Automorphism invariant inner product in Hilbert spaces of holomorphic functions on the unit ball of \(\mathbb{C}^ n\)
Let \(B\) be the open unit ball in complex \(n\)-space \(\mathbb{C}^n\) and \(\Aut(B)\) denote the space of holomorphic automorphisms of \(B\) defined by the Möbius transformations of \(B\) onto itself. A result of \textit{K. Zhu} [Trans. Am. Math. Soc. 323, No. 2, 823-842 (1991; Zbl 0739.46009)] says that there is a unique Hilbert space \(H\) of holomorphic functions on \(B\) which is \(\Aut(B)\) invariant. In the present paper, the author gives two explicit integral formulas for the \(\Aut(B)\) invariant inner product in \(H\). These formulas are first derived on the space \(P(B)\) of holomorphic polynomial functions from \(B\) to \(C\) and then extended to the Hilbert space completion \(H\) of \(P(B)\).
scientific article; zbMATH DE number 880170