# Article

 Title: Product spaces generated by bilinear maps and duality (English) Author: Sánchez Pérez, Enrique A. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 65 Issue: 3 Year: 2015 Pages: 801-817 Summary lang: English . Category: math . Summary: In this paper we analyse a definition of a product of Banach spaces that is naturally associated by duality with a space of operators that can be considered as a generalization of the notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of functional analysis that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown, regarding pointwise products of Banach function spaces, spaces of integrable functions with respect to vector measures, spaces of operators, multipliers on Banach spaces of analytic functions and spaces of Lipschitz functions. (English) Keyword: Banach space Keyword: product Keyword: multiplication operator Keyword: duality Keyword: Banach function space Keyword: Hadamard product Keyword: Lipschitz map Keyword: integration Keyword: vector measure MSC: 46A32 MSC: 46B10 MSC: 46E30 MSC: 47A30 idZBL: Zbl 06537693 idMR: MR3407606 DOI: 10.1007/s10587-015-0209-y . Date available: 2015-10-04T18:21:13Z Last updated: 2017-10-02 Stable URL: http://hdl.handle.net/10338.dmlcz/144444 . Reference: [1] Blasco, Ó., Pavlović, M.: Coefficient multipliers on Banach spaces of analytic functions.Rev. Mat. Iberoam. 27 (2011), 415-447. Zbl 1235.42004, MR 2848526, 10.4171/RMI/642 Reference: [2] Calabuig, J. M., Delgado, O., Pérez, E. A. Sánchez: Generalized perfect spaces.Indag. Math., New Ser. 19 (2008), 359-378. MR 2513056, 10.1016/S0019-3577(09)00008-1 Reference: [3] Defant, A., Floret, K.: Tensor Norms and Operator Ideals.North-Holland Mathematics Studies 176 North-Holland, Amsterdam (1993). Zbl 0774.46018, MR 1209438 Reference: [4] Delgado, O., Pérez, E. A. Sánchez: Summability properties for multiplication operators on Banach function spaces.Integral Equations Oper. Theory 66 (2010), 197-214. MR 2595653, 10.1007/s00020-010-1741-7 Reference: [5] Diestel, J., Uhl, J. J., Jr., \rm: Vector Measures.Mathematical Surveys 15 American Mathematical Society, Providence (1977). MR 0453964 Reference: [6] Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez-Pérez, E. A.: Spaces of {$p$}-integrable functions with respect to a vector measure.Positivity 10 (2006), 1-16. Zbl 1111.46018, MR 2223581, 10.1007/s11117-005-0016-z Reference: [7] Ferrando, I., Pérez, E. A. Sánchez: Tensor product representation of the (pre)dual of the {$L^p$}-space of a vector measure.J. Aust. Math. Soc. 87 (2009), 211-225. MR 2551119, 10.1017/S1446788709000196 Reference: [8] Ferrando, I., Rodríguez, J.: The weak topology on $L^p$ of a vector measure.Topology Appl. 155 (2008), 1439-1444. Zbl 1151.28014, MR 2427417, 10.1016/j.topol.2007.12.014 Reference: [9] Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization.J. Funct. Anal. 266 (2014), 616-659. Zbl 1308.46039, MR 3132723, 10.1016/j.jfa.2013.10.028 Reference: [10] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. II: Function Spaces.Ergebnisse der Mathematik und ihrer Grenzgebiete 97 Springer, Berlin (1979). Zbl 0403.46022, MR 0540367 Reference: [11] Mastyło, M., Sánchez-Pérez, E. A.: Köthe dual of Banach lattices generated by vector measures.Monatsh. Math. 173 (2014), 541-557. Zbl 1305.46021, 10.1007/s00605-013-0560-8 Reference: [12] Okada, S., Ricker, W. J., Pérez, E. A. Sánchez: Optimal Domain and Integral Extension of Operators: Acting in Function Spaces.Operator Theory: Advances and Applications 180 Birkhäuser, Basel (2008). MR 2418751 Reference: [13] Pérez, E. A. Sánchez: Factorization theorems for multiplication operators on Banach function spaces.Integral Equations Oper. Theory 80 (2014), 117-135. MR 3248477, 10.1007/s00020-014-2169-2 Reference: [14] Pérez, E. A. Sánchez: Vector measure duality and tensor product representations of {$L_p$}-spaces of vector measures.Proc. Am. Math. Soc. 132 (2004), 3319-3326. MR 2073308, 10.1090/S0002-9939-04-07521-5 Reference: [15] Pérez, E. A. Sánchez: Compactness arguments for spaces of {$p$}-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces.Ill. J. Math. 45 (2001), 907-923. MR 1879243 Reference: [16] Rueda, P., Pérez, E. A. Sánchez: Compactness in spaces of \mbox{$p$-integrable} functions with respect to a vector measure.Topol. Methods Nonlinear Anal. 45 (2015), 641-654. MR 3408839, 10.12775/TMNA.2015.030 Reference: [17] Schep, A. R.: Products and factors of Banach function spaces.Positivity 14 (2010), 301-319. Zbl 1216.46028, MR 2657636, 10.1007/s11117-009-0019-2 Reference: [18] Sukochev, F., Tomskova, A.: $(E,F)$-Schur multipliers and applications.Stud. Math. 216 (2013), 111-129. Zbl 1281.47023, MR 3085499, 10.4064/sm216-2-2 .

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