Bloch functions on the unit ball of an infinite dimensional Hilbert space

Abstract

Abstract. The Bloch space has been studied on the open unit disk of C and some homogeneous domains of C n . We define Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that-modulo the constant functionsis automorphism invariant as well. All bounded analytic functions on BE are also Bloch functions.

Bloch type spaces on the unit ball of a Hilbert space

“…In [11], S. G. Krantz and D. Ma considered function theoretic and functional analytic properties of Bloch functions on strongly pseudoconvex domain. To have a more complete insight on the theory of the Bloch space in the finite dimensional space, see the book [25] by Zhu. Recently, Bloch functions on the unit ball of an infinite-dimensional complex Hilbert space have been studied by Blasco, Galindo and Miralles [1]. In this article, we shall continue the study in [1] and consider Bloch type spaces on the unit ball of a Hilbert space.…”

Bloch functions on the unit ball of a Banach space

“…In particular, this study includes the unit euclidean ball B n and the polydisc D n . The study of Bloch functions on bounded symmetric domains of infinite dimensional Banach spaces was introduced by Blasco, Galindo and Miralles (see [4]) for the Hilbert case and by Chu, Hamada, Honda and Kohr for general bounded symmetric domains by means of the Kobayashi metric (see [7]). If we consider these domains as the unit ball B E of a JB * −triple E, the corresponding Bloch space is the set of holomorphic functions on B E which satisfy that sup x∈B E Q f (z) < ∞.…”

“…K. T. Hahn and R. M. Timoney extended the notion of Bloch function by considering bounded homogeneous domains in C n , such as the unit ball B n and the polydisk D n (see [11,17,18]). O. Blasco, P. Galindo and A. Miralles extended the notion to the infinite dimensional setting by considering Bloch functions on the unit ball of an infinite dimensional Hilbert space (see [4,5,6]) and C. Chu, H. Hamada, T. Honda and G. Kohr considered Bloch functions on bounded symmetric domains which may be also infinite dimensional (see [7]).…”

Operators from the Hardy space to the α‐Bloch space

“…When the target is the unit disc in , Chu, Hamada, Honda and Kohr obtained the following Schwarz Pick lemma (cf. [, Theorem 4.2], [, Theorem 4.6]). Lemma Let be such that .…”