7,825 research outputs found

    Extensions of c0c_0

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    If XX is a closed subspace of a Banach space LL which embeds into a Banach lattice not containing β„“βˆžn\ell_\infty^n's uniformly and L/XL/X contains β„“βˆžn\ell_\infty^n's uniformly, then XX cannot have local unconditional structure in the sense of Gordon-Lewis (GL-{\sl l.u.st.})

    A Schauder basis for L1(0,∞)L_1(0,\infty) consisting of non-negative functions

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    We construct a Schauder basis for L1L_1 consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in LpL_p, 1≀p<∞1\le p < \infty

    Universal Non-Completely-Continuous Operators

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    A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L1L_1 into an arbitrary Banach space, namely, the operator from L1L_1 into β„“βˆž\ell_\infty defined by T0(f)=(∫rnf dΞΌ)nβ‰₯0Β , T_0 (f) =\left( \int r_n f \, d\mu \right)_{n\ge 0} \ , where rnr_n is the nthn^{\text{th}} Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space

    Extension of Operators from Weakβˆ—^*-closed Subspaces of β„“1\ell_1

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    It is proved that every operator from a weakβˆ—^*-closed subspace of β„“1\ell_1 into a space C(K)C(K) of continuous functions on a compact Hausdorff space KK can be extended to an operator from β„“1\ell_1 to C(K)C(K)

    The Complete Continuity Property and Finite Dimensional Decompositions

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    A Banach space \X has the complete continuity property (CCP) if each bounded linear operator from L1L_1 into \X is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP (resp., failing the CCP and failing cotype) has a subspace with a finite dimensional decomposition (resp., basis) which fails the CCP

    Subspaces of LpL_p that embed into Lp(ΞΌ)L_p(\mu) with ΞΌ\mu finite

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    Enflo and Rosenthal proved that β„“p(β„΅1)\ell_p(\aleph_1), 1<p<21 < p < 2, does not (isomorphically) embed into Lp(ΞΌ)L_p(\mu) with ΞΌ\mu a finite measure. We prove that if XX is a subspace of an LpL_p space, 1<p<21< p < 2, and β„“p(β„΅1)\ell_p(\aleph_1) does not embed into XX, then XX embeds into Lp(ΞΌ)L_p(\mu) for some finite measure ΞΌ\mu

    Computing p-summing norms with few vectors

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    It is shown that the p-summing norm of any operator with n-dimensional domain can be well-aproximated using only ``few" vectors in the definition of the p-summing norm. Except for constants independent of n and log n factors, ``few" means n if 1<p<2 and n^{p/2} if 2<p<infinity

    Polynomial Schur and Polynomial Dunford-Pettis Properties

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    A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the PNP_N Dunford--Pettis property if and only if every weakly compact Nβˆ’N-homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric condition, involving estimates on spreading models and implied by nontrivial type, is shown to be sufficient to imply that a space is polynomially Schur

    The proportional UAP characterizes weak Hilbert spaces

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    We prove that a Banach space has the uniform approximation property with proportional growth of the uniformity function iff it is a weak Hilbert space

    Commutators on β„“βˆž\ell_{\infty}

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    The operators on β„“βˆž\ell_{\infty} which are commutators are those not of the form Ξ»I+S\lambda I + S with Ξ»β‰ 0\lambda\neq 0 and SS strictly singular.Comment: 15 pages. Submitted to the Bulletin of the London Mathematical Society The proof of Theorem 3.3 was corrected. Other minor changes were mad
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