Eigenfunctions for hyperbolic

composition operators--redux



Joel H. Shapiro


Operator Theory: Advances and Applications 202 (2010) 519--534.


 

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 H2 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.


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