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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 $H^\infty$, westudy the problem of approximation of a general function in $\mathcal{H}(b)$ bya function with an extension to the unit circle $\mathbb{T}$ of some degree ofsmoothness, for instance satisfying H\"older estimates or being differentiable.We will exhibit connections between this question and the theory of subnormaloperators and, in particular, we will tie the possibility of smoothapproximations to properties of invariant subspaces of a certain subnormaloperator. This leads us to several computable conditions on $b$ which arenecessary for such approximations to be possible. For a large class of extremepoints $b$ we use our result to obtain explicit necessary and sufficientconditions on the symbol $b$ which guarantee the density of functions withdifferentiable boundary values in the space $\mathcal{H}(b)$. These conditionsinclude an interplay between the modulus of $b$ on $\mathbb{T}$ and thespectrum of its inner factor.

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