Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
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2011-09-30Resumen
Let (X,d) be a pointed compact metric space, let 0 < α < 1, and let ϕ : X → X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator C_ϕ induced by the symbol ϕ on the spaces (lip_0(X),d^α) and (Lip_0(X),d^α) is given by the formula ||C_ϕ||_e=lim_{t→0}sup_{0<d(x,y)<t}d(ϕ(x),ϕ(y))^α/d(x y)^α whenever the dual space (lip_0(X), d^α)* has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space.