Eigenfunctions of the hyperbolic composition operator

Abstract

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform ϕ of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V ^∞ _n=−∞ C ⁿ_φ u contains nonconstant eigenfunctions of the composition operatorC _ϕ. This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC _ϕ (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of ϕ.