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Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 901-917, 2012
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Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space $H^2$

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Mahsa Fatehi, Bahram Khani Robati

**Abstract:** In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi}$, when $\varphi$ is a linear-fractional self-map of $\mathbb{D}$. In this paper first, we investigate the essential normality problem for the operator $T_wC_{\varphi}$ on the Hardy space $H^2$, where $w$ is a bounded measurable function on $\partial\mathbb{D}$ which is continuous at each point of $F(\varphi)$, $\varphi\in{\cal S}(2)$, and $T_w$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on $H^2$.

**Keywords:** Hardy spaces, essentially normal, composition operator, linear-fractional transformation

**Classification (MSC 2010):** 47B33

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