On reproducing kernels and invariant subspaces of the Bergman shift

We denote by $L2aD$ the classical Bergman space of all square integrable analytic functions with respect to the Lebesgue area measure on the unit disc. We set ζ( z) = z, z ∈ $D$ and by M_ζ we denote the operator of multiplication by ζ on $L2aD$. Additionally, we suppose that M is a multiplier invariant subspace of $L2aD$; that is, M_ζf ∈ M for all f ∈ M and moreover we assume that ind M ≡ dim (M $\ominus$ ζM) = 1.; If G is a unit vector in M $\ominus$ ζM, then there is a positive definite sesquianalytic kernel $llz$ defined on $D\times D$ such that $1-lzllz1-lz2$ is the reproducing kernel for $MG$ (this space is the closure of the analytic polynomials in $L2aG2D$). As a consequence, one checks that $llz$ defines the space M uniquely. Hence, it is natural to ask about the properties, the boundary behavior and the structure of $llz$. It is within this context that this study has been undertaken.; We define the rank of a positive definite sesquianalytic kernel and we study its properties for a larger class of Hilbert spaces which contains $L2aD$. In the case of $L2aD$, we set to be the spectrum of $M*z$ restricted to M⊥ and we consider a conjecture which is due to H. Hedenmalm and which states that rank $ll$ equals the cardinality of . We show that; cardinalitys M^*z&vbm0;M^⊥ ∩D ≤rankll≤cardinality sM^*z&vbm0; M^⊥ .; Additionally, we prove that the conjecture is true whenever M is a nontrivial, zero based invariant subspace of $L2aD$.; Furthermore, it is shown that if I =