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2-convexity and 2-concavity in Schatten ideals

  • G. J. O. Jameson (a1)

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

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[G-TJ] D. J. H. Garling and N. Tomczak-Jaegermann . The cotype and uniform convexity of unitary ideals. Israel J. Math. 45 (1983), 175197.

[L-Tz II] J. Lindenstrauss and L. Tzafriri . Classical Banach spaces II (Springer, 1979).

[LP-P] F. Lust-Piquard and G. Pisier . Non-commutative Khintchine and Paley inequalities. Arkiv för Matematik 29 (1991), 241260.

[R] S. Reisner . A factorization theorem in Banach lattices and its application to Lorentz spaces. Ann. Inst. Fourier 31 (1981), 239255.

[Sch] C. Schütt . Lorentz spaces that are isomorphic to subspaces of L1. Trans. Amer. Math. Soc. 314 (1989). 583595.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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