# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  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  F. Behrouzi, Riesz and quasi-compact endomorphisms of Lipschitz algebras , Houst. J. Math. 36 (2010), 793-802. · Zbl 1218.47056  H.G. Dales, Banach algebras and automatic continuity , Lond. Math.Soc. Mono. 24 , The Clarendon Press, Oxford, 2000. · Zbl 0981.46043  J.F. Feinstein and H. Kamowitz, Quasicompact and Riesz endomorphisms of Banach algebras , J. Funct. Anal. 225 (2005), 427-438. · Zbl 1089.46029  —-, Quasicompact endomorphisms of commutative semiprime Banach algebras , Banach Center Publ. 91 (2010), 159-167. · Zbl 1244.47030  H.G. Heuser, Functional analysis , John Wiley and Sons, New York, 1982. · Zbl 0465.47001  T.G. Honary and H. Mahyar, Approximation in Lipschitz algebras , Quaest. Math. 23 (2000), 13-19. · Zbl 0963.46034  H. Kamowitz and S. Scheinberg, Some properties of endomorphisms of Lipschitz algebras , Stud. Math. 24 (1990), 383-391. · Zbl 0713.47030  H. Mahyar, Quasicompact and Riesz endomorphisms of infinitely differentiable Lipschitz algebras , Southeast Asian Bull. Math. 35 (2011), 249-259. · Zbl 1240.46073  H. Mahyar and A. H. Sanatpour, Quasicompact endomorphisms of Lipschitz algebras of analytic functions , Publ. Math. Debr, 76 (2010), 135-145. · Zbl 1274.46105  —-, Compact and quasicompact homomorphisms between differentiable Lipschitz algebras , Bull. Belg. Math. Soc. Simon-Stevin 17 (2010), 485-497. · Zbl 1213.47042  D.R. Sherbert, Banach algebras of Lipschitz functions , Pac. J. Math. 13 (1963), 1387-1399. · Zbl 0121.10203  —-, The structure of ideals and point derivations in Banach algebras of Lipschitz functions , Trans. Amer. Math. Soc. 111 (1964), 240-272. · Zbl 0121.10204  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  N. Weaver, Lipschitz algebras , World Scientific, Singapore, 1999. · Zbl 0936.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.