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

For large classes of (finite and) infinite dimensional complex Banach spaces Z , B its open unit ball and f : B → B a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W ( f ), of f in ∂ 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 ( f ) of ( f n )n map B into ∂ B (for any topology finer than the topology of pointwise convergence on B). The target set of f is T ( f ) = ⋃ g∈ ( f ) g(B). To locate T ( f ), we use a concept of closed convex holomorphic hull, Ch(x) ⊂ ∂ B for each x ∈ ∂ B and define a distinguished Wolff hull W ( f ). We show that the Wolff hull intersects all hulls from T ( f ), namely 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 J B∗-triples). These include many well-known operator spaces, for example, L(H , K ), where either H or K is finite dimensional.