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Spectra of composition operators on the Bloch and Bergman spaces. (English) Zbl 1024.47009

Let \(U\) be the unit disc in the complex plane and \(\varphi\) be an analytic map from \(U\) to itself. In [J. Am. Math. Soc. 10, 299-325 (1997; Zbl 0870.30018)], P. S. Bourdon and J. H. Shapiro give the essential spectral radius formula on the Hardy space \(H^p\), for any analytic \(\varphi\) and \(0<p<\infty\). Using the Calderón theory of complex interpolation, the authors obtain an analogous result on the Bergman space \(A^p\), namely that for \(1<p<\infty\), \[ \bigl( r_{e,A^p}(C_\varphi) \bigr)^p = \bigl( r_{e,A^2}(C_\varphi) \bigr)^2. \] They also obtain the spectrum of the composition operator \(C_\varphi\) on the Bergman space, Bloch space, BMOA and Hardy space for \(\varphi\) univalent, not an automorphism, with fixed point in \(U\).
Reviewer: Jinkee Lee (Pusan)

MSC:

47B33 Linear composition operators
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A36 Bergman spaces of functions in several complex variables

Citations:

Zbl 0870.30018
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References:

[1] J. Arazy, S. Fisher and J. Peetre,Möbius invariant function spaces, Journal für die reine und angewandte Mathematik363 (1985), 110–145. · Zbl 0566.30042
[2] S. Axler,Bergman spaces and their operators, inSurveys of Some Recent Results in Operator Theory Vol. 1 (J. B. Conway and B. B Morrel, eds.), Pitman Research Notes in Mathematics171 (1988), 1–50.
[3] J. Bergh and J. Löfström,Interpolation Spaces, Springer-Verlag, Berlin, 1976.
[4] P. Bourdon,Essential angular derivatives and maximum growth of Koenigs eigen-functions, Journal of Functional Analysis160 (1998), 561–580. · Zbl 0926.30015
[5] P. Bourdon, J. Cima and A. Matheson,Compact composition operators on BMOA, Transactions of the American Mathematical Society351 (1999), 2183–2196. · Zbl 0920.47029
[6] P. Bourdon and J. Shapiro,Mean growth of Koenigs eigenfunctions, Journal of the American Mathematical Society10 (1997), 299–325. · Zbl 0870.30018
[7] A. Calderón,Intermediate spaces and interpolation, the complex method, Studia Mathematica24 (1964), 113–190. · Zbl 0204.13703
[8] C. Cowen and B. MacCluer,Spectra of some composition operators, Journal of Functional Analysis125 (1994), 223–251. · Zbl 0814.47040
[9] C. Cowen and B. MacCluer,Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. · Zbl 0873.47017
[10] M. Cwikel, M. Milman and Y. Sagher,Complex interpolation of some quasi-Banach spaces, Journal of Functional Analysis65 (1986), 339–347. · Zbl 0586.46054
[11] R. Donaway,Norm and essential norm estimates of composition operators on Besov type spaces, Ph.D. Dissertation, University of Virginia, 1999.
[12] H. Kamowitz,The spectra of endomorphisms of the disc algebra, Pacific Journal of Mathematics46 (1973), 433–440. · Zbl 0261.46058
[13] K. Madigan and A. Matheson,Compact composition operators on the Bloch space, Transactions of the American Mathematical Society347 (1995), 2679–2687. · Zbl 0826.47023
[14] A. Montes-Rodríguez,The essential norm of a composition operator on the Bloch spaces, Pacific Journal of Mathematics188 (1999), 339–351. · Zbl 0932.30034
[15] P. Poggi-Corradini,The Hardy class of Koenigs maps, The Michigan Mathematical Journal44 (1997), 495–507. · Zbl 0897.30014
[16] P. Poggi-Corradini,The essential norm of composition operators revisited, inStudies on Composition Operators (Laramie, WY, 1996), Contemporary Mathematics, Vol. 213, American Mathematical Society, Providence, RI, 1998, pp. 167–173. · Zbl 0936.47010
[17] W. Rudin,Function Theory in the Unit Ball of \(\mathbb{C}\) n,Springer-Verlag, Berlin, 1980. · Zbl 0495.32001
[18] K. Saxe,Compactness-like properties preserved by complex interpolation, Arkiv för Matematik35 (1997), 353–362. · Zbl 0927.46054
[19] J. Shapiro,The essential norm of a composition operator, Annals of Mathematics125 (1987), 375–404. · Zbl 0642.47027
[20] W. Smith,Composition operators between Bergman and Hardy spaces, Transactions of the American Mathematical Society348 (1996), 2331–2348. · Zbl 0857.47020
[21] M. Tjani,Compact composition operators on some Möbius invariant Banach spaces, Ph.D. Dissertation, Michigan State University, 1996. · Zbl 0963.47023
[22] D. Vukotić, A sharp estimate forA a p functions in \(\mathbb{C}\)n, Proceedings of the American Mathematical Society117 (1993), 753–756. · Zbl 0773.32004
[23] T. Wolff,A note on interpolation spaces, inHarmonic Analysis (Minneapolis, MN, 1981), Lecture Notes in Mathematics908, Springer-Verlag, Berlin, 1982, pp. 199–204.
[24] L. Zheng, Essential norm and spectrum of composition operators onH preprint.
[25] K. Zhu,Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. · Zbl 0706.47019
[26] K. Zhu and T. MacGregor,Coefficient multipliers between Bergman and Hardy spaces, Mathematika42 (1995), 413–426. · Zbl 0844.30025
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