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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.

MSC:
 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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References:
 [1] W.G. Bade, P.C. Curtis, Jr., and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras , Proc. Lond. Math. Soc. 55 (1987), 359-377. · Zbl 0634.46042 [2] F. Behrouzi, Riesz and quasi-compact endomorphisms of Lipschitz algebras , Houst. J. Math. 36 (2010), 793-802. · Zbl 1218.47056 [3] H.G. Dales, Banach algebras and automatic continuity , Lond. Math.Soc. Mono. 24 , The Clarendon Press, Oxford, 2000. · Zbl 0981.46043 [4] J.F. Feinstein and H. Kamowitz, Quasicompact and Riesz endomorphisms of Banach algebras , J. Funct. Anal. 225 (2005), 427-438. · Zbl 1089.46029 [5] —-, Quasicompact endomorphisms of commutative semiprime Banach algebras , Banach Center Publ. 91 (2010), 159-167. · Zbl 1244.47030 [6] H.G. Heuser, Functional analysis , John Wiley and Sons, New York, 1982. · Zbl 0465.47001 [7] T.G. Honary and H. Mahyar, Approximation in Lipschitz algebras , Quaest. Math. 23 (2000), 13-19. · Zbl 0963.46034 [8] H. Kamowitz and S. Scheinberg, Some properties of endomorphisms of Lipschitz algebras , Stud. Math. 24 (1990), 383-391. · Zbl 0713.47030 [9] H. Mahyar, Quasicompact and Riesz endomorphisms of infinitely differentiable Lipschitz algebras , Southeast Asian Bull. Math. 35 (2011), 249-259. · Zbl 1240.46073 [10] H. Mahyar and A. H. Sanatpour, Quasicompact endomorphisms of Lipschitz algebras of analytic functions , Publ. Math. Debr, 76 (2010), 135-145. · Zbl 1274.46105 [11] —-, Compact and quasicompact homomorphisms between differentiable Lipschitz algebras , Bull. Belg. Math. Soc. Simon-Stevin 17 (2010), 485-497. · Zbl 1213.47042 [12] D.R. Sherbert, Banach algebras of Lipschitz functions , Pac. J. Math. 13 (1963), 1387-1399. · Zbl 0121.10203 [13] —-, The structure of ideals and point derivations in Banach algebras of Lipschitz functions , Trans. Amer. Math. Soc. 111 (1964), 240-272. · Zbl 0121.10204 [14] A.J. Vargas, M. Lacruz and M.V. Vallecillos, Essential norm of composition operators on Banach spaces of Hölder functions , Hindawi Publishing Corporation, Abstract and Applied Analysis (2011), Article ID 590853, 13 pages. · Zbl 1230.47049 [15] N. Weaver, Lipschitz algebras , World Scientific, Singapore, 1999. · Zbl 0936.46002
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