurely, non-technical, treatment of the Douglas-Shapiro-Shields and Aleksandrov results. In § 5, we focus on some new results, based on techniques in [4], which give an alternative description of certain invariant subspaces of H. As a consequence, we eventually wind up characterizing the weakly closed invariant subspaces of H. In § 6, we make some remarks about the invariant subspaces of the standard Bergman spaces La when 0 < p < 1.