# Article

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Keywords:
Banach spaces; compact operator; asymptotic isometric copy of $\ell _1$
Summary:
For the Azimi-Hagler spaces more geometric and topological properties are investigated. Any constructed space is denoted by $X_{\alpha ,p}$. We show \item {(i)} The subspace $[(e_{n_k})]$ generated by a subsequence $(e_{n_k})$ of $(e_n)$ is complemented. \item {(ii)} The identity operator from $X_{\alpha ,p}$ to $X_{\alpha ,q}$ when $p>q$ is unbounded. \item {(iii)} Every bounded linear operator on some subspace of $X_{\alpha ,p}$ is compact. It is known that if any $X_{\alpha ,p}$ is a dual space, then \item {(iv)} duals of $X_{\alpha ,1}$ spaces contain isometric copies of $\ell _{\infty }$ and their preduals contain asymptotically isometric copies of $c_0$. \item {(v)} We investigate the properties of the operators from $X_{\alpha ,p}$ spaces to their predual.
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