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SPECTRA OF LINEAR FRACTIONAL COMPOSITION OPERATORS ON THE GROWTH SPACE AND BLOCH SPACE OF THE UPPER HALF-PLANE

Published online by Cambridge University Press:  29 October 2018

SHI-AN HAN
Affiliation:
School of Mathematics, Tianjin University, Tianjin 300354, PR China email [email protected]
ZE-HUA ZHOU*
Affiliation:
School of Mathematics, Tianjin University, Tianjin 300354, PR China email [email protected], [email protected]
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Abstract

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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.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

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

References

Aron, R. and Lindström, M., ‘Spectra of weighted composition operators on weighted Banach spaces of analytic functions’, Israel J. Math. 141(1) (2004), 263276.Google Scholar
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.Google Scholar
Bourdon, P. and Shapiro, J. H., Cyclic Phenomena for Composition Operators, Memoirs of the American Mathematical Society, 596 (American Mathematical Society, Providence, RI, 1997).Google Scholar
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.Google Scholar
Cowen, C. C., ‘Composition operators on H 2 ’, J. Operator Theory 9 (1983), 77106.Google Scholar
Cowen, C. C., ‘Linear fractional composition operators on H 2 ’, Integral Equations Operator Theory 11(2) (1988), 151160.Google Scholar
Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).Google Scholar
Donaway, R., ‘Norm and essential norm estimates of composition operators on Besov type spaces’, PhD Thesis, The University of Virginia, 1999.Google Scholar
Elliott, S. and Jury, M. T., ‘Composition operators on Hardy spaces of a half-plane’, Bull. Lond. Math. Soc. 44(3) (2012), 489495.Google Scholar
Elliott, S. and Wynn, A., ‘Composition operators on weighted Bergman spaces of a half-plane’, Proc. Edinb. Math. Soc. 54(2) (2011), 373379.Google Scholar
Gallardo-Gutiérrez, E. A. and Montes-Rodríguez, A., ‘Adjoints of linear fractional composition operators on the Dirichlet space’, Math. Ann. 327(1) (2003), 117134.Google Scholar
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.Google Scholar
Gunatillake, G., ‘Invertible weighted composition operators’, J. Funct. Anal. 261(3) (2011), 831860.Google Scholar
Higdon, W. M., ‘The spectra of composition operators from linear fractional maps acting upon the Dirichlet space’, J. Funct. Anal. 220(1) (2005), 5575.Google Scholar
Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups (American Mathematical Society, Providence, RI, 1996).Google Scholar
Hyvärinen, O., Lindström, M., Nieminen, I. and Saukko, E., ‘Spectra of weighted composition operators with automorphic symbols’, J. Funct. Anal. 265(8) (2013), 17491777.Google Scholar
Matache, V., ‘Composition operators on Hardy spaces of a half-plane’, Proc. Amer. Math. Soc. 127(5) (1999), 14831491.Google Scholar
Matache, V., ‘Weighted composition operators on H 2 and applications’, Complex Anal. Oper. Theory 2(1) (2008), 169197.Google Scholar
Matache, V., ‘Invertible and normal composition operators on the Hilbert Hardy space of a half-plane’, Concr. Oper. 3(1) (2016), 7784.Google Scholar
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.Google Scholar
Shapiro, J. H., Composition Operators and Classic Function Theory (Springer, New York, 1993).Google Scholar
Shapiro, J. H. and Smith, W., ‘Hardy spaces that support no compact composition operators’, J. Funct. Anal. 205(1) (2003), 6289.Google Scholar
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.Google Scholar
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.Google Scholar