The Invariant Subspace Problem via Composition Operators-redux

Abstract

The Invariant Subspace Problem (“ISP”) for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H ² by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results: If φ is a hyperbolic disc automorphism with fixed points at a and β (both necessarily on the unit circle), and C _φ the composition operator it induces on H ², then for every $f \in \sqrt {(z - \alpha )(z - \beta )}$ H ², the doubly C _φ-cyclic subspace generated by f contains many independent eigenvectors; more precisely, the point spectrum of C _φ’s restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C _φ is hypercyclic (some forward orbit is dense). Under the stronger restriction $f \in \sqrt {(z - \alpha )(z - \beta )}$ H ^p for some p > 2, the point spectrum of the restricted operator contains an open annulus centered at the origin.