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 $$f:B\rightarrow B$$ f : B → B a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in $$\partial B$$ ∂ B and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates $$(f^n)_n$$ ( 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 $$(f^n)_n$$ ( 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 $$\Gamma (f)$$ Γ ( f ) of $$(f^n)_n$$ ( f n ) n map B into $$\partial B$$ ∂ B (for any topology finer than the topology of pointwise convergence on B). The target set of f is $$\begin{aligned} T(f)=\bigcup _{g \in \Gamma (f)} g(B). \end{aligned}$$ T ( f ) = ⋃ g ∈ Γ ( f ) g ( B ) . To locate T(f), we use a concept of closed convex holomorphic hull, $${\text {Ch}}(x) \subset \partial B$$ Ch ( x ) ⊂ ∂ B for each $$x \in \partial B$$ x ∈ ∂ B and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely $$\begin{aligned} W(f) \cap {\text {Ch}}(x)\ne \emptyset \ \ \hbox {for all}\ \ x \in T(f). \end{aligned}$$ W ( f ) ∩ Ch ( x ) ≠ ∅ for all x ∈ T ( f ) . 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 $$JB^*$$ J B ∗ -triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional.