Difference of composition operators over the half-plane
HTML articles powered by AMS MathViewer
- by Boo Rim Choe, Hyungwoon Koo and Wayne Smith PDF
- Trans. Amer. Math. Soc. 369 (2017), 3173-3205 Request permission
Abstract:
We study the differences of composition operators acting on weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is Hilbert-Schmidt.References
- Earl Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981), no. 2, 230–232. MR 593463, DOI 10.1090/S0002-9939-1981-0593463-0
- Boo Rim Choe, Takuya Hosokawa, and Hyungwoon Koo, Hilbert-Schmidt differences of composition operators on the Bergman space, Math. Z. 269 (2011), no. 3-4, 751–775. MR 2860263, DOI 10.1007/s00209-010-0757-7
- Boo Rim Choe, Hyungwoon Koo, and Inyoung Park, Compact differences of composition operators over polydisks, Integral Equations Operator Theory 73 (2012), no. 1, 57–91. MR 2913660, DOI 10.1007/s00020-012-1962-z
- Boo Rim Choe, Hyungwoon Koo, and Inyoung Park, Compact differences of composition operators on the Bergman spaces over the ball, Potential Anal. 40 (2014), no. 1, 81–102. MR 3146510, DOI 10.1007/s11118-013-9343-z
- Boo Rim Choe, Hyungwoon Koo, and Heungsu Yi, Positive Toeplitz operators between the harmonic Bergman spaces, Potential Anal. 17 (2002), no. 4, 307–335. MR 1918239, DOI 10.1023/A:1016356229211
- Boo Rim Choe and Heungsu Yi, Representations and interpolations of harmonic Bergman functions on half-spaces, Nagoya Math. J. 151 (1998), 51–89. MR 1650336, DOI 10.1017/S0027763000025174
- Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
- C. Cowen and B. D. MacCluer, Composition operators on Hardy spaces of a half-plane, Bull. London. Math. Soc. 44 (2012), 489–495.
- Sam J. Elliott and Andrew Wynn, Composition operators on weighted Bergman spaces of a half-plane, Proc. Edinb. Math. Soc. (2) 54 (2011), no. 2, 373–379. MR 2794660, DOI 10.1017/S0013091509001412
- Eva A. Gallardo-Gutiérrez, María J. González, Pekka J. Nieminen, and Eero Saksman, On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008), no. 3, 986–1001. MR 2442059, DOI 10.1016/j.aim.2008.06.005
- Paul R. Halmos, Naive set theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1974. Reprint of the 1960 edition. MR 0453532
- Katherine Heller, Barbara D. MacCluer, and Rachel J. Weir, Compact differences of composition operators in several variables, Integral Equations Operator Theory 69 (2011), no. 2, 247–268. MR 2765588, DOI 10.1007/s00020-010-1840-5
- Herbert Hunziker, Hans Jarchow, and Vania Mascioni, Some topologies on the space of analytic self-maps of the unit disk, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 133–148. MR 1110192
- Liangying Jiang and Caiheng Ouyang, Compact differences of composition operators on holomorphic function spaces in the unit ball, Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 5, 1679–1693. MR 2884942, DOI 10.1016/S0252-9602(11)60353-6
- Thomas Kriete and Jennifer Moorhouse, Linear relations in the Calkin algebra for composition operators, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2915–2944. MR 2286063, DOI 10.1090/S0002-9947-07-04166-9
- Hyungwoon Koo and Maofa Wang, Joint Carleson measure and the difference of composition operators on $A_\alpha ^p(\mathbf {B}_n)$, J. Math. Anal. Appl. 419 (2014), no. 2, 1119–1142. MR 3225424, DOI 10.1016/j.jmaa.2014.05.037
- Jennifer Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), no. 1, 70–92. MR 2108359, DOI 10.1016/j.jfa.2004.01.012
- Pekka J. Nieminen and Eero Saksman, On compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004), no. 2, 501–522. MR 2086972, DOI 10.1016/j.jmaa.2004.05.024
- Erno Saukko, Difference of composition operators between standard weighted Bergman spaces, J. Math. Anal. Appl. 381 (2011), no. 2, 789–798. MR 2802114, DOI 10.1016/j.jmaa.2011.03.058
- Erno Saukko, An application of atomic decomposition in Bergman spaces to the study of differences of composition operators, J. Funct. Anal. 262 (2012), no. 9, 3872–3890. MR 2899981, DOI 10.1016/j.jfa.2012.02.003
- Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
- Joel H. Shapiro and Wayne Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal. 205 (2003), no. 1, 62–89. MR 2020208, DOI 10.1016/S0022-1236(03)00215-5
- Joel H. Shapiro and Carl Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no. 1, 117–152. MR 1066401
- Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954
- Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138
Additional Information
- Boo Rim Choe
- Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
- MR Author ID: 251281
- Email: cbr@korea.ac.kr
- Hyungwoon Koo
- Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
- MR Author ID: 606733
- Email: koohw@korea.ac.kr
- Wayne Smith
- Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
- MR Author ID: 213832
- Email: wayne@math.hawaii.edu
- Received by editor(s): April 14, 2014
- Received by editor(s) in revised form: April 27, 2015
- Published electronically: September 1, 2016
- Additional Notes: The first author was supported by NRF(2015R1D1A1A01057685) of Korea
The second author was supported by NRF(2014R1A1A2054145) of Korea - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3173-3205
- MSC (2010): Primary 47B33; Secondary 30H20
- DOI: https://doi.org/10.1090/tran/6742
- MathSciNet review: 3605968