Abstract
Let ψ and φ be analytic functions on the unit disk D such that φ(D) ⊂ D. We characterize the boundedness and compactness of the weighted composition operators f ↦ ψ · (f oφ) on BMOA, the space of analytic functions on D that have bounded mean oscillation on ∂D, and its subspace VMOA. We also provide estimates for the norm of a weighted composition operator on BMOA and its essential norm on VMOA. Finally, we use the above results to show that boundedness or compactness of a weighted composition operator on BMOA implies its boundedness or compactness on the Bloch space B, respectively.
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J. Arazy, S. D Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145.
A. Baernstein II, Analytic functions of bounded mean oscillation, in: D. A. Brannan and J. G. Clunie (eds.), Aspects of Contemporary Complex Analysis, Academic Press, London, 1980, pp. 3–36.
P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183–2196.
J. A. Cima and A. L. Matheson, Weakly compact composition operators on VMO, Rocky Mountain J. Math. 32 (2002), 937–951.
M. D. Contreras and A.G. Hernández-Díaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), 224–233.
M. D. Contreras and A.G. Hernández-Díaz, Weighted composition operators between different Hardy spaces, Integral Equations Operator Theory 46 (2003), 165–188.
C. C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
Ž. Čučković and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499–511.
Ž. Čučković and R. Zhao, Weighted composition operators between different weighted Bergman and Hardy spaces, Illinois J. Math. 51 (2007), 479–498.
R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC, Boca Raton, 2003.
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
D. Girela, Analytic functions of bounded mean oscillation, in: R. Aulaskari (ed.), Complex Function Spaces (Mekrijärvi, 1999), Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu, 2001, pp. 61–170.
S. Janson, On functions with conditions on the mean oscillation, Ark. Math. 14 (1976), 189–196.
J. Laitila, Composition operators and vector-valued BMOA, Integral Equations Operator Theory 58 (2007), 487–502.
M. Lindström, S. Makhmutov and J. Taskinen, The essential norm of a Bloch-to-Qp composition operator, Canad. Math. Bull. 47 (2004), 49–59.
B. D. MacCluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437–1458.
S. Makhmutov and M. Tjani, Composition operators on some Möbius invariant Banach spaces, Bull. Austral. Math. Soc. 62 (2000), 1–19.
A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific J. Math. 188 (1999), 339–351.
S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc. 63 (2001), 177–185.
J. M. Ortega and J. Fàbrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier (Grenoble) 46 (1996), 111–137.
J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125 (1987), 375–404.
J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York 1993.
W. Smith, Compactness of composition operators on BMOA, Proc. Amer. Math. Soc. 127 (1999), 2715–2725.
D. A. Stegenga, Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), 573–589.
K. Stephenson, Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), 565–577.
M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University, 1996.
M. Wang and P. Liu, Weighted composition operators between Hardy spaces, Math. Appl. (Wuhan) 16 (2003), 130–135.
K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990
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The author was supported by the Finnish Academy of Science and Letters (Vilho, Yrjö and Kalle Väisälä Foundation) and the Academy of Finland, projects 53893, 210970 and 118422.
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Laitila, J. Weighted Composition Operators on BMOA. Comput. Methods Funct. Theory 9, 27–46 (2009). https://doi.org/10.1007/BF03321712
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DOI: https://doi.org/10.1007/BF03321712