Positivity (2010) 14:215–223
On disjoint function systems and complemented
Francisco L. Hernandez · Evgeny M. Semenov
Received: 4 May 2007 / Accepted: 6 May 2008 / Published online: 1 September 2008
© Birkhäuser Verlag Basel/Switzerland 2010
Abstract A sufﬁcient condition for the complementability of subspaces generated
by disjoint function systems in rearrangement invariant spaces is given. Orthogo-
nal projections in L
-spaces are extended to certain rearrangement invariant spaces.
Applications to Lorentz spaces are given.
Keywords Orthogonal projections · Rearrangement invariant spaces
Mathematics Subject Classiﬁcation (2000) 46E30
of disjoint supported functions (i.e. supp x
∩ supp x
for i = j)inL
,1≤ p < ∞, the closed subspace [x
} is always complemented in L
, i. e. there exists a bounded linear operator
acting from L
] such that its restriction on [x
] is the identical operator (the
“orthogonal” projection). This simple property characterizes the L
(μ) spaces. Thus
T. Ando in (seealso Thm 1.b.8) proved the following result: Let E be a Banach
lattice (of dimension ≥ 3), then E is order isometric to L
(μ) ,forsome 1≤ p < ∞
To the memory of Grigorii Lozanovsky.
F. L. Hernandez was partially supported by the Spanish grants MTM 2005–00082 and PR34/07-15837.
E. M. Semenov was partially supported by RFBR (Russia), grant 05–01–00629, and a
Santander-Complutense grant 2005.
F. L. Hernandez (
Dpto. de Análisis Matemático, Madrid Complutense University, 28040 Madrid, Spain
E. M. Semenov
Department of Mathematics, Voronezh State University, Voronezh 394006, Russia