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    Contreras, Manuel D. Díaz-Madrigal, Santiago and Vukotić, Dragan 2014. Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces. Journal of Mathematical Analysis and Applications, Vol. 415, Issue. 2, p. 713.


    Galindo, Pablo Laitila, Jussi and Lindström, Mikael 2013. Essential norm estimates for composition operators on BMOA. Journal of Functional Analysis, Vol. 265, Issue. 4, p. 629.


    Laitila, Jussi Nieminen, Pekka J. Saksman, Eero and Tylli, Hans-Olav 2013. Compact and Weakly Compact Composition Operators on BMOA. Complex Analysis and Operator Theory, Vol. 7, Issue. 1, p. 163.


    Doubtsov, Evgueni 2012. Hyperbolic BMOA classes. Journal of Mathematical Analysis and Applications, Vol. 391, Issue. 1, p. 57.


    Laitila, Jussi 2009. Weighted Composition Operators on BMOA. Computational Methods and Function Theory, Vol. 9, Issue. 1, p. 27.


    Li, Songxiao and Wulan, Hasi 2007. Composition operators on <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mi>Q</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math> spaces. Journal of Mathematical Analysis and Applications, Vol. 327, Issue. 2, p. 948.


    Tjani, Maria 2006. Distance of a Bloch Function to the Little Bloch Space. Bulletin of the Australian Mathematical Society, Vol. 74, Issue. 01, p. 101.


    Kwon, E.G. 2005. Hyperbolic g-function and Bloch pullback operators. Journal of Mathematical Analysis and Applications, Vol. 309, Issue. 2, p. 626.


    Laitila, Jussi 2005. Weakly compact composition operators on vector-valued BMOA. Journal of Mathematical Analysis and Applications, Vol. 308, Issue. 2, p. 730.


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  • Bulletin of the Australian Mathematical Society, Volume 62, Issue 1
  • August 2000, pp. 1-19

Composition operators on some Möbius invariant Banach spaces

  • Shamil Makhmutov (a1) and Maria Tjani (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700018426
  • Published online: 01 April 2009
Abstract

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

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[3]R. Aulaskari , D.A. Stegenga and J. Xiao , ‘Some subclasses of BMOA and their characterization in terms of Carleson measures’, Rocky Mountain J. Math. 26 (1996), 485506.

[6]B.R. Choe , W. Ramey and D. Ullrich , ‘Bloch-to-BMOA pullbacks on the disk’, Proc. Amer. Math. Soc. 125 (1997), 29872996.

[7]J.B. Conway , A course in functional analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1985).

[10]P.K. Madigan and A. Matheson , ‘Compact composition operators on the Bloch space’, Trans. Amer. Math. Soc. 347 (1995), 26792687.

[11]S. Makhmutov , ‘Hyperbolic Besov functions and Bloch-to-Besov composition operators’, Hokkaido Math. J. 26 (1997), 699711.

[12]R. Nevanlinna , Analytic functions, (second edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1970).

[13]C. Pommerenke , ‘Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation’, Comment. Math. Helv. 52 (1977), 591602.

[14]L.A. Rubel and R.M. Timoney , ‘An extremal property of the Bloch space’, Proc. Amer. Math. Soc. 75 (1979), 4549.

[16]J.H. Shapiro , ‘The essential norm of a coposition operator’, Ann of Math. 125 (1987), 375404.

[17]J.H. Shapiro , Composition operators and classical function theory (Springer-Verlag, Berlin, Heidelberg, New York, 1993).

[18]W. Smith and R. Zhao , ‘Composition operators mapping into the Qp spaces’, Analysis 17 (1997), 239263.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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