Abstract view
$1$complemented subspaces of spaces with $1$unconditional bases


Published:19971201
Printed: Dec 1997
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 normone 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 normone
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.