The Denjoy–Wolff Theorem in the Open Unit Ball of a Strictly Convex Banach Space

The Denjoy–Wolff Theorem for Condensing Holomorphic Mappings

“…Observe that Lemma 2.1 is a generalization to all Banach spaces of the analogous result obtained by Yang [51] in the case of C n (see also [1] and [29]). …”

“…(xii) When X is a Hilbert space the following equality is valid [29] (see also [1] and [51] for X=C n ): …”

A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces

“…Using Całka's theorem [6], one arrives at the following basic result. [20], [21] and [26]) Let D be a bounded and convex domain in a complex Banach space …”

“…Recall that this is valid for open unit balls in complex Banach spaces ( [5], [20], [21], [26], [27], [29]; see also [1], [2] and [3] for the case of bounded and convex domains in C k ). Now we prove this fact in the special case where the sequence {x n } stems from a compact, fixed-point-free and k D -nonexpansive self-mapping f of D; in particular, from a compact, fixed-point-free and holomorphic self-mapping of D. In contrast with [4], we no longer assume that the complex Banach space X is reflexive.…”

The Denjoy-Wolff Theorem for condensing mappings in a bounded and strictly convex domain in a complex Banach space

“…Namely, Stachura shows that in the complex infinite dimensional Hilbert space H = l 2 the convergence result fails even for biholomorphic self-maps in the open unit ball B H ( [69], see also [28]). So, in order to obtain a generalization of the Denjoy-Wolff theorem we have to impose some restrictions on the holomorphic self-mapping f : D → D. Then, using the methods similar to those which were applied in [12], the following Denjoy-Wolff theorem for compact mappings was proved in [16] (in case of the open and strictly convex unit ball B in a complex Banach space (X, · ), see [22,37,44,65]). Theorem 1.4.…”