Minimal invariant subspaces for composition operators

Abstract

A remarkable result by Nordgren, Rosenthal and Wintrobe states that a positive answer to the Invariant Subspace Problem is equivalent to the statement that any minimal invariant subspace for a composition operator C phi, induced by a hyperbolic automorphism phi of the unit disc D in the Hardy space H(2) is one dimensional. Motivated by this result, for f epsilon H(2) we consider the space K(f), which is the closed subspace generated by the orbit of f. We obtain two results, one for functions with radial limit zero, and one for functions without radial limit zero, but tending to zero on a sequence of iterates. More precisely, for those functions f epsilon H(2) with radial limit zero and continuous at the fixed points of phi, we provide a construction of a function g epsilon K(f) such that f is a cluster point of the sequence of iterates {g o phi(_)n}. In case f is in the disc algebra, we have K(g) subset of K(f) subset of (span) over bar {g o phi(n): n epsilon Z). For a function f epsilon H(2) tending to zero on a sequence of iterates {phi(n) (z(0))) at a point z(0) with vertical bar z(0)vertical bar < 1, but having no radial limit at the attractive fixed point, we establish the existence of certain functions in the space and show that unless f is constant on the sequence of iterates {phi(n) (z(0))}, the space K(f) is not minimal.