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On the Argyros-Haydon space every operator is a compact perturbation of a scalar multiple of the identity, and hence every quasinilpotent operator is compact.


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Question 2 has a negative answer. Suppose $\ell_p$ contains uniformly complemented copies $E_N$ of $\text{span}(d_n)_{n=1}^N$. Take an ultra power to get a copy $E$ of the completion of $\text{span}(d_n)_{n=1}^\infty$ in an $L_p$ space. The corresponding copies of $E_n$ in the $L_p$ space are uniformly complemented in the $L_p$ space. When $p>1$, this ...


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