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

TL;DR

This paper investigates the completeness of biorthogonal systems of reproducing kernels in certain Hilbert spaces of analytic functions, providing conditions for completeness and constructing examples of non-complete biorthogonal systems.

Contribution

It identifies classes of spaces where biorthogonal systems are always complete and constructs explicit examples where they are not, answering a question by N.K. Nikolski.

Findings

In some spaces, biorthogonal systems are always complete.

Constructed a model subspace with a non-complete biorthogonal system.

Provided conditions under which completeness fails or holds.

Abstract

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.