2024/08/08 by Bera, Santu, Chavan, Sameer, Jain, Shubham · 2 citations
#46E22 #47A13 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 32A36 #Secondary 32A10
paper · doi:10.48550/arxiv.2408.04384
Let σ: \mathbb Cd → \mathbb Cd be an affine-linear involution such that Jσ= -1 and let U, V be two domains in \mathbb Cd. Let ϕ: U → V be a σ-invariant 2-proper map such that Jϕ is affine-linear and let \mathscr H(U) be a σ-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on U. It is shown that the space \mathscr Hϕ(V):=\f ∈ Hol(V) : Jϕ⋅ f ∘ ϕ∈ \mathscr H(U)\ endowed with the norm ‖f‖ϕ:=‖Jϕ⋅ f ∘ ϕ‖\mathscr H(U) is a reproducing kernel Hilbert space and the linear mapping \varGammaϕ defined by \varGammaϕ(f) = Jϕ⋅ f ∘ ϕ, f ∈ Hol(V), is a unitary from \mathscr Hϕ(V) onto \f ∈ \mathscr H(U) : f = -f ∘ σ\. Moreover, a neat formula for the reproducing kernel κϕ of \mathscr Hϕ(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.