Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?

Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems of reproducing kernels in a reproducing kernel Hilbert space $\mathcal{H}$ of analytic functions. We study the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_\Theta$ of the Hardy space. We find a class of spaces where systems biorthogonal to complete systems of reproducing kernels are always complete, and show that in general this is not true. In particular we answer the question posed by N.K. Nikolski and construct a model subspace with a non-complete biorthogonal system.