A Denjoy–Wolff theorem for compact holomorphic mappings in reflexive Banach spaces

Abstract

a b s t r a c tUsing the Kobayashi distance, we establish a Denjoy-Wolff theorem for compact holomorphic self-mappings of a bounded and strictly convex domain in a complex reflexive Banach space.

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

“…[15,38,46] Let D be a bounded and strictly convex domain in a complex Banach space (X, · ). Let {x j } j∈J and {y j } j∈J be two nets in D which converge in norm to ξ ∈ ∂D and to η ∈ D, respectively.…”

“…Theorem 1.4. ( [16], see also [15]) If D is a bounded and strictly convex domain in a complex Banach space (X, · ), and f : D → D is compact, holomorphic and fixed-point-free, then there exists a point ξ ∈ ∂D such that the sequence {f n } of the iterates of f converges in the bounded-open topology to the constant map taking the value ξ, i.e., on each k D -bounded subset C of D, the sequence {f n } tends uniformly to ξ.…”

“…The main result of our paper is the Denjoy-Wolff theorem for condensing k Dnonexpansive mapping (k D denotes the Kobayashi distance in D) in the bounded and strictly convex domain D (in case of the open and strictly convex unit ball B in (X, · ), see [38,48] and Remark 3.1 in [15]). Note that the class of all condensing mappings contains the class of all compact mappings.…”

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

“…The following version of the Denjoy-Wolff theorem ( [8], [33], [34] and [35]) for bounded and strictly convex domains in complex and reflexive Banach spaces has recently been established in [4] (see [3] and [5] for an up-to-date list of references regarding this topic). Since the assumption that the complex Banach space (X, ∥ · ∥) is reflexive is essential in [4], it is natural to ask if Theorem 1.1 holds in all Banach spaces.…”

“…Since the assumption that the complex Banach space (X, ∥ · ∥) is reflexive is essential in [4], it is natural to ask if Theorem 1.1 holds in all Banach spaces. In the present paper we answer this question in the affirmative.…”