A transference principle for involution-invariant functional Hilbert spaces

Let σ : Cᵈ → Cᵈ be an affine-linear involution such that J_σ = -1 and let $U, V$ be two domains in Cᵈ. Let φ : U → V be a σ-invariant $2$-proper map such that J_φ is affine-linear and let H(U) be a σ-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space H_φ(V):= ∈ Hol(V) : J_φ · f ∘ φ ∈ H(U)\ endowed with the norm \|f\|_φ :=\|J_φ · f ∘ φ\|H(U) is a reproducing kernel Hilbert space and the linear mapping _φ defined by _φ(f) = J_φ · f ∘ φ, f ∈ Hol(V), is a unitary from H_φ(V) onto ∈ H(U) : f = -f ∘ σ\. Moreover, a neat formula for the reproducing kernel κφ of H_φ(V) in terms of the reproducing kernel of 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.