The Wolff hull of a compact holomorphic self-map on an infinite dimensional ball

Abstract

For large classes of (finite and) infinite dimensional complex Banach spaces Z, B its open unit ball and \(f:B\rightarrow B\) a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in \(\partial B\) and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates \((f^n)_n\) of f. The Wolff hull generalises the concept of a Wolff point, where such a point can no longer be uniquely determined, and coincides with the Wolff point if Z is a Hilbert space. Recall that \((f^n)_n\) does not generally converge even in finite dimensions, compactness of f (i.e. f(B) is relatively compact) is necessary for convergence in the infinite dimensional Hilbert ball and all accumulation points \(\Gamma (f)\) of \((f^n)_n\) map B into \(\partial B\) (for any topology finer than the topology of pointwise convergence on B). The target set of f is

$$\begin{aligned} T(f)=\bigcup _{g \in \Gamma (f)} g(B). \end{aligned}$$

To locate T(f), we use a concept of closed convex holomorphic hull, \({\text {Ch}}(x) \subset \partial B\) for each \(x \in \partial B\) and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely

$$\begin{aligned} W(f) \cap {\text {Ch}}(x)\ne \emptyset \ \ \hbox {for all}\ \ x \in T(f). \end{aligned}$$

If B is the Hilbert ball, W(f) is the Wolff point, and this is the usual Denjoy–Wolff result. Our results are for all reflexive Banach spaces having a homogeneous ball (or equivalently, for all finite rank \(JB^*\)-triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional.