Some Sufficient Conditions for the Division Property of Invariant Subspaces in Weighted Bergman Spaces

A weighted Bergman spaceBis a Banach space of the formLp(μ)∩Hol(Ω), whereμis a Borel measure carried by the bounded regionΩin the complex plane. We consider closed subspacesMofBthat are invariant for multiplication by the independent variablez. We sayMhas the division property, if dimM/(z−λ)M=1 for eachλ∈Ω. In terms of the local boundary behavior of the functions inMwe give several conditions which imply the division property. For example, this happens ifMis generated by functions that extend analytically near a fixed boundary point and if ∂Ωis nice near this point. “Analytic” may be replaced by “locally Nevanlinna.” For the standard weights (1−|z|)αdAon the unit disc we show thatMhas the division property if it contains one function that is locally Nevanlinna near a boundary point. Furthermore, in the unweighted case (α=0) the invariant subspace generated by two functions that areLrrespectivelyLs(1/r+1/s=1/p) near some boundary point, has the division property.