Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Abstract

Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

Norm-attaining Composition Operators on Lipschitz Spaces

“…We prepare its proof with two lemmas whose proofs use the same methods applied in the proofs of Lemma 3.2 and Theorem 3.1 in [11] for the special case of spaces lip 0 (X α ) with 0 < α < 1.…”

“…We will follow the proof of [11,Theorem 3.1] to prove the next lemma, but we include it because that adaptation is not immediate. Lemma 4.7.…”

“…Proof. According to the proof of [11,Theorem 3.1], we first note that and obtain two sequences {x n } and {y n } in X satisfying that…”