Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Banach spaces; compact operator; asymptotic isometric copy of $\ell _1$
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.
References:
[1] Azimi, P.: A new class of Banach sequence spaces. Bull. Iran. Math. Soc. 28 (2002), 57-68. MR 1992259 | Zbl 1035.46006
[2] Azimi, P.: On geometric and topological properties of the classes of hereditarily $\ell_p$ Banach spaces. Taiwanese J. Math. 10 (2006), 713-722. DOI 10.11650/twjm/1500403857 | MR 2206324 | Zbl 1108.46009
[3] Azimi, P., Hagler, J.: Example of hereditarily $\ell_p$ Banach spaces failing the Schur property. Pac. J. Math. 122 (1987), 287-297. DOI 10.2140/pjm.1986.122.287 | MR 0831114
[4] Chen, D.: Asymptotically isometric copy of $c_0$ and $\ell_1$ in certain Banach spaces. J. Math. Anal. Appl. 284 (2003), 618-625. DOI 10.1016/S0022-247X(03)00368-8 | MR 1998656
[5] Chen, S., Lin, B. L.: Dual action of asymptotically isometric copies of $\ell_p$ $(1\leq p<\infty)$ and $c_0$. Collect. Math. 48 (1997), 449-458. MR 1602639 | Zbl 0892.46014
[6] Diestel, J.: Sequence and Series in Banach Spaces. Springer New York (1983). MR 0737004
[7] Dowling, P. N.: Isometric copies of $c_0$ and $\ell_{\infty}$ in duals of Banach spaces. J. Math. Anal. Appl. 244 (2000), 223-227. DOI 10.1006/jmaa.2000.6714 | MR 1746799 | Zbl 0955.46011
[8] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces. Springer Berlin (1977). MR 0500056 | Zbl 0362.46013
[9] Morrison, T. J.: Functional Analysis: An Introduction to Banach Space Theory. John Wiley & Sons (2001). MR 1885114 | Zbl 1005.46004
[10] Pelczynski, A.: Projections in certain Banach spaces. Stud. Math. 19 (1960), 209-228. DOI 10.4064/sm-19-2-209-228 | MR 0126145 | Zbl 0104.08503
[11] Popov, M. M.: More examples of hereditarily $\ell_p$ Banach spaces. Ukrainian Math. Bull. 2 (2005), 95-111. MR 2172327 | Zbl 1166.46304
Search
Partner of
