Galindo, P.; Gamelin, T. W.; Lindström, M. Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets. (English) Zbl 1118.47014 Trans. Am. Math. Soc. 359, No. 5, 2109-2121 (2007). Let \(M_A\) be the spectrum of a uniform algebra \(A\) and let \(\Phi\) be a self-map of \(M_A\) that induces a composition operator \(C_\Phi:A\to A\). The authors relate their earlier notion of hyperbolic boundedness [J. Korean Math.Soc.41, No.1, 1–20 (2004; Zbl 1049.46031)] to properties of the essential spectrum of \(C_\Phi\). For example, they show that the spectral radius of \(C_\Phi\) is \(<1\) iff the image of \(M_A\) under some iterate of \(\Phi\) is hyperbolically bounded. They also partition the set of all composition operators into so-called hyperbolic vicinities each of which is clopen with respect to the essential norm. They show that this partition is related to an analogous partition with respect to the uniform operator norm. Reviewer: Hans Jarchow (Zürich) Cited in 2 ReviewsCited in 5 Documents MSC: 47B33 Linear composition operators 47B48 Linear operators on Banach algebras 46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces 46J10 Banach algebras of continuous functions, function algebras Keywords:composition operator; hyperbolical boundedness; Gleason part; essential norm; uniform algebra Citations:Zbl 1049.46031 PDFBibTeX XMLCite \textit{P. Galindo} et al., Trans. Am. Math. Soc. 359, No. 5, 2109--2121 (2007; Zbl 1118.47014) Full Text: DOI References: [1] Richard Aron, Pablo Galindo, and Mikael Lindström, Connected components in the space of composition operators in \?^{\infty } functions of many variables, Integral Equations Operator Theory 45 (2003), no. 1, 1 – 14. · Zbl 1029.46053 · doi:10.1007/BF02789591 [2] C.-H. Chu, R. V. Hügli, and M. Mackey, The identity is isolated among composition operators, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3305 – 3308. · Zbl 1066.47025 [3] A. M. Davie, Linear extension operators for spaces and algebras of functions, Amer. J. Math. 94 (1972), 156 – 172. · Zbl 0242.46041 · doi:10.2307/2373598 [4] P. Galindo, T. W. Gamelin, and M. Lindström, Composition operators on uniform algebras and the pseudohyperbolic metric, J. Korean Math. Soc. 41 (2004), no. 1, 1 – 20. Satellite Conference on Infinite Dimensional Function Theory. · Zbl 1049.46031 · doi:10.4134/JKMS.2004.41.1.001 [5] Pablo Galindo and Mikael Lindström, Factorization of homomorphisms through \?^{\infty }(\?), J. Math. Anal. Appl. 280 (2003), no. 2, 375 – 386. · Zbl 1029.46070 · doi:10.1016/S0022-247X(03)00065-9 [6] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. · Zbl 0213.40401 [7] T. W. Gamelin, Embedding Riemann surfaces in maximal ideal spaces, J. Functional Analysis 2 (1968), 123 – 146. · Zbl 0172.18102 [8] T. W. Gamelin, Uniform algebras on plane sets, Approximation theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973) Academic Press, New York, 1973, pp. 101 – 149. · Zbl 0325.30035 [9] T. W. Gamelin, Homomorphisms of uniform algebras, Recent progress in functional analysis (Valencia, 2000) North-Holland Math. Stud., vol. 189, North-Holland, Amsterdam, 2001, pp. 95 – 105. · Zbl 1032.46520 · doi:10.1016/S0304-0208(01)80038-4 [10] Pamela Gorkin and Raymond Mortini, Norms and essential norms of linear combinations of endomorphisms, Trans. Amer. Math. Soc. 358 (2006), no. 2, 553 – 571. · Zbl 1081.47029 [11] Takuya Hosokawa, Keiji Izuchi, and Dechao Zheng, Isolated points and essential components of composition operators on \?^{\infty }, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1765 – 1773. · Zbl 1008.47031 [12] Herbert Kamowitz, Compact endomorphisms of Banach algebras, Pacific J. Math. 89 (1980), no. 2, 313 – 325. · Zbl 0465.46047 [13] Udo Klein, Kompakte multiplikative Operatoren auf uniformen Algebren, Mitt. Math. Sem. Giessen 232 (1997), iv+120 (German). · Zbl 0933.47017 [14] Heinz König, Zur abstrakten Theorie der analytischen Funktionen. II, Math. Ann. 163 (1966), 9 – 17 (German). · Zbl 0135.35502 · doi:10.1007/BF02052482 [15] Lixin Zheng, The essential norms and spectra of composition operators on \?^{\infty }, Pacific J. Math. 203 (2002), no. 2, 503 – 510. · Zbl 1053.47022 · doi:10.2140/pjm.2002.203.503 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.