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Essential normality of linear fractional composition operators in the unit ball of ℂN

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Abstract

Let φ be a linear fractional self-map of the ball B N with a boundary fixed point e 1, we show that

$$ 1 - \operatorname{Re} \phi _1 \left( z \right) \sim \operatorname{Re} \left( {1 - z_1 } \right) $$

holds in a neighborhood of e 1 on \( {B_N } \). Applying this result we give a positive answer for a conjecture by MacCluer and Weir, and improve their results relating to the essential normality of composition operators on H 2(B N ) and A 2 γ (B N ) (γ > −1). Combining this with other related results in MacCluer & Weir, Integral Equations Operator Theory, 2005, we characterize the essential normality of composition operators induced by parabolic or hyperbolic linear fractional self-maps of B 2. Some of them indicate a difference between one variable and several variables.

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, China

    LiangYing Jiang

  2. Department of Applied Mathematics, Shanghai Finance University, Shanghai, 201209, China

    LiangYing Jiang

  3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China

    CaiHeng Ouyang

Authors
  1. LiangYing Jiang
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  2. CaiHeng Ouyang
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Corresponding author

Correspondence to CaiHeng Ouyang.

Additional information

Dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday

This work was supported by National Natural Science Foundation of China (Grant No. 10571044)

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Jiang, L., Ouyang, C. Essential normality of linear fractional composition operators in the unit ball of ℂN . Sci. China Ser. A-Math. 52, 2668–2678 (2009). https://doi.org/10.1007/s11425-009-0055-1

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  • DOI: https://doi.org/10.1007/s11425-009-0055-1

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