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Schatten class composition operators
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- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 26 February 2025, pp. 1-8
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Let
$C_{\varphi }$ be a composition operator on the Bergman space 
$A^2$ of the unit disc. A well-known problem asks whether the condition 
$\int _D\big ({1-|z|^2\over 1-|\varphi (z)|^2}\big )^pd\lambda (z) < \infty $ is equivalent to the membership of 
$C_\varphi $ in the Schatten class 
${\mathcal {C}}_p$, 
$1 < p < \infty $. This was settled in the negative for the case 
$2 < p < \infty $ in [3]. When 
$2 < p < \infty $, this condition is not sufficient for 
$C_\varphi \in {\mathcal {C}}_p$. In this article, we take up the case 
$1 < p < 2$. We show that when 
$1 < p < 2$, this condition is not necessary for 
$C_\varphi \in {\mathcal {C}}_p$.
COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASS 𝒮p
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 81 / Issue 3 / June 2010
- Published online by Cambridge University Press:
- 17 March 2010, pp. 465-472
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- June 2010
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