A transference principle for involution-invariant functional Hilbert spaces

Let $\sigma :{\mathbb{C}}^d\to {\mathbb{C}}^d$ be an affine-linear involution such that $J_{\sigma }=-1$ and let $U,V$ be two domains in ${\mathbb{C}}^d\mathrm{.}$ Let $\phi :U\to V$ be a $\sigma$-invariant $2$-proper map such that $J_{\phi }$ is affine-linear and let $\mathcal{H}(U)$ be a $\sigma$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U\mathrm{.}$ It is shown that the space ${\mathcal{H}}_{\phi }(V):=\{f\in \mathrm{H}\mathrm{o}\mathrm{l}(V):J_{\phi }\cdot f\circ \phi \in \mathcal{H}(U)\}$ endowed with the norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{\phi }:=\mathrm{\parallel }{J}_{\phi }\cdot f\circ \phi {\mathrm{\parallel }}_{\mathcal{H}(U)}$ is a reproducing kernel Hilbert space and the linear mapping ${\Gamma }_{\phi }$ defined by ${\Gamma }_{\phi }(f)=J_{\phi }\cdot f\circ \phi ,$ $f\in \mathrm{H}\mathrm{o}\mathrm{l}(V),$ is a unitary from ${\mathcal{H}}_{\phi }(V)$ onto $\{f\in \mathcal{H}(U):f=-f\circ \sigma \}\mathrm{.}$ Moreover, a neat formula for the reproducing kernel ${\kappa }_{\phi }$ of ${\mathcal{H}}_{\phi }(V)$ in terms of the reproducing kernel of $\mathcal{H}(U)$ 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.