Essential spectral radius of quasicompact endomorphisms of Lipschitz algebras

Abstract

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]).

Quasicompact composition operators and power-contractive selfmaps

“…But, note that by releasing this condition on X we get the next theorem only for the algebra Lip(X, d), not for its subalgebras A. (See also [4,Corollary 2.4], where the result of the next theorem is obtained using a different approach based on the essential spectral radius estimates.) It is worth mentioning that Section 3 is based on applying the result of Theorem 2.3, which is valid not only for the algebra Lip(X) but also for certain subalgebras A of Lip(X).…”