| Title:
|
The Grothendieck property for injective tensor products of Banach spaces (English) |
| Author:
|
Ji, Donghai |
| Author:
|
Xue, Xiaoping |
| Author:
|
Bu, Qingying |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
1153-1159 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of $X$ and $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from $X^\ast $ to $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$ has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from $X^\ast $ to $Y$ is compact. (English) |
| Keyword:
|
Banach space |
| Keyword:
|
Grothendieck property |
| Keyword:
|
tensor product |
| MSC:
|
46B28 |
| MSC:
|
46M05 |
| idZBL:
|
Zbl 1224.46034 |
| idMR:
|
MR2738976 |
| . |
| Date available:
|
2010-11-20T14:03:51Z |
| Last updated:
|
2023-07-17 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140813 |
| . |
| Reference:
|
[1] Bu, Q., Emmanuele, G.: The projective and injective tensor products of $L^p[0,1]$ and $X$ being Grothendieck spaces.Rocky Mt. J. Math. 35 (2005), 713-726. MR 2150306, 10.1216/rmjm/1181069704 |
| Reference:
|
[2] Defant, A., Floret, K.: Tensor Norms and Operator Ideals.North-Holland Amsterdam (1993). Zbl 0774.46018, MR 1209438 |
| Reference:
|
[3] Diestel, J., Uhl, J. J.: Vector Measures. Mathematical Surveys No. 15.American Mathematical Society (AMS) Providence (1977). MR 0453964 |
| Reference:
|
[4] Dunford, N., Schwartz, J. T.: Linear Operators. Part I: General Theory.John Wiley & Sons New York (1988). Zbl 0635.47001, MR 1009162 |
| Reference:
|
[5] González, M., Gutiérrez, J. M.: Polynomial Grothendieck properties.Glasg. Math. J. 37 (1995), 211-219. MR 1333740, 10.1017/S0017089500031116 |
| Reference:
|
[6] Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires.Mem. Am. Math. Soc. 16 (1955), French. Zbl 0123.30301, MR 0075539 |
| Reference:
|
[7] Kalton, N. J.: Schauder decompositions and completeness.Bull. Lond. Math. Soc. 2 (1970), 34-36. Zbl 0196.13601, MR 0259547, 10.1112/blms/2.1.34 |
| Reference:
|
[8] Kalton, N. J.: Spaces of compact operators.Math. Ann. 208 (1974), 267-278. Zbl 0266.47038, MR 0341154, 10.1007/BF01432152 |
| Reference:
|
[9] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Sequence Spaces.Springer Berlin-Heidelberg-London (1977). Zbl 0362.46013, MR 0500056 |
| Reference:
|
[10] Ryan, R. A.: Introduction to Tensor Products of Banach Spaces.Springer London (2002). Zbl 1090.46001, MR 1888309 |
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