Canonical divisors and invariant subspaces in weighted Bergman spaces.

The weighted Bergman space Apa , for 0 p f(z) analytic in the unit disk D⊂C for which |f(z)| p is integrable with respect to the weight w alpha(z) = (alpha + 1)(1 - |z| 2)alpha. In this dissertation, we study the properties of certain contractive divisors which are fundamental to an understanding of the structure of the invariant subspaces of Apa . Hedenmalm showed that in A2 such divisors can be constructed as solutions of extremal problems. This result was extended to Ap for 0 p 1. This leaves open a natural question. Namely, for which values of alpha do canonical divisors exist in Apa for p ≠ 2? To begin to answer this question, for alpha > -1 and 1 ≤ p p. The formula is valid whenever the hypergeometric function is nonvanishing in D . We establish that this hypergeometric function is nonvanishing when alpha = 1, 2, 3. Our numerical evidence indicates the full range of alpha for which it is nonvanishing, depending on p. Whenever the hypergeometric function does not have zeros in the disk, we show that an arbitrary extremal function has no extraneous zeros in the unit disk, assuming that the weighted biharmonic Green function corresponding to the weight w alpha is positive. We also show that if the extremal function is continuous in D and acts as a contractive divisor, then it must map all points on 6D to points outside D . Our numerical evidence suggests that the single-point extremal function can have this property only when -1 Ap do act as contractive divisors, Duren et al. established an integral formula involving the extremal function and the unweighted biharmonic Green function. We establish an analogous integral formula in Ap1 .PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/124202/2/3001064.pd