Topics in Operator Theory pp 519-534 | Cite as

# 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.

## Keywords

Composition operator hyperbolic automorphism Invariant Subspace Problem## Mathematics Subject Classification (2000)

Primary 47B33 Secondary 47A15## Preview

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