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On symmetric basic sequences in lorentz sequence spaces II

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Abstract

It is shown that if {y n} is a block of type I of a symmetric basis {x n} in a Banach spaceX, then {y n} is equivalent to {x n} if and only if the closed linear span [y n] of {y n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x n,f n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f n] has a complemented subspace isomorphic tol p (respectively,l q, 1/p+1/q=1 when 1<p<+∞ andc 0 whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f n] are obtained. We also obtain necessary and sufficient conditions such that [f n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x n} such that every symmetric block basic sequence of {x n} spans a complemented subspace inX butX is not isomorphic to eitherc 0 orl p, 1≤p<+∞.

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Authors and Affiliations

  1. Department of Mathematics, The University of Alabama Huntsville, Alabama, U. S. A.

    P. G. Casazza & Bor-Luh Lin

  2. Department of Mathematics, The University of Iowa, Iowa City, Iowa, U. S. A.

    P. G. Casazza & Bor-Luh Lin

Authors
  1. P. G. Casazza
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  2. Bor-Luh Lin
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Casazza, P.G., Lin, BL. On symmetric basic sequences in lorentz sequence spaces II. Israel J. Math. 17, 191–218 (1974). https://doi.org/10.1007/BF02882238

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  • DOI: https://doi.org/10.1007/BF02882238

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