A transference principle for involution-invariant functional Hilbert spaces

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 $\mathscr H_\phi(V):=\{f \in \mathrm{Hol}(V) : J_\phi \cdot f \circ \phi \in \mathscr H(U)\}$ endowed with the norm $\|f\|_\phi :=\|J_\phi \cdot f \circ \phi\|_{\mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear mapping $\varGamma_\phi$ defined by $\varGamma_\phi(f) = J_\phi \cdot f \circ \phi,$ $f \in \mathrm{Hol}(V),$ 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.