38 research outputs found
A disjointness type property of conditional expectation operators
We give a characterization of conditional expectation operators through a
disjointness type property similar to band preserving operators. We say that
the operator on a Banach lattice is semi band preserving if and
only if for all , implies that . We prove
that when
is a purely atomic Banach lattice, then an operator on is a
weighted conditional expectation operator if and only if is semi band
preserving.Comment: 11 page
1-complemented subspaces of spaces with 1-unconditional bases
We prove that if is a complex strictly monotone sequence space with
-unconditional basis, has no bands isometric to
and is the range of norm-one projection from , then is a closed
linear span a family of mutually disjoint vectors in .
We completely characterize -complemented subspaces and norm-one
projections in complex spaces for .
Finally we give a full description of the subspaces that are spanned by a
family of disjointly supported vectors and which are -complemented in (real
or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or
Lorentz space is not isomorphic to for some
then the only subspaces of which are -complemented and disjointly
supported are the closed linear spans of block bases with constant
coefficients
On the structure of level sets of uniform and Lipschitz quotient mappings from to
We study two questions posed by Johnson, Lindenstrauss, Preiss, and
Schechtman, concerning the structure of level sets of uniform and Lipschitz
quotient maps from . We show that if , , is a
uniform quotient map then for every , has a bounded number
of components, each component of separates and the upper
bound of the number of components depends only on and the moduli of
co-uniform and uniform continuity of . Next we obtain a characterization of
the form of any closed, hereditarily locally connected, locally compact,
connected set with no end points and containing no simple closed curve, and we
apply it to describe the structure of level sets of co-Lipschitz uniformly
continuous mappings . We prove that all level sets of any
co-Lipschitz uniformly continuous map from to are locally connected,
and we show that for every pair of a constant and a function
with , there exists a natural number ,
so that for every co-Lipschitz uniformly continuous map with a
co-Lipschitz constant and a modulus of uniform continuity , there
exists a natural number and a finite set with
\card(T_f)\leq n(f)-1 so that for all , has
exactly components, has exactly
components and each component of is homeomorphic with the real line
and separates the plane into exactly 2 components. The number and form of
components of for are also described - they have a
finite graph structure. We give an example of a uniform quotient map from
which has non-locally connected level sets.Comment: 34 pages, 10 figure
On isometric stability of complemented subspaces of
We show that Rudin-Plotkin isometry extension theorem in implies that
when and are isometric subspaces of and is not an even
integer, , then is complemented in if and only if
is; moreover the constants of complementation of and are equal. We
provide examples demonstrating that this fact fails when is an even integer
larger than 2
A note on Banach--Mazur problem
We prove that if is a real Banach space, with , which
contains a subspace of codimension 1 which is 1-complemented in and whose
group of isometries is almost transitive then is isometric to a Hilbert
space. This partially answers the Banach-Mazur rotation problem and generalizes
some recent related results.Comment: 8 pages, 2 figures but one of the figures doesn't run well in TeX so
it is not included here. The ps file of this paper which includes all figures
is available at http://www.users.muohio.edu/randrib/bm3.ps. to appear in
Glasgow J. Math. (2002
Isometric classification of norms in rearrangement-invariant function spaces
Suppose that a real nonatomic function space on is equipped with two
re\-arran\-ge\-ment-invariant norms and . We study the
question whether or not the fact that is isometric to
implies that for all in . We show
that in strictly monotone Orlicz and Lorentz spaces this is equivalent to
asking whether or not the norms are defined by equal Orlicz functions, resp.
Lorentz weights.
We show that the above implication holds true in most rearrangement-invariant
spaces, but we also identify a class of Orlicz spaces where it fails. We
provide a complete description of Orlicz functions with the
property that and are isometric
Contractive projections in Orlicz sequence spaces
We characterize norm one complemented subspaces of Orlicz sequence spaces
equipped with either Luxemburg or Orlicz norm, provided that the
Orlicz function is sufficiently smooth and sufficiently different from the
square function. This paper concentrates on the more difficult real case, the
complex case follows from previously known results.Comment: 14 page
Injective isometries in Orlicz spaces
We show that injective isometries in Orlicz space have to preserve
disjointness, provided that Orlicz function satisfies -condition,
has a continuous second derivative , satisfies another ``smoothness type''
condition and either
or and for all .
The fact that surjective isometries of any rearrangement-invariant function
space have to preserve disjointness has been determined before. However
dropping the assumption of surjectivity invalidates the general method. In this
paper we use a differential technique.Comment: 20 pages, 2 figures, to appear in the Proceedings of the Third
Conference on Function Spaces held in Edwardsville in May 1998, Contemporary
Mat
One-complemented subspaces of real sequence spaces
Characterizations are given for 1-complemented hyperplanes of strictly
monotone real Lorentz spaces and 1-complemented finite codimensional subspaces
(which contain at least one basis element) of real Orlicz spaces equipped with
either Luxemburg or Orlicz norm
Contractive projections and isometries in sequence spaces
We characterize 1-complemented subspaces of finite codimension in strictly
monotone one--convex, sequence spaces. Next we describe, up to
isometric isomorphism, all possible types of 1-unconditional structures in
sequence spaces with few surjective isometries. We also give a new example of a
class of real sequence spaces with few surjective isometries