Abstract
We study an \((n+1)\)-tensor norm \(\alpha ^C_{\mathbf {r}}\) extending to \((n+1)\)-fold tensor products a tensor norm defined by Michor when \(n=1\) by convexification of a certain s-norm. We characterize the maps of the minimal and the maximal multilinear operator ideals related to \(\alpha ^C_{\mathbf {r}}\) in the sense of Defant, Floret and Hunfeld.
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References
Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces. Academic, New York (1978)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic, New York (1985)
Day, M.M.: Normed Linear Spaces. Springer, Berlin (1973)
Defant, A., Floret, K.: Tensor Norms and Operator Ideals, North Holland Math. Studies 176. North Holland, Amsterdam (1993)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, Cambridge Studies in Advanced Maths. 43. Camdridge University Press, Cambridge (1995)
Floret, K.: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17, 153–188 (1997)
Floret, K., Hunfeld, S.: Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces. Proc. Am. Math. Soc. 130(5), 1425–1435 (2001)
Haydon, R., Levy, M., Raynaud, Y.: Randomly Normed Spaces. Herman, París (1991)
Heinrich, S.: Ultraproducts in Banach space theory. J. Reine Angew. Math. 313, 72–104 (1980)
Kürsten, K.D.: Local Duality of Ultraproducts of Banach Lattices, Lecture Notes in Mathematics 991, pp. 137–142. Springer, Berlin (1983)
Lacey, H.E.: The Isometric Theory of Banach Spaces. Springer, Berlin (1974)
López Molina, J.A.: Multilinear operator ideals associated to Saphar type \((n+1)\)-tensor norms. Positivity 11, 95–117 (2007)
López Molina, J.A.: \((n+1)\)-tensor norms of Lapresté’s type. Glasg. Math. J. 54, 665–692 (2012)
López Molina, J.A., Puerta, M.E., Rivera, M.J.: Ultraproduts of real interpolation spaces between \(L^p\)-spaces. Bull. Braz. Math. Soc. 37(2), 191–216 (2006)
Maharam, D.: The representation of abstract measure functions. Trans. Am. Math. Soc. 65, 279–330 (1949)
Maharam, D.: Decompositions of measure algebras and spaces. Trans. Am. Math. Soc. 69, 142–160 (1950)
Mazur, S., Ulam, S.: Sur les transformations isométriques d’espaces vectoriels. C. R. Acad. Sci. Paris 194, 946–948 (1932)
Michor, P.W.: Functors and Categories of Banach Spaces, Lecture Notes on Mathematics, 651. Springer, Berlin (1978)
Pietsch, A.: Ideals of multilinear functionals (designs of a theory). In: Proceeding of the 2nd International Conference on Operator Algebras, Ideals and their applications in Theoretical Physics, Leipzig 1983, pp. 185–199. Teubner-Texte (1983)
Popov, M.M.: An isomorphic classification of the \(L_p(\mu )\) for \(0<p<1\). J. Sov. Math. 48(6), 674–681 (1990)
Schreiber, M.: Quelques remarques sur les charactérisations des spaces \(L^p, 0\le p<1\). Ann. Inst. Henri Poincaré VIII(1), 83–92 (1972)
Wnuk, W., Wiatrowski, B.: When are ultrapowers lattices discrete or continuous? In: Proceedings Positiviy IV—Theory and Applications, pp. 173–182, Dresden (2006)
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López Molina, J.A. The minimal and maximal operator ideals associated to \((n+1)\)-tensor norms of Michor’s type. Positivity 22, 1109–1142 (2018). https://doi.org/10.1007/s11117-018-0563-8
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DOI: https://doi.org/10.1007/s11117-018-0563-8
Keywords
- \((n+1)\)-fold tensor products
- \(\alpha ^C_{\mathbf {r}}\)-nuclear and \(\alpha ^C_{\mathbf {r}}\)-integral n-linear operators
- Ultraproducts