Denjoy-Wolff theory and spectral properties of weighted composition operators on \(\text{Hol}(\mathbb{D})\). (English) Zbl 07632818
The paper studies weighted composition operators induced by an analytic multiplier and an analytic self-map of the unit disk D. The analytic self-map is not supposed to be an elliptic automorphism and the operators act on the full space H(D) of all analytic functions on D. Various results about the spectrum and point spectrum of these weighted composition operators are obtained.
Reviewer: Kehe Zhu (Albany)
MSC:
| 47B33 | Linear composition operators |
| 47A10 | Spectrum, resolvent |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
References:
| [1] | J. Agler and J. Mac Carthy, Pick Interpolation and Hilbert Function Spaces, Grad. Stud. Math. 44, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1010.47001 · doi:10.1090/gsm/044 |
| [2] | J. Aaronson, Ergodic theory for inner functions of the upper half plane, Ann Inst. H. Poincaré Sect. B (N.S.) 14 (1978), no. 3, 33-253. · Zbl 0378.28009 |
| [3] | W. Arendt, E. Bernard, B. Célariès, and I. Chalendar, Spectral properties of weighted composition operators induced by a rotation on \[ \text{Hol}(\mathbb{D})\], Indiana Univ. Math. J., accepted, arXiv:2108.08270 [math.FA]. · Zbl 07632818 · doi:10.48550/arXiv.2108.08270 |
| [4] | W. Arendt, B. Célariès, and I. Chalendar, In Koenigs’ footsteps: Diagonalization of composition operators, J. Funct. Anal. 278 (2020), no. 2, 108313, 24. · Zbl 1436.30049 · doi:10.1016/j.jfa.2019.108313 |
| [5] | I. N. Baker and Ch. Pommerenke, On the iteration of analytic functions in a half-plane II, J. London Math. Soc. 20 (1979), no. 2, 255-258, 1979. · Zbl 0419.30025 · doi:10.1112/jlms/s2-20.2.255 |
| [6] | F. Bracci, M. D. Contreras, and S. Díaz-Madrigal, Continuous Semigroups of Holomorphic Self-Maps of the Unit Disc, Springer Monogr. Math., Springer, Cham, 2020. · Zbl 1441.30001 · doi:10.1007/978-3-030-36782-4 |
| [7] | J. Bonet, P. Galindo, and M. Lindström, Spectra and essential spectral radii of composition operators on weighted Banach spaces of analytic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 884-891. · Zbl 1138.47017 · doi:10.1016/j.jmaa.2007.09.006 |
| [8] | P. S. Bourdon, Spectra of some composition operators and associated weighted composition, J. Operator Theory 67 (2012), no. 2, 537-560. · Zbl 1262.47036 |
| [9] | P. S. Bourdon, V. Matache, and J. H. Shapiro, On convergence to the Denjoy-Wolff point, Illinois J. Math. 49 (2005), no. 2, 405-430. · Zbl 1088.30015 · doi:10.1215/ijm/1258138025 |
| [10] | F. Bracci and P. Poggi-Corradini, “On Valiron’s theorem” in Future Trends in Geometric Function Theory, Rep. Univ. Jyväskylä, Dep. Math. Stat. 92, Univ. Jyväskylä, Jyväskylä, Finland, 2003, 39-55. · Zbl 1043.30010 |
| [11] | I. Chalendar, E. A. Gallardo-Gutiérrez, and J. R. Partington, Weighted composition operators on the Dirichlet space: Boundedness and spectral properties, Math. Ann. 363 (2015), nos. 3-4, 1265-1279. · Zbl 1390.47004 · doi:10.1007/s00208-015-1195-y |
| [12] | M. D. Contreras, S. Díaz-Madrigal, and C. Pommerenke, Boundary behavior of the iterates of a self-map of the unit disk, J. Math. Anal. Appl. 396 (2012), no. 1. 93-97. · Zbl 1263.30010 · doi:10.1016/j.jmaa.2012.05.062 |
| [13] | C. C. Cowen, Composition operators on \[{H^2} \], J. Operator Theory 9 (1983), no. 1, 77-106. · Zbl 0504.47032 |
| [14] | C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math. CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017 |
| [15] | P. Galindo, M. Lindström, and N. Wikman, Spectra of weighted compostion operators on analytic function spaces, Mediterr. J. Math. 17 (2020), no. 1, Paper No. 34. · Zbl 1498.47064 · doi:10.1007/s00009-019-1465-0 |
| [16] | O. Hyvärinen, M. Lindström, I. Nieminen, and E. Saukko, Spectra of weighted composition operators with automorphisms symbols, J. Funct. Anal. 265 (2013), no. 8, 1749-1777. · Zbl 1325.47054 · doi:10.1016/j.jfa.2013.06.003 |
| [17] | H. Kamowitz, The spectra of a class of operators on the disc algebra, Indiana Univ. Math. J. 27 (1978), no. 4, 581-610. · Zbl 0385.47016 · doi:10.1512/iumj.1978.27.27039 |
| [18] | J. L. Kelley and I. Namioka, Linear Topological Spaces, Grad. Texts Math. 36, Springer, Cham, 1963. · Zbl 0115.09902 |
| [19] | G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 1 (1884), 3-41. · JFM 16.0376.01 |
| [20] | C. Pommerenke, On the iteration of analytic functions in a half-plane I, J. London Math. Soc. (2) 19 (1979), no. 3, 439-447. · doi:10.1112/jlms/s2-19.3.439 |
| [21] | H. Queffélec and M. Queffélec, Analyse complexe et applications, Calvage & Mounet, Paris, 2017. |
| [22] | W. Rudin, Real and complex analysis, 3rd Ed., McGraw-Hill, New York, 1987. · Zbl 0925.00005 |
| [23] | J. H. Shapiro, Composition operators and classical function theory, Springer, New York, 1993. · Zbl 0791.30033 |
| [24] | G. Valiron, Sur l’itération des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. 55 (1931), no. 2, 105-128 · JFM 57.0381.03 |
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