| Title:
|
Copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}(C[ 0,1],X) $ (English) |
| Author:
|
Popa, Dumitru |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
457-467 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the presence of copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}( C[0,1] ,X)$. By using Dvoretzky's theorem we deduce that if $X$ is an infinite-dimensional Banach space, then $\Pi _{2}( C[ 0,1] ,X) $ contains $\lambda \sqrt {2}$-uniformly copies of $l_{\infty }^{n}$'s and $\Pi _{1}( C[ 0,1] ,X) $ contains $\lambda $-uniformly copies of $l_{2}^{n}$'s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}( C[ 0,1] ,X) $ are distinct, extending the well-known result that the spaces $\Pi _{2}( C[ 0,1],X) $ and $\mathcal {N}( C[ 0,1] ,X) $ are distinct. (English) |
| Keyword:
|
$p$-summing linear operators |
| Keyword:
|
copies of $l_{p}^{n}$'s uniformly |
| Keyword:
|
local structure of a Banach space |
| Keyword:
|
multiplication operator |
| Keyword:
|
average |
| MSC:
|
46B07 |
| MSC:
|
46B28 |
| MSC:
|
47B10 |
| MSC:
|
47L20 |
| idZBL:
|
Zbl 06738531 |
| idMR:
|
MR3661053 |
| DOI:
|
10.21136/CMJ.2017.0009-16 |
| . |
| Date available:
|
2017-06-01T14:30:05Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146768 |
| . |
| Reference:
|
[1] Costara, C., Popa, D.: Exercises in Functional Analysis.Kluwer Texts in the Mathematical Sciences 26, Kluwer Academic Publishers Group, Dordrecht (2003). Zbl 1070.46001, MR 2027363, 10.1007/978-94-017-0223-2 |
| Reference:
|
[2] Defant, A., Floret, K.: Tensor Norms and Operator Ideals.North-Holland Mathematics Studies 176, North-Holland Publishing, Amsterdam (1993). Zbl 0774.46018, MR 1209438, 10.1016/s0304-0208(08)x7019-7 |
| Reference:
|
[3] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators.Cambridge Studies in Advanced Mathematics 43, Cambridge University Press, Cambridge (1995). Zbl 0855.47016, MR 1342297, 10.1017/CBO9780511526138 |
| Reference:
|
[4] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Mathematical Surveys 15, American Mathematical Society, Providence (1977). Zbl 0369.46039, MR 0453964, 10.1090/surv/015 |
| Reference:
|
[5] Lima, Å., Lima, V., Oja, E.: Absolutely summing operators on $C[0,1]$ as a tree space and the bounded approximation property.J. Funct. Anal. 259 (2010), 2886-2901. Zbl 1207.46019, MR 2719278, 10.1016/j.jfa.2010.07.017 |
| Reference:
|
[6] Pietsch, A.: Operator Ideals.Mathematische Monographien 16, VEB Deutscher der Wissenschaften, Berlin (1978). Zbl 0399.47039, MR 0519680 |
| Reference:
|
[7] Popa, D.: Examples of operators on $C[0,1]$ distinguishing certain operator ideals.Arch. Math. 88 (2007), 349-357. Zbl 1124.47013, MR 2311842, 10.1007/s00013-006-1916-2 |
| Reference:
|
[8] Popa, D.: Khinchin's inequality, Dunford-Pettis and compact operators on the space $C([0,1],X)$.Proc. Indian Acad. Sci., Math. Sci. 117 (2007), 13-30. Zbl 1124.47023, MR 2300675, 10.1007/s12044-007-0002-4 |
| Reference:
|
[9] Popa, D.: Averages and compact, absolutely summing and nuclear operators on $C(\Omega)$.J. Korean Math. Soc. 47 (2010), 899-924. Zbl 1214.47023, MR 2722999, 10.4134/JKMS.2010.47.5.899 |
| Reference:
|
[10] Sofi, M. A.: Factoring operators over Hilbert-Schmidt maps and vector measures.Indag. Math., New Ser. 20 (2009), 273-284. Zbl 1193.46005, MR 2599817, 10.1016/S0019-3577(09)80014-1 |
| . |
