AbstractThe aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let$$\mathcal {H}$$Hand$$\mathcal {K}$$Kbe real Hilbert spaces,$$b \in \mathcal {K}$$b∈Kand$$T \in \mathcal {B} (\mathcal {H},\mathcal {K})$$T∈B(H,K)a linear operator with closed range and Moore–Penrose inverse$$T^\dagger $$T†. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator$$\mathrm {Prox}:\mathcal {K}\rightarrow \mathcal {K}$$Prox:K→Kthe operator$$T^\dagger \, \mathrm {Prox}( T \cdot + b)$$T†Prox(T·+b)is a proximity operator on$$\mathcal {H}$$Hequipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator$$\mathrm {Prox}= S_{\lambda }:\ell _2 \rightarrow \ell _2$$Prox=Sλ:ℓ2→ℓ2and any frame analysis operator$$T:\mathcal {H}\rightarrow \ell _2$$T:H→ℓ2that the frame shrinkage operator$$T^\dagger \, S_\lambda \, T$$T†SλTis a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on$$\mathbb R^d$$Rdequipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. In particular, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training of such networks. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.