Cyclic vectors and invariant subspaces for the backward shift operator

Abstract

The operator U of multiplication by z on the Hardy space H 2 of square summable power series has been studied by many authors. In this paper we make a similar study of the adjoint operator U * (the “backward shift”). Let K f denote the cyclic subspace generated by f ( f ∈ H 2 ) , that is, the smallest closed subspace of H 2 that contains { U * n f } ( n ≥ 0 ) . If K f = H 2 , then f is called a cyclic vector for U * . Theorem : f is a cyclic vector if and only if there is a function g , meromorphic and of bounded Nevanlinna characteristic in the region 1 < | z | = ∞ , such that the radical limits of f and g coincide almost everywhere on the boundary | z | = 1 . Such a g is called a “pseudo analytic continuation” of f . Other results include the following. If f has a power series with Hadamard gaps, then f is a cyclic vector. If f is not cyclic, and if f can be continued analytically across some boundary point, then every function h ∈ K f can be continued across this same point. The set of all the non-cyclic vectors is a dense F σ set of the first category that is also a vector subspace of H 2 . In addition we study the relationship of cyclic vectors to inner functions, and to approximation by rational functions.