Thin Sequences and Their Role in Model Spaces and Douglas Algebras

We study thin interpolating sequences ${lambda_n}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_Theta = H^2 ominus Theta H^2$, where $Theta$ is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that $Theta(lambda_n) to 0$ the interpolation properties in $H^2$ are essentially the same as those in $K_Theta$.

Thin sequences and the Gram matrix

We provide a new proof of Volbergs Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-$2$ class perturbation of a unitary operator. We extend this characterization from $2$ to $p$, where $2 le p le infty$.