Eigenfunctions for Hyperbolic Composition Operators--Redux

TL;DR

This paper investigates the invariant subspace problem for hyperbolic composition operators on the Hardy space, showing that certain subspaces contain many eigenvectors and that the operators exhibit hypercyclicity, thus contributing to operator theory and invariant subspace research.

Contribution

The paper revisits and simplifies known results on hyperbolic composition operators, demonstrating the abundance of eigenvectors and hypercyclicity in specific subspaces, advancing understanding of the invariant subspace problem.

Findings

Subspaces generated by functions with specific zeros contain many eigenvectors.

Restrictions of hyperbolic composition operators are hypercyclic.

The point spectrum intersects the unit circle in a set of positive measure.

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 f is a hyperbolic disc automorphism with fixed points at a and b (both necessarily on the unit circle), and C_f the composition operator it induces on H^2, then for every function g in the subspace [{(z-a)(z-a)]^(1/2)H^2, the doubly C_f-cyclic subspace generated by g contains many independent eigenvectors; more precisely, the point spectrum of C_f's restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C_f is hypercyclic (some forward orbit is dense).