Adjoints of composition operators with irrational symbol
HTML articles powered by AMS MathViewer
- by Caixing Gu, Erin Rizzie and Jonathan Shapiro PDF
- Proc. Amer. Math. Soc. 148 (2020), 145-155 Request permission
Abstract:
In this paper we derive formulas for the adjoints of a class of composition operators with irrational symbol, in particular, the $n$-th root functions. We discuss these formulas on both the Hardy space and the Bergman space.References
- Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
- Estelle L. Basor and Dylan Q. Retsek, Extremal non-compactness of composition operators with linear fractional symbol, J. Math. Anal. Appl. 322 (2006), no. 2, 749–763. MR 2250613, DOI 10.1016/j.jmaa.2005.09.018
- Paul S. Bourdon and Barbara D. MacCluer, Selfcommutators of automorphic composition operators, Complex Var. Elliptic Equ. 52 (2007), no. 1, 85–104. MR 2275399, DOI 10.1080/17476930600797440
- Paul S. Bourdon, David Levi, Sivaram K. Narayan, and Joel H. Shapiro, Which linear-fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), no. 1, 30–53. MR 1972190, DOI 10.1016/S0022-247X(03)00005-2
- Paul S. Bourdon and Joel H. Shapiro, Adjoints of rationally induced composition operators, J. Funct. Anal. 255 (2008), no. 8, 1995–2012. MR 2462584, DOI 10.1016/j.jfa.2008.07.002
- Carl C. Cowen, Linear fractional composition operators on $H^2$, Integral Equations Operator Theory 11 (1988), no. 2, 151–160. MR 928479, DOI 10.1007/BF01272115
- Carl C. Cowen and Eva A. Gallardo-Gutiérrez, A new class of operators and a description of adjoints of composition operators, J. Funct. Anal. 238 (2006), no. 2, 447–462. MR 2253727, DOI 10.1016/j.jfa.2006.04.031
- 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
- Carl C. Cowen and Barbara D. MacCluer, Linear fractional maps of the ball and their composition operators, Acta Sci. Math. (Szeged) 66 (2000), no. 1-2, 351–376. MR 1768872
- Sean Effinger-Dean, Alan Johnson, Joseph Reed, and Jonathan Shapiro, Norms of composition operators with rational symbol, J. Math. Anal. Appl. 324 (2006), no. 2, 1062–1072. MR 2265101, DOI 10.1016/j.jmaa.2006.01.005
- Sam Elliott, Adjoints of composition operators on Hardy spaces of the half-plane, J. Funct. Anal. 256 (2009), no. 12, 4162–4186. MR 2521923, DOI 10.1016/j.jfa.2008.11.025
- Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodríguez, Adjoints of linear fractional composition operators on the Dirichlet space, Math. Ann. 327 (2003), no. 1, 117–134. MR 2005124, DOI 10.1007/s00208-003-0442-9
- Caixing Gu, Functional calculus for $m$-isometries and related operators on Hilbert spaces and Banach spaces, Acta Sci. Math. (Szeged) 81 (2015), no. 3-4, 605–641. MR 3443775, DOI 10.14232/actasm-014-550-3
- Caixing Gu and Zhengli Chen, A model for $(n,p)$-hypercontractions on Banach space, Indag. Math. (N.S.) 26 (2015), no. 3, 485–494. MR 3341810, DOI 10.1016/j.indag.2015.02.003
- Christopher Hammond, Jennifer Moorhouse, and Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008), no. 1, 626–639. MR 2394110, DOI 10.1016/j.jmaa.2007.10.039
- Katherine Heller, Adjoints of linear fractional composition operators on $S^2(\Bbb {D})$, J. Math. Anal. Appl. 394 (2012), no. 2, 724–737. MR 2927493, DOI 10.1016/j.jmaa.2012.05.006
- Paul R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68 (1997), no. 6, 503–513. MR 1444662, DOI 10.1007/s000130050083
- Thomas L. Kriete, Barbara D. MacCluer, and Jennifer L. Moorhouse, Toeplitz-composition $C^*$-algebras, J. Operator Theory 58 (2007), no. 1, 135–156. MR 2336048
- María J. Martín and Dragan Vukotić, Adjoints of composition operators on Hilbert spaces of analytic functions, J. Funct. Anal. 238 (2006), no. 1, 298–312. MR 2253017, DOI 10.1016/j.jfa.2006.04.024
- John N. McDonald, Adjoints of a class of composition operators, Proc. Amer. Math. Soc. 131 (2003), no. 2, 601–606. MR 1933352, DOI 10.1090/S0002-9939-02-06590-5
- Erin E. M. Rizzie, Adjoints of Composition Operators with a Broader Class of Symbols, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–Purdue University. MR 3450312
- 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
Additional Information
- Caixing Gu
- Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
- MR Author ID: 236909
- ORCID: 0000-0001-6289-7755
- Email: cgu@calpoly.edu
- Erin Rizzie
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
- Email: erin.rizzie@uconn.edu
- Jonathan Shapiro
- Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
- MR Author ID: 622125
- Email: jshapiro@calpoly.edu
- Received by editor(s): May 19, 2016
- Received by editor(s) in revised form: June 8, 2016
- Published electronically: October 3, 2019
- Additional Notes: The authors would like to thank the referee for providing many useful suggestions.
- Communicated by: Stephan Ramon Garcia
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 145-155
- MSC (2010): Primary 47B33, 47A05
- DOI: https://doi.org/10.1090/proc/13340
- MathSciNet review: 4042838