Let \(\sigma : {\mathbb {C}}^d \rightarrow {\mathbb {C}}^d\) be an affine-linear involution such that \(J_\sigma = -1\) and let U, V be two domains in \({\mathbb {C}}^d.\) Let \(\phi : U \rightarrow V\) be a \(\sigma \) -invariant 2-proper map such that \(J_\phi \) is affine-linear and let \({\mathscr {H}}(U)\) be a \(\sigma \) -invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on U. It is shown that the space \({\mathscr {H}}_\phi (V):=\{f \in \textrm{Hol}(V): J_\phi \cdot f \circ \phi \in {\mathscr {H}}(U)\}\) endowed with the norm \(\Vert f\Vert _\phi :=\Vert J_\phi \cdot f \circ \phi \Vert _{{\mathscr {H}}(U)}\) is a reproducing kernel Hilbert space and the linear mapping \(\varGamma _\phi \) defined by \(\varGamma _\phi (f) = J_\phi \cdot f \circ \phi ,\) \(f \in \textrm{Hol}(V),\) is a unitary from \({\mathscr {H}}_\phi (V)\) onto \(\{f \in {\mathscr {H}}(U): f = -f \circ \sigma \}.\) Moreover, a neat formula for the reproducing kernel \(\kappa _{\phi }\) of \({\mathscr {H}}_\phi (V)\) in terms of the reproducing kernel of \({\mathscr {H}}(U)\) is given. The above scheme is applicable to symmetrized bidisc, tetrablock, d-dimensional fat Hartogs triangle and d-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann’s inequality for contractive tuples naturally associated with these domains.
