On the problem of smooth approximations in de Branges-Rovnyak spaces and connections to subnormal operators

Abstract: For the class of de Branges-Rovnyak spaces $\mathcal{H}(b)$ of the unit disk $\mathbb{D}$ defined by extreme points $b$ of the unit ball of $H^\infty$, we study the problem of approximation of a general function in $\mathcal{H}(b)$ by a function with an extension to the unit circle $\mathbb{T}$ of some degree of smoothness, for instance satisfying H\"older estimates or being differentiable. We will exhibit connections between this question and the theory of subnormal operators and, in particular, we will tie the possibility of smooth approximations to properties of invariant subspaces of a certain subnormal operator. This leads us to several computable conditions on $b$ which are necessary for such approximations to be possible. For a large class of extreme points $b$ we use our result to obtain explicit necessary and sufficient conditions on the symbol $b$ which guarantee the density of functions with differentiable boundary values in the space $\mathcal{H}(b)$. These conditions include an interplay between the modulus of $b$ on $\mathbb{T}$ and the spectrum of its inner factor.