Reproducing kernels and invariant subspaces of the Bergman shift

In this article we consider index 1 invariant subspaces M of the operator of multiplication by ζ(z) = z, M_ζ, on the Bergman space L_a²(double struck D sign) of the unit disc. It turns out that there is a positive sesquianalytic kernel l_λ defined on double struck D sign × double struck D sign which determines M uniquely. We set σ(M*_ζ|M^⊥) to be the spectrum of M*_ζ restricted to M^⊥, and we consider a conjecture due to Hedenmalm which states that if M ≠ L_a ²(double struck D sign), then rank l_λ equals the cardinality of σ(M*_ζ|M^⊥). In this direction we show that cardinality (σ(M*_ζ|M ^⊥) ∩ double struck D sign ≤ rank l_λ ≤ cardinality σ(M*_ζ|M^⊥) and furthermore, we resolve the conjecture in the case of zero based invariant subspaces. Moreover, we describe the structure of l_λ for finite zero based invariant subspaces.