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$1$-complemented subspaces of spaces with $1$-unconditional bases

  Published:1997-12-01
 Printed: Dec 1997
  • Beata Randrianantoanina
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Abstract

We prove that if $X$ is a complex strictly monotone sequence space with $1$-un\-con\-di\-tion\-al basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.
MSC Classifications: 46B20, 46B45, 41A65 show english descriptions Geometry and structure of normed linear spaces
Banach sequence spaces [See also 46A45]
Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 - Geometry and structure of normed linear spaces
46B45 - Banach sequence spaces [See also 46A45]
41A65 - Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
 

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