Structure of the kernels associated with 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 double-struck D sign. 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. Here we study the boundary behaviour and some of the basic properties of the kernel l_λ. Among other things, we show that if the lower zero set of M, Z(M), is nonempty, the kernel l_λ for fixed λ ∈ double-struck D sign has a meromorphic continuation across double-struck T sign/Z(M), where double-struck T struck is the unit circle. Furthermore we consider some special types of kernels l_λ and by studying their structure we obtain information for the invariant subspaces related to them. Lastly, and after introducing the general vector valued setting, we discuss some analogous results for the case of ⊕ L_a²(double-struck D sign;), where m is a positive integer.