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Let $\chi_1$ be the map on the unit circle defined by $\chi_1(e^{it})=e^{it}$. Let $T_{\chi_1}$ be the corresponding Toeplitz operator. Consider the map $T_{\chi_1}^* T_{\chi_1}- T_{\chi_1} T_{\chi_1}^*$ where $T_{\chi_1}^*$ is the adjoint of $T_{\chi_1}$. The book I am reading says the map $T_{\chi_1}^* T_{\chi_1}- T_{\chi_1} T_{\chi_1}^*$ is a nonzero rank one. It is easy to see it is nonzero but I cannot see why it is rank one.

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