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The Essential Norm of a Bloch-to-${{Q}_{p}}$ Composition Operator

Published online by Cambridge University Press:  20 November 2018

Mikael Lindstróm
Affiliation:
Department of Mathematics Åbo Akademi University FIN-20500 Åbo Finland, e-mail: mlindstr@abo.fi
Shamil Makhmutov
Affiliation:
Department of Mathematics College of Science Sultan Qaboos University PO Box 36 Al-Khod, PC 123 Sultanate of Oman, e-mail: makhm@squ.edu.om
Jari Taskinen
Affiliation:
Department of Mathematics University of Joensuu FIN-80101 Joensuu Finland, e-mail: jari.taskinen@joensuu.fi
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Abstract

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The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$, in particular from the Bloch space into $\text{BMOA}$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

References

[AXZ] Aulaskari, R., Xiao, J. and Zhao, R., On subspaces and subsets of BMOA and UBC. Analysis 15 (1995), 101121.Google Scholar
[ASX] Aulaskari, R., Stegenga, D. and Xiao, J., Some subclasses of BMOA and their characterization in terms of Carleson measure. Rocky Mountain J. Math. 26 (1996), 485506.Google Scholar
[BDL] Bonet, J., Domański, P. and Lindström, M., Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Canad.Math. Bull. 42 (1999), 139148.Google Scholar
[BDLT] Bonet, J., Domański, P., Lindström, M. and Taskinen, J., Composition operators between weighted Banach spaces of analytic functions. J. Austral.Math. Soc. 64 (1998), 101118.Google Scholar
[C] Conway, J., A course in Functional Analysis. Springer Verlag, New York, 1990.Google Scholar
[CM] Cowen, C. and MacCluer, B., Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, 1995.Google Scholar
[JR] Jarchow, H. and Riedl, R., Factorization of composition operators through Bloch type spaces. Illinois J. Math. 39 (1995), 431440.Google Scholar
[MM] Madigan, K. M. and Matheson, A., Compact composition operators on Bloch spaces. Trans. Amer. Math. Soc. 347 (1995), 26792687.Google Scholar
[MR] Montes-Rodriguez, A., The essential norm of a composition operator on Bloch spaces. Pacific J. Math. 188 (1999), 339351.Google Scholar
[MT] Makhmutov, S. and Tjani, M., Composition operators on some Móbius invariant Banach spaces. Bull. Austral. Math. Soc. 62 (2000), 119.Google Scholar
[N] Ng, K., On a theorem of Dixmier. Math. Scand. 29 (1971), 279280.Google Scholar
[P] Pommerenke, Ch., On Bloch functions. J. LondonMath. Soc. 2 (1970), 689695.Google Scholar
[R] Rosenthal, H. P., On relatively disjoint families of measures with some applications to Banach space theory. Studia Math. 37 (1970), 155190.Google Scholar
[Sh1] Shapiro, J. O., The essential norm of a composition operator. Ann. Math. 125 (1987), 375404.Google Scholar
[Sh2] Shapiro, J. O., Composition Operators and Classical Function Theory. Universitext, Springer-Verlag, New York, 1993.Google Scholar
[SW] Shields, A. L. and Williams, D. L., Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162 (1971), 287302.Google Scholar
[SZ] Smith, W. and Zhao, R., Composition operators mapping into the Qp spaces. Analysis 17 (1997), 239263.Google Scholar
[X] Xiao, J., Composition operators: Nα to the Bloch space to Qp . Studia Math. 139 (2000), 245260.Google Scholar
[Z] Zhu, K., Operator Theory in Functions Spaces. Dekker, 1990.Google Scholar