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Essential spectral radius of quasicompact endomorphisms of Lipschitz algebras. (English) Zbl 1354.47004
Let \((X, d)\) be a compact metric space with infinitely many points and \(0 < \alpha \leq 1\). For a function \(f:X\to{\mathbb C}\), let \(p_{\alpha}=\sup\{ \frac{|f(x)-f(y)|}{d(x,y)^{\alpha}}; \; x, t\in X,\;x\neq y\}\). The Lipschitz algebras of order \(\alpha\) are Banach algebras given by \[ \mathrm{Lip}(A,\alpha)=\{ f:X\to {\mathbb C};\; \sup_{x\in X}|f(x)|+p_{\alpha}(f)<\infty\} \] and \[ \mathrm{lip}(X,\alpha)=\{ f\in \mathrm{Lip}(X,\alpha);\; \lim_{d(x,y)\to 0}\frac{|f(x)-f(y)|}{d(x,y)^{\alpha}}=0\}. \] For an endomorphism \(T\) of a Lipschitz algebra \(A\), let \(\| T\|_e\) denote the essential norm, i.e., the norm of \(T+K(A)\) in the Calkin algebra \(B(A)/K(A)\), where \(B(A)\) denotes the Banach algebra of all bounded operators on \(A\) and \(K(A)\) is the ideal of all compact operators on \(A\). Then the essential spectral radius of \(T\) is given by \(r_e(T)=\lim_{n\to \infty}\| T^n\|_{e}^{1/n}.\) If \(r_e(T)<1\), then \(T\) is said to be quasicompact, and if \(r_e(T)=0\), then it is a Riesz operator.
If \(\varphi: X \to X\) is a mapping satisfying some suitable condition, then \( C_\varphi: f \mapsto f\circ \varphi\) defines an endomorphism on a Lipschitz algebra \(A\). In this paper, quasicompact and Riesz operators \(C_\varphi\) on \(A\) are studied. A formula for the essential spectral radius is given in terms of \(\varphi\) and a necessary and sufficient condition for such an operator to be Riesz is given. In the last part, the spectrum and the point spectrum of quasicompact operators \(C_\varphi\) are considered.

47A10 Spectrum, resolvent
46J10 Banach algebras of continuous functions, function algebras
47B48 Linear operators on Banach algebras
47B38 Linear operators on function spaces (general)
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