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We establish a formula for the essential spectral radius of an endomorphism T of Lipschitz algebras under a condition which is equivalent to the quasicompactness of the endomorphism T . We also conclude a necessary and sufficient condition for an endomorphism of these algebras to be Riesz. Finally, we get a relation for the spectrum and the set of eigenvalues of a quasicompact and Riesz endomorphism of these algebras.1. Introduction. Let (X, d) be a compact metric space with infinitely many points and 0 < α ≤ 1. The Lipschitz algebra of order α, Lip(X, α), is the algebra of all complex-valued functions f on X for whichThe subalgebra of those functions f withis denoted by lip(X, α). These Lipschitz algebras were first studied by Sherbert [12, 13]. The algebras Lip(X, α) for 0 < α ≤ 1 and lip(X, α) for 0 < α < 1 are natural Banach function algebras on X under the norm ∥f ∥ α = ∥f ∥ X +p α (f ), where ∥f ∥ X = sup x∈X |f (x)|. Recall that a function algebra A on a compact Hausdorff space X is called natural if every nonzero complex homomorphism on A is an evaluation homomorphism at some point of X [3, Definition 4.1.3]. We note that Lip(X, 1) ⊆ lip(X, α) ⊆ Lip(X, α) (see [1, 7]).

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