Subspaces of de Branges spaces with prescribed growth

The growth properties of de Branges spaces and their subspaces are studied. It is shown that, for each given pair of growth functions $\lambda(r)=O(r)$ and $\lambda_1=o(\lambda)$, there exist de Branges spaces of growth $\lambda$ that have a de Branges subspace of growth $\lambda_1$. This phenomenon cannot occur for a class of de Branges spaces that, in a certain sense, behave regularly along the real axis.