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    p-summing operators on injective tensor products of spaces

    Montgomery-Smith, Stephen, 1963-
    Saab, Paulette, 1952-
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    [PDF] PSummingOperators.pdf (163.5Kb)
    Date
    2011
    Format
    Preprint
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    Abstract
    Let X, Y and Z be Banach spaces, and let Πp(Y,Z) (1 ≤ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a £∞-space, then a bounded linear operator T:X⊗εY→Z is 1-summing if and only if a naturally associated operator T#:X→Π1(Y,Z) is 1-summing. This result need not be true if X is not a £∞-space. For p > 1, several examples are given with X = C[0,1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on C[0,1]⊗εl1 whose associated operator T# is 2-summing, but for all N∈N, there exists an N-dimensional subspace U of C[0,1]⊗εl1 such that T restricted to U is equivalent to the identity operator on l∞N. Finally, we show that there is a compact Hausdorff space K and a bounded linear operator T:C(K)⊗εl1→l2 for which T#:C(K)→Π1(l1,l2) is not 2-summing.
    URI
    http://hdl.handle.net/10355/9838
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    Mathematics publications (MU)
    Rights
    OpenAccess.
    This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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