Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Banach space; Grothendieck property; tensor product
Banach space; Grothendieck property; tensor product
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.
References:
[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. DOI 10.1216/rmjm/1181069704 | MR 2150306
[2] Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Amsterdam (1993). MR 1209438 | Zbl 0774.46018
[3] Diestel, J., Uhl, J. J.: Vector Measures. Mathematical Surveys No. 15. American Mathematical Society (AMS) Providence (1977). MR 0453964
[4] Dunford, N., Schwartz, J. T.: Linear Operators. Part I: General Theory. John Wiley & Sons New York (1988). MR 1009162 | Zbl 0635.47001
[5] González, M., Gutiérrez, J. M.: Polynomial Grothendieck properties. Glasg. Math. J. 37 (1995), 211-219. DOI 10.1017/S0017089500031116 | MR 1333740
[6] Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16 (1955), French. MR 0075539 | Zbl 0123.30301
[7] Kalton, N. J.: Schauder decompositions and completeness. Bull. Lond. Math. Soc. 2 (1970), 34-36. DOI 10.1112/blms/2.1.34 | MR 0259547 | Zbl 0196.13601
[8] Kalton, N. J.: Spaces of compact operators. Math. Ann. 208 (1974), 267-278. DOI 10.1007/BF01432152 | MR 0341154 | Zbl 0266.47038
[9] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Sequence Spaces. Springer Berlin-Heidelberg-London (1977). MR 0500056 | Zbl 0362.46013
[10] Ryan, R. A.: Introduction to Tensor Products of Banach Spaces. Springer London (2002). MR 1888309 | Zbl 1090.46001
Search
Partner of
