Möbius invariant Hilbert spaces of holomorphic functions in the unit ball of 𝐶ⁿ

Zhu, Ke He

doi:10.1090/S0002-9947-1991-0982233-8

Möbius invariant Hilbert spaces of holomorphic functions in the unit ball of $\textbf {C}^ n$
HTML articles powered by AMS MathViewer

by Ke He Zhu PDF

Trans. Amer. Math. Soc. 323 (1991), 823-842 Request permission

Abstract:

We prove that there exists a unique Hilbert space of holomorphic functions in the open unit ball of ${\mathbb {C}^n}$ whose (semi-) inner product is invariant under Möbius transformations.

References

J. Arazy and S. D. Fisher, The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math. 29 (1985), no. 3, 449–462. MR 786732, DOI 10.1215/ijm/1256045634
J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145. MR 814017, DOI 10.1007/BFb0078341
J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), no. 6, 989–1053. MR 970119, DOI 10.2307/2374685
C. A. Berger, L. A. Coburn, and K. H. Zhu, Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus, Amer. J. Math. 110 (1988), no. 5, 921–953. MR 961500, DOI 10.2307/2374698

in the Bergman metric on bounded symmetric domains

Alexander Nagel and Walter Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J. 43 (1976), no. 4, 841–865. MR 425178
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
Ke He Zhu, Hilbert-Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 721–730. MR 1013987, DOI 10.1090/S0002-9939-1990-1013987-7