A transference principle for involution-invariant functional Hilbert spaces

Let: Cᵈ Cᵈ be an affine-linear involution such that J_ = -1 and let U, V be two domains in Cᵈ. Let: U V be a -invariant 2-proper map such that J_ is affine-linear and let H (U) be a -invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on U. It is shown that the space H_ (V): =\f Hol (V): J_ f H (U) \ endowed with the norm \|f\|_: =\|J_ f \| ₇ (ₔ) is a reproducing kernel Hilbert space and the linear mapping _ defined by _ (f) = J_ f, f Hol (V), is a unitary from H_ (V) onto \f H (U): f = -f \. Moreover, a neat formula for the reproducing kernel _ of H_ (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.