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 f : B → B a compact holomorphic fixed-point free map, we introduce and define the Wolff hull , W ( f ), of f in ∂ B ∂ B and prove that W ( f ) is proximal to the images of all subsequential limits of the sequences of iterates (fⁿ)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 (fⁿ)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 Γ (f) Γ ( f ) of (fⁿ)ₙ ( f n ) n map B into ∂ B ∂ B (for any topology finer than the topology of pointwise convergence on B ). The target set of f is aligned T(f)= g ∈ Γ (f) g(B). aligned T ( f ) = ⋃ g ∈ Γ ( f ) g ( B ) . To locate T ( f ), we use a concept of closed convex holomorphic hull, Ch(x) ⊂ ∂ B Ch ( x ) ⊂ ∂ B for each x ∈ ∂ B x ∈ ∂ B and define a distinguished Wolff hull W ( f ). We show that the Wolff hull intersects all hulls from T ( f ), namely aligned W(f) ∩ Ch(x)≠ ∅ \ \ for all\ \ x ∈ T(f). 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.