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 $\mathbb C^d$ with $U$ being $\sigma$-invariant. 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. This recovers some known results. Our result not only yields a candidate for Hardy spaces but also an analog of von Neumann's inequality for contractive tuples naturally associated with these domains. Unlike the existing techniques, we capitalize on the methods from several complex variables.