Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces

A weighted Bergman space B is a Banach space of the form L(p)(mu)boolean AND Ho1(Omega), where mu is a Borel measure carried by the bounded region Omega in the complex plane. We consider closed subspaces. M of B that are invariant for multiplication by the independent variable z. We say M has the division property, if dim M/(z - lambda) M = 1 for each lambda epsilon Omega. In terms of the local boundary behavior of the functions in M we give several conditions which imply the division property. For example, this happens if M is generated by functions that extend analytically near a fixed boundary point and if partial derivative Omega is nice near this point. ''Analytic'' may be replaced by ''locally Nevanlinna.'' For the standard weights (1-\z\)(alpha) dA on the unit disc we show that M has the division property if it contains one function that is locally Nevanlinna near a boundary point. Furthermore, in the unweighted case (alpha = 0) the invariant subspace generated by two functions that are L(r) respectively L(s)(1/r + 1/s = 1/p) near some boundary point, has the division property. (C) 1997 Academic Press.