Extremal functions in invariant subspaces of Bergman spaces
Some recent results on contractive zero-divisors in the Bergman spaces \(A^p\) \((0 < p < \infty)\) are partially generalized to arbitrary subspaces invariant under multiplication by polynomials. It is shown that each invariant subspace of \(A^p\) generated by an inner function has a unique extremal function, represented in terms of the reproducing kernel of a certain weighted \(A^2\) space. The extremal function is shown to have an analytic continuation across each arc of the unit circle that lies outside the singular set of the generator. It is also shown that the canonical divisor in \(A^p\) of a Blaschke sequence generates the invariant subspace determined by the zero-set, as does the associated Blaschke product. Thus the canonical divisor of a Blaschke sequence has an analytic continuation over each arc of the unit circle containing no cluster points of the zeros.
scientific article; zbMATH DE number 926190