I am working with a non-Euclidean proximity operator defined by a Bregman distance function $D(\cdot, \cdot)$: $$ \operatorname{prox}_f(x) = \operatorname*{argmin}_u \{ f(u) + D(u, x) \} $$
Is anybody aware of a result similar to Moreau decomposition for such proximal operators?
The only thing I found is a paper by Combettes et. al. - Moreau’s Decomposition in Banach Spaces about such an extension of Moreau decomposition for Banach Spaces. But I am wondering if there is something simpler which is true for Hilbert spaces, since I don't need something as general as a Banach space.