The Feichtinger conjecture for reproducing kernels in model subspaces

We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace $K_\Theta = H^2\ominus \Theta H^2$ of the Hardy space $H^2$, where $\Theta$ is an inner function. First, we verify the Feichtinger conjecture for the kernels $ \tilde k_{\lambda_n} = k_{\lambda_n}/\|k_{\lambda_n}\|$ under the assumption that $\sup_n |\Theta(\lambda_n)|<1$. Secondly, we prove the Feichtinger conjecture in the case where $\Theta$ is a one-component inner function, meaning that the set $\{z:|\Theta(z)|<\varepsilon\}$ is connected for some $\varepsilon\in(0,1)$.