Skip to main content


  • SHI-AN HAN (a1) and ZE-HUA ZHOU (a2)

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

Corresponding author
Hide All

The work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771323; 11371276).

Hide All
[1] Aron, R. and Lindström, M., ‘Spectra of weighted composition operators on weighted Banach spaces of analytic functions’, Israel J. Math. 141(1) (2004), 263276.
[2] Bonet, J., Galindo, P. and Lindström, M., ‘Spectra and essential spectral radii of composition operators on weighted Banach spaces of analytic functions’, J. Math. Anal. Appl. 340(2) (2008), 884891.
[3] Bourdon, P. and Shapiro, J. H., Cyclic Phenomena for Composition Operators, Memoirs of the American Mathematical Society, 596 (American Mathematical Society, Providence, RI, 1997).
[4] Contreras, M. D. and Hernandez-Diaz, A. G., ‘Weighted composition operators in weighted Banach spaces of analytic functions’, J. Aust. Math. Soc. 69(1) (2000), 4160.
[5] Cowen, C. C., ‘Composition operators on H 2 ’, J. Operator Theory 9 (1983), 77106.
[6] Cowen, C. C., ‘Linear fractional composition operators on H 2 ’, Integral Equations Operator Theory 11(2) (1988), 151160.
[7] Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).
[8] Donaway, R., ‘Norm and essential norm estimates of composition operators on Besov type spaces’, PhD Thesis, The University of Virginia, 1999.
[9] Elliott, S. and Jury, M. T., ‘Composition operators on Hardy spaces of a half-plane’, Bull. Lond. Math. Soc. 44(3) (2012), 489495.
[10] Elliott, S. and Wynn, A., ‘Composition operators on weighted Bergman spaces of a half-plane’, Proc. Edinb. Math. Soc. 54(2) (2011), 373379.
[11] Gallardo-Gutiérrez, E. A. and Montes-Rodríguez, A., ‘Adjoints of linear fractional composition operators on the Dirichlet space’, Math. Ann. 327(1) (2003), 117134.
[12] Gallardo-Gutiérrez, E. A. and Schroderus, R., ‘The spectra of linear fractional composition operators on weighted Dirichlet spaces’, J. Funct. Anal. 271(3) (2016), 720745.
[13] Gunatillake, G., ‘Invertible weighted composition operators’, J. Funct. Anal. 261(3) (2011), 831860.
[14] Higdon, W. M., ‘The spectra of composition operators from linear fractional maps acting upon the Dirichlet space’, J. Funct. Anal. 220(1) (2005), 5575.
[15] Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups (American Mathematical Society, Providence, RI, 1996).
[16] Hyvärinen, O., Lindström, M., Nieminen, I. and Saukko, E., ‘Spectra of weighted composition operators with automorphic symbols’, J. Funct. Anal. 265(8) (2013), 17491777.
[17] Matache, V., ‘Composition operators on Hardy spaces of a half-plane’, Proc. Amer. Math. Soc. 127(5) (1999), 14831491.
[18] Matache, V., ‘Weighted composition operators on H 2 and applications’, Complex Anal. Oper. Theory 2(1) (2008), 169197.
[19] Matache, V., ‘Invertible and normal composition operators on the Hilbert Hardy space of a half-plane’, Concr. Oper. 3(1) (2016), 7784.
[20] Schroderus, R., ‘Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane’, J. Math. Anal. Appl. 447(2) (2017), 817833.
[21] Shapiro, J. H., Composition Operators and Classic Function Theory (Springer, New York, 1993).
[22] Shapiro, J. H. and Smith, W., ‘Hardy spaces that support no compact composition operators’, J. Funct. Anal. 205(1) (2003), 6289.
[23] Sharma, S. D., Sharma, A. K. and Ahmed, S., ‘Composition operators between Hardy and Bloch-type spaces of the upper half-plane’, Bull. Korean Math. Soc. 43(3) (2007), 475482.
[24] Sharma, A. K. and Ueki, S. I., ‘Compact composition operators on the Bloch space and the growth space of the upper half-plane’, Mediterr. J. Math. 14(2) (2017), Article 76.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed