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1,000 |
Factorizations of natural embeddings of l_p^n int L_r
|
math.FA
|
This is a continuation of the paper [FJS] with a similar title. Several
results from there are strengthened, in particular:
1. If T is a "natural" embedding of l_2^n into L_1 then, for any well-bounded
factorization of T through an L_1 space in the form T=uv with v of norm one, u
well-preserves a copy of l_1^k with k exponential in n.
2. Any norm one operator from a C(K) space which well-preserves a copy of
l_2^n also well-preserves a copy of l_{\infty}^k with k exponential in n.
As an application of these and other results we show the existence, for any
n, of an n-dimensional space which well-embeds into a space with an
unconditional basis only if the latter contains a copy of l_{\infty}^k with k
exponential in n.
|
math
|
1,001 |
The Rademacher cotype of operators from $l_\infty^N$
|
math.FA
|
We show that for any operator $T:l_\infty^N\to Y$, where $Y$ is a Banach
space, that its cotype 2 constant, $K_2(T)$, is related to its $(2,1)$-summing
norm, $\pi_{2,1}(T)$, by $K_2(T) \le c \log\log N \pi_{2,1}(T) $. Thus, we can
show that there is an operator $T:C(K)\to Y$ that has cotype 2, but is not
2-summing.
|
math
|
1,002 |
Operators which factor through Banach lattices not containing c_0
|
math.FA
|
In this supplement to [GJ1], [GJ3], we give an intrinsic characterization of
(bounded, linear) operators on Banach lattices which factor through Banach
lattices not containing a copy of $c_0$ which complements the characterization
of [GJ1], [GJ3] that an operator admits such a factorization if and only if it
can be written as the product of two operators neither of which preserves a
copy of $c_0$. The intrinsic characterization is that the restriction of the
second adjoint of the operator to the ideal generated by the lattice in its
bidual does not preserve a copy of $c_0$. This property of an operator was
introduced by C. Niculescu [N2] under the name ``strong type B".
|
math
|
1,003 |
Integral Operators on Spaces of Continuous Vector-valued Functions
|
math.FA
|
Let $X$ be a compact Hausdorff space, let $E$ be a Banach space, and let
$C(X,E)$ stand for the Banach space of $E$-valued continuous functions on $X$
under the uniform norm. In this paper we characterize Integral operators (in
the sense of Grothendieck) on $C(X,E)$ spaces in term of their representing
vector measures. This is then used to give some applications to Nuclear
operators on $C(X,E)$ spaces.
|
math
|
1,004 |
Nuclear operators on spaces of continuous vector-valued functions
|
math.FA
|
Let $\Omega$ be a compact Hausdorff space, let $E$ be a Banach space, and let
$C(\Omega, E)$ stand for the Banach space of all $E$-valued continuous
functions on $\Omega$ under supnorm. In this paper we study when nuclear
operators on $C(\Omega, E)$ spaces can be completely characterized in terms of
properties of their representing vector measures. We also show that if $F$ is a
Banach space and if $T:\ C(\Omega, E)\rightarrow F$ is a nuclear operator, then
$T$ induces a bounded linear operator $T^\#$ from the space $C(\Omega)$ of
scalar valued continuous functions on $\Omega$ into $\slN(E,F)$ the space of
nuclear operators from $E$ to $F$, in this case we show that $E^*$ has the
Radon-Nikodym property if and only if $T^\#$ is nuclear whenever $T$ is
nuclear.
|
math
|
1,005 |
Complemented subspaces of spaces obtained by interpolation
|
math.FA
|
If Z is a quotient of a subspace of a separable Banach space X, and V is any
separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0
and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using
the real or complex interpolation method contains a complemented subspace
isomorphic to Z. Thus many properties of Banach spaces, including having
non-trivial cotype, having the Radon-Nikodym property, and having the analytic
unconditional martingale difference sequence property, do not pass to
intermediate spaces.
|
math
|
1,006 |
Permutations of the Haar system
|
math.FA
|
General permutations acting on the Haar system are investigated. We give a
necessary and sufficient condition for permutations to induce an isomorphism on
dyadic BMO. Extensions of this characterization to Lipschitz spaces $\lip,
(0<p\leq1)$ are obtained. When specialized to permutations which act on one
level of the Haar system only, our approach leads to a short straightforward
proof of a result due to E.M.Semyonov and B.Stoeckert.
|
math
|
1,007 |
On the complemented subspaces of X_p
|
math.FA
|
In this paper we prove some results related to the problem of isomorphically
classifying the complemented subspaces of $X_{p}$. We characterize the
complemented subspaces of $X_{p}$ which are isomorphic to $X_{p}$ by showing
that such a space must contain a canonical complemented subspace isomorphic to
$X_{p}.$ We also give some characterizations of complemented subspaces of
$X_{p}$ isomorphic to $\ell_{p}\oplus \ell_{2}.$
|
math
|
1,008 |
p-summing operators on injective tensor products of spaces
|
math.FA
|
Let $X,Y$ and $Z$ be Banach spaces, and let $\prod_p(Y,Z) (1\leq p<\infty)$
denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$
is a {\it \$}$_\infty$-space, then a bounded linear operator $T: X\hat
\otimes_\epsilon Y\longrightarrow Z$ is 1-summing if and only if a naturally
associated operator $T^#: X\longrightarrow \prod_1(Y,Z)$ is 1-summing. This
result need not be true if $X$ is not a {\it \$}$_\infty$-space. For $p>1$,
several examples are given with $X=C[0,1]$ to show that $T^#$ can be
$p$-summing without $T$ being $p$-summing. Indeed, there is an operator $T$ on
$C[0,1]\hat \otimes_\epsilon \ell_1$ whose associated operator $T^#$ is
2-summing, but for all $N\in \N$, there exists an $N$-dimensional subspace $U$
of $C[0,1]\hat \otimes_\epsilon \ell_1$ such that $T$ restricted to $U$ is
equivalent to the identity operator on $\ell^N_\infty$. Finally, we show that
there is a compact Hausdorff space $K$ and a bounded linear operator $T:\
C(K)\hat \otimes_\epsilon \ell_1\longrightarrow \ell_2$ for which $T^#:\
C(K)\longrightarrow \prod_1(\ell_1, \ell_2)$ is not 2-summing.
|
math
|
1,009 |
Some deviation inequalities
|
math.FA
|
We introduce a concentration property for probability measures on
$\scriptstyle{R^n}$, which we call Property~($\scriptstyle\tau$); we show that
this property has an interesting stability under products and contractions
(Lemmas 1,~2,~3). Using property~($\scriptstyle\tau$), we give a short proof
for a recent deviation inequality due to Talagrand. In a third section, we also
recover known concentration results for Gaussian measures using our approach.}
|
math
|
1,010 |
On quotients of Banach spaces having shrinking unconditional bases
|
math.FA
|
It is proved that if a Banach space $Y$ is a quotient of a Banach space
having a shrinking unconditional basis, then every normalized weakly null
sequence in $Y$ has an unconditional subsequence. The proof yields the
corollary that every quotient of Schreier's space is $c_o$-saturated.
|
math
|
1,011 |
The proportional UAP characterizes weak Hilbert spaces
|
math.FA
|
We prove that a Banach space has the uniform approximation property with
proportional growth of the uniformity function iff it is a weak Hilbert space.
|
math
|
1,012 |
Comparison of Orlicz-Lorentz spaces
|
math.FA
|
Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and
Lorentz spaces. They have been studied by many authors, including Masty\l o,
Maligranda, and Kami\'nska. In this paper, we consider the problem of comparing
the Orlicz-Lorentz norms, and establish necessary and sufficient conditions for
them to be equivalent. As a corollary, we give necessary and sufficient
conditions for a Lorentz-Sharpley space to be equivalent to an Orlicz space,
extending results of Lorentz and Raynaud. We also give an example of a
rearrangement invariant space that is not an Orlicz-Lorentz space.
|
math
|
1,013 |
Non dentable sets in Banach spaces with separable dual
|
math.FA
|
A non RNP Banach space E is constructed such that $E^{*}$ is separable and
RNP is equivalent to PCP on the subsets of E.
|
math
|
1,014 |
Level sets and the uniqueness of measures
|
math.FA
|
A result of Nymann is extended to show that a positive $\sigma$-finite
measure with range an interval is determined by its level sets. An example is
given of two finite positive measures with range the same finite union of
intervals but with the property that one is determined by its level sets and
the other is not.
|
math
|
1,015 |
On Schreier unconditional sequences
|
math.FA
|
Let $(x_n)$ be a normalized weakly null sequence in a Banach space and let
$\varep>0$. We show that there exists a subsequence $(y_n)$ with the following
property: $$\hbox{ if }\ (a_i)\subseteq \IR\ \hbox{ and }\ F\subseteq \nat$$
satisfies $\min F\le |F|$ then $$\big\|\sum_{i\in F} a_i y_i\big\| \le
(2+\varep) \big\| \sum a_iy_i\big\|\ . $$
|
math
|
1,016 |
An arbitrarily distortable Banach space
|
math.FA
|
In this work we construct a ``Tsirelson like Banach space'' which is
arbitrarily distortable.
|
math
|
1,017 |
Interpolation of operators when the extreme spaces are $L^\infty$
|
math.FA
|
In this paper, equivalence between interpolation properties of linear
operators and monotonicity conditions are studied, for a pair $(X_0,X_1)$ of
rearrangement invariant quasi Banach spaces, when the extreme spaces of the
interpolation are $L^\infty$ and a pair $(A_0,A_1)$ under some assumptions.
Weak and restricted weak intermediate spaces fall in our context. Applications
to classical Lorentz and Lorentz-Orlicz spaces are given.
|
math
|
1,018 |
A simple proof of a theorem of Jean Bourgain
|
math.FA
|
We give a simple proof of Bourgain's disc algebra version of Grothendieck's
theorem, i.e. that every operator on the disc algebra with values in $L_1$ or
$L_2$ is 2-absolutely summing and hence extends to an operator defined on the
whole of $C$. This implies Bourgain's result that $L_1/H^1$ is of cotype 2. We
also prove more generally that $L_r/H^r$ is of cotype 2 for $0<r< 1$.
|
math
|
1,019 |
Interpolation between H^p spaces and non-commutative generalizations, I
|
math.FA
|
We give an elementary proof that the $H^p$ spaces over the unit disc (or the
upper half plane) are the interpolation spaces for the real method of
interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter
Jones. The proof uses only the boundedness of the Hilbert transform and the
classical factorisation of a function in $H^p$ as a product of two functions in
$H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra
difficulty to the non-commutative setting and to several Banach space valued
extensions of $H^p$ spaces. In particular, this proof easily extends to the
couple $H^{p_0}(\ell_{q_0}),H^{p_1}(\ell_{q_1})$, with $1\leq p_0, p_1, q_0,
q_1 \leq \infty$. In that situation, we prove that the real interpolation
spaces and the K-functional are induced ( up to equivalence of norms ) by the
same objects for the couple $L_{p_0}(\ell_{q_0}), L_{p_1}(\ell_{q_1})$. In
another direction, let us denote by $C_p$ the space of all compact operators
$x$ on Hilbert space such that $tr(|x|^p) <\infty$. Let $T_p$ be the subspace
of all upper triangular matrices relative to the canonical basis. If
$p=\infty$, $C_p$ is just the space of all compact operators. Our proof allows
us to show for instance that the space $H^p(C_p)$ (resp. $T_p$) is the
interpolation space of parameter $(1/p,p)$ between $H^1(C_1)$ (resp. $T_1$) and
$H^\infty(C_\infty)$ (resp. $T_\i$). We also prove a similar result for the
complex interpolation method. Moreover, extending a recent result of
Kaftal-Larson and Weiss, we prove that the distance to the subspace of upper
triangular matrices in $C_1$ and $C_\infty$ can be essentially realized
simultaneously by the same element.
|
math
|
1,020 |
Banach spaces with Property (w)
|
math.FA
|
A Banach space E is said to have Property (w) if every (bounded linear)
operator from E into E' is weakly compact. We give some interesting examples of
James type Banach spaces with Property (w). We also consider the passing of
Property (w) from E to C(K,E).
|
math
|
1,021 |
A Gordon-Chevet type Inequality
|
math.FA
|
We prove a new inequality for Gaussian processes, this inequality implies the
Gordon-Chevet inequality. Some remarks on Gaussian proofs of Dvoretzky's
theorem are given.
|
math
|
1,022 |
The K_t-functional for the interpolation couple L_1(A_0),L_infinity(A_1)
|
math.FA
|
Let (A_0,A_1) be a compatible couple of Banach spaces in the interpolation
theory sense. We give a formula for the K_t-functional of the interpolation
couples (l_1(A_0),c_0(A_1)) or (l_1(A_0),l_infinity(A_1)) and
(L_1(A_0),L_infinity(A_1)).
|
math
|
1,023 |
On J. Borwein's concept of sequentially reflexive Banach spaces
|
math.FA
|
A Banach space $X$ is reflexive if the Mackey topology $\tau(X^*,X)$ on $X^*$
agrees with the norm topology on $X^*$. Borwein [B] calls a Banach space $X$
{\it sequentially reflexive\/} provided that every $\tau(X^*,X)$ convergent
{\it sequence\/} in $X^*$ is norm convergent. The main result in [B] is that
$X$ is sequentially reflexive if every separable subspace of $X$ has separable
dual, and Borwein asks for a characterization of sequentially reflexive spaces.
Here we answer that question by proving
\proclaim Theorem. {\sl A Banach space $X$ is sequentially reflexive if and
only if $\ell_1$ is not isomorphic to a subspace of $X$.}
|
math
|
1,024 |
Analytic Disks in Fibers over the Unit Ball of a Banach Space
|
math.FA
|
We study biorthogonal sequences with special properties, such as weak or
weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig
theorem. This result is applied to embed analytic disks in the fiber over 0 of
the spectrum of H^infinity (B), the algebra of bounded analytic functions on
the unit ball B of an arbitrary infinite dimensional Banach space. Various
other embedding theorems are obtained. For instance, if the Banach space is
superreflexive, then the unit ball of a Hilbert space of uncountable dimension
can be embedded analytically in the fiber over 0 via an embedding which is
uniformly bicontinuous with respect to the Gleason metric.
|
math
|
1,025 |
On the distribution of Sidon series
|
math.FA
|
Let B denote an arbitrary Banach space, G a compact abelian group with Haar
measure $\mu$ and dual group $\Gamma$. Let E be a Sidon subset of $\Gamma$ with
Sidon constant S(E). Let r_n denote the n-th Rademacher function on [0, 1]. We
show that there is a constant c, depending only on S(E), such that, for all
$\alpha > 0$: c^{-1}P[| \sum_{n=1}^Na_nr_n| >= c \alpha ] <= \mu[|
\sum_{n=1}^Na_n\gamma_n| >= \alpha ] <= cP [|\sum_{n=1}^Na_nr_n| >= c^{-1}
\alpha ]
|
math
|
1,026 |
On certain classes of Baire-1 functions with applications to Banach space theory
|
math.FA
|
Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric
space $K$, are defined and characterized. Some applications to Banach spaces
are given.
|
math
|
1,027 |
Isomorphisms of certain weak L^p spaces
|
math.FA
|
It is shown that the weak $L^p$ spaces $\ell^{p,\infty}, L^{p,\infty}[0,1]$,
and $L^{p,\infty}[0,\infty)$ are isomorphic as Banach spaces.
|
math
|
1,028 |
On the integration of vector-valued functions
|
math.FA
|
We discuss relationships between the McShane, Pettis, Talagrand and Bochner
integrals. A large number of different methods of integration of
Banach-space-valued functions have been introduced, based on the various
possible constructions of the Lebesgue integral. They commonly run fairly
closely together when the range space is separable (or has w^*-separable dual)
and diverge more or less sharply for general range spaces. The McShane integral
as described by [Go] is derived from the `gauge-limit' integral of [McS]. Here
we give both positive and negative results concerning it and the other three
integrals listed above.
|
math
|
1,029 |
Lower estimates of random unconditional constants of Walsh-Paley martingales with values in banach spaces
|
math.FA
|
For a Banach space X we define RUMD_n(X) to be the infimum of all c>0 such
that (AVE_{\epsilon_k =\pm 1} || \sum_1^n epsilon_k (M_k - M_{k-1}
)||_{L_2^X}^2 )^{1/2} <= c || M_n ||_{L_2^X} holds for all Walsh-Paley
martingales {M_k}_0^n subset L_2^X with M_0 =0. We relate the asymptotic
behaviour of the sequence {RUMD(X)}_{n=1}^{infinity} to geometrical properties
of the Banach space X such as K-convexity and superreflexivity.
|
math
|
1,030 |
Complexity of weakly null sequences
|
math.FA
|
We introduce an ordinal index which measures the complexity of a weakly null
sequence, and show that a construction due to J. Schreier can be iterated to
produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in
C(omega^{omega^{alpha}})) with complexity alpha. As in the Schreier example
each of these is a sequence of indicator functions which is a suppression-1
unconditional basic sequence. These sequences are used to construct
Tsirelson-like spaces of large index. We also show that this new ordinal index
is related to the Lavrentiev index of a Baire-1 function and use the index to
sharpen some results of Alspach and Odell on averaging weakly null sequences.
|
math
|
1,031 |
Structure of local Banach spaces of locally convex spaces
|
math.FA
|
We show that a continuous bilinear mapping P: C(I) \times C(I) \to C(I) can
be presented in the form P(f,g) = B((Af)(Ag)), where A and B are bounded linear
operators on C(I) and multiplication is defined pointwise, if and only if for
all t in I the bilinear form (f,g) -> P(f,g)(t) is integral on C(I) times C(I)
and depends in a sense continuously on t. To this end we construct a continuous
surjection phi : I \to I^2 admitting a regular averaging operator in the sense
of Pelczynski.
|
math
|
1,032 |
The volume of the intersection of a convex body with its translates
|
math.FA
|
It is proved that for a symmetric convex body K in R^n, if for some tau > 0,
|K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of
the proof, smoothness properties of convolution bodies ls are studied.
|
math
|
1,033 |
The Distorion Problem
|
math.FA
|
We prove that Hilbert space is distortable and, in fact, arbitrarily
distortable. This means that for all lambda >1 there exists an equivalent norm
|.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there
exist x,y in Y with ||x||_2 = ||y||_2 =1 yet |x| >lambda |y|.
We also prove that if X is any infinite dimensional Banach space with an
unconditional basis then the unit sphere of X and the unit sphere of l_1 are
uniformly homeomorphic if and only if X does not contain l_infty^n's uniformly.
|
math
|
1,034 |
A Note on Unconditional Structures in Weak Hilbert Spaces
|
math.FA
|
We prove that if a non-atomic separable Banach lattice in a weak Hilbert
space, then it is lattice isomorphic to $L_2(0,1)$.
|
math
|
1,035 |
A l_1-predual which is not isometric to a quotient of C(alpha)
|
math.FA
|
About twenty years ago Johnson and Zippin showed that every separable
L_1(mu)-predual was isometric to a quotient of C(Delta ), where Delta is the
Cantor set. In this note we will show that the natural analogue of the theorem
for l_1-preduals does not hold. We will show that there are many l_1-preduals
which are not isometric to a quotient of any C(K)-space with K a countable
compact metric space. We also prove some general results about the relationship
between l_1-preduals and quotients of C(K)-spaces with K a countable compact
metric space.
The results in this paper were presented at the Workshop on Banach Space
Theory in Merida, Venezuela, January 1992.
|
math
|
1,036 |
Jean Bourgain's analytic partition of unity via holomorphic martingales
|
math.FA
|
Using stopping time arguments on holomorphic martingales we present a soft
way of constructing J. Bourgain's analytic partitions of unity. Applications to
Marcinkiewicz interploation in weighted Hardy spaces are discussed.
|
math
|
1,037 |
The Compact Approximation Property does not imply the Approximation Property
|
math.FA
|
It is shown how to construct, given a Banach space which does not have the
approximation property, another Banach space which does not have the
approximation property but which does have the compact approximation property.
|
math
|
1,038 |
The unconditional basic sequence problem
|
math.FA
|
We construct a Banach space that does not contain any infinite unconditional
basic sequence.
|
math
|
1,039 |
On $c_0$-saturated Banach spaces
|
math.FA
|
A Banach space E is c_0-saturated if every closed infinite dimensional
subspace of E contains an isomorph of c_0. A c_0-saturated Banach space with an
unconditional basis which has a quotient space isomorphic to l^2 is
constructed.
|
math
|
1,040 |
Set-functions and factorization
|
math.FA
|
If $\phi$ is a submeasure satisfying an appropriate lower estimate we give a
quantitative result on the total mass of a measure $\mu$ satisfying
$0\le\mu\le\phi.$ We give a dual result for supermeasures and then use these
results to investigate convexity on non-locally convex quasi-Banach lattices.
We then show how to use these results to extend some factorization theorems due
to Pisier to the setting of quasi-Banach spaces. We conclude by showing that if
$X$ is a quasi-Banach space of cotype two then any operator $T:C(\Omega)\to X$
is 2-absolutely summing and factors through a Hilbert space and discussing
general factorization theorems for cotype two spaces.
|
math
|
1,041 |
Some Questions Arising from the Homogeneous Banach Space Problem
|
math.FA
|
We review the current state of the homogeneous Banach space problem. We then
formulate several questions which arise naturally from this problem, some of
which seem to be fundamental but new. We give many examples defining the bounds
on the problem. We end with a simple construction showing that every infinite
dimensional Banach space contains a subspace on which weak properties have
become stable (under passing to further subspaces). Implications of this
construction are considered.
|
math
|
1,042 |
The distribution of vector-valued Rademacher series
|
math.FA
|
Let $X=\sum \epsilon_n x_n$ be a Rademacher series with vector-valued
coefficients. We obtain an approximate formula for the distribution of the
random variable $||X||$ in terms of its mean and a certain quantity derived
from the K-functional of interpolation theory. Several applications of the
formula are given.
|
math
|
1,043 |
On nonatomic Banach lattices and Hardy spaces
|
math.FA
|
We are interested in the question when a Banach space $X$ with an
unconditional basis is isomorphic (as a Banach space) to an order-continuous
nonatomic Banach lattice. We show that this is the case if and only if $X$ is
isomorphic as a Banach space with $X(\ell_2)$. This and results of J. Bourgain
are used to show that spaces $H_1(\bold T^n)$ are not isomorphic to nonatomic
Banach lattices. We also show that tent spaces introduced in \cite{4} are
isomorphic to Rad $H_1$.
|
math
|
1,044 |
More smoothly real compact spaces
|
math.FA
|
A topological space $X$ is called $\Cal A$-real compact, if every algebra
homomorphism from $\Cal A$ to the reals is an evaluation at some point of $X$,
where $\Cal A$ is an algebra of continuous functions. Our main interest lies on
algebras of smooth functions. In \cite{AdR} it was shown that any separable
Banach space is smoothly real compact. Here we generalize this result to a huge
class of locally convex spaces including arbitrary products of separable
Fr\'echet spaces.
|
math
|
1,045 |
Banach Spaces Of The Type Of Tsirelson
|
math.FA
|
To any pair ( M , theta ) where M is a family of finite subsets of N compact
in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type
Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is
greater than n and theta >{1/n} then T_M^theta is reflexive. Moreover, if the
Cantor-Bendixson index of M is greater than omega then T_M^theta does not
contain any l^p, while if the Cantor-Bendixson index of M is finite
thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N :
|A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p .
|
math
|
1,046 |
On Weakly Null FDD's in Banach Spaces
|
math.FA
|
In this paper we show that every sequence (F_n) of finite dimensional
subspaces of a real or complex Banach space with increasing dimensions can be
``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so
that either every bounded sequence (x_n), with x_n in G_n for n in N, is weakly
null, or every normalized sequence (x_n), with x_n in G_n for n in N, is
equivalent to the unit vector basis of l_1.
Crucial to the proof are two stabilization results concerning Lipschitz
functions on finite dimensional normed spaces. These results also lead to other
applications. We show, for example, that every infinite dimensional Banach
space X contains an F.D.D. (F_n), with lim_{n to infty} dim (F_n)=infty, so
that all normalized sequences (x_n), with x_n in F_n, n in N, have the same
spreading model over X. This spreading model must necessarily be
1-unconditional over X.
|
math
|
1,047 |
On Uniform Homeomorphisms of the Unit Spheres of Certain Banach Lattices
|
math.FA
|
We prove that if X is an infinite dimensional Banach lattice with a weak unit
then there exists a probability space (Omega, Sigma,mu) so that the unit sphere
S(L_1(Omega, Sigma, mu) is uniformly homeomorphic to the unit sphere S(X) if
and only if X does not contain l_{infty}^n's uniformly.
|
math
|
1,048 |
Vector-valued L_p convergence of orthogonal series and Lagrange interpolation
|
math.FA
|
We give necessary and sufficient conditions for interpolation inequalities of
the type considered by Marcinkiewicz and Zygmund to be true in the case of
Banach space-valued polynomials and Jacobi weights and nodes. We also study the
vector-valued expansion problem of $L_p$-functions in terms of Jacobi
polynomials and consider the question of unconditional convergence. The notion
of type $p$ with respect to orthonormal systems leads to some characterizations
of Hilbert spaces. It is also shown that various vector-valued Jacobi means are
equivalent.
|
math
|
1,049 |
Vector-valued Lagrange interpolation and mean convergence of Hermite series
|
math.FA
|
Let X be a Banach space and $1\le p<\infty$. We prove interpolation
inequalities of Marcinkiewicz-Zygmund type for X-valued polynomials g of degree
$\le n$ on $R$,
\[c_p (\sum\limits_{i=1}^{n+1} \mu_i \| g(t_i)e^{-t_i^2 /2} \|^p)^{1/p} \le
(\int\limits_{\RR}^{} \|g(t)e^{-t^2 /2} \|^p dt)^{1/p} \le d_p
(\sum\limits_{i=1}^{n+1} \mu_i \|g(t_i)e^{-t_i^2 /2} \|^p)^{1/p}\;\;,\]
where $(t_i)_1^{n+1}$ are the zeros of the Hermite polynomial $H_{n+1}$ and
$(\mu_i)_1^{n+1}$ are suitable weights. The validity of the right inequality
requires $1<p<4$ and X being a UMD-space. This implies a mean convergence
theorem for the Lagrange interpolation polynomials of continuous functions on
$R$ taken at the zeros of the Hermite polynomials. In the scalar case, this
improves a result of Nevai $[$N$]$. Moreover, we give vector-valued extensions
of the mean convergence results of Askey-Wainger $[$AW$]$ in the case of
Hermite expansions.
|
math
|
1,050 |
Amenability of Banach algebras of compact operators
|
math.FA
|
In this paper we study conditions on a Banach space X that ensure that the
Banach algebra K(X) of compact operators is amenable. We give a symmetrized
approximation property of X which is proved to be such a condition. This
property is satisfied by a wide range of Banach spaces including all the
classical spaces. We then investigate which constructions of new Banach spaces
from old ones preserve the property of carrying amenable algebras of compact
operators. Roughly speaking, dual spaces, predual spaces and certain tensor
products do inherit this property and direct sums do not. For direct sums this
question is closely related to factorization of linear operators. In the final
section we discuss some open questions, in particular, the converse problem of
what properties of X are implied by the amenability of K(X).
|
math
|
1,051 |
The Distribution of Non-Commutative Rademacher Series
|
math.FA
|
We give a formula for the tail of the distribution of the non-commutative
Rademacher series, which generalizes the result that is already available in
the commutative case. As a result, we are able to calculate the norm of these
series in many rearrangement invariant spaces, generalizing work of Pisier and
Rodin and Semyonov.
|
math
|
1,052 |
The theorems of Caratheodory and Gluskin for $0<p<1$
|
math.FA
|
In this note we investigate some aspects of the local structure of finite
dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp
lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved
that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is
to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper
bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional
$p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show
that such bound is optimum.
|
math
|
1,053 |
Asymptotic $l_p$ spaces and bounded distortions
|
math.FA
|
The new class of Banach spaces, so-called asymptotic $l_p$ spaces, is
introduced and it is shown that every Banach space with bounded distortions
contains a subspace from this class.
The proof is based on an investigation of certain functions, called
enveloping functions, which are intimately connected with stabilization
properties of the norm.
|
math
|
1,054 |
Computing p-summing norms with few vectors
|
math.FA
|
It is shown that the p-summing norm of any operator with n-dimensional domain
can be well-aproximated using only ``few" vectors in the definition of the
p-summing norm. Except for constants independent of n and log n factors, ``few"
means n if 1<p<2 and n^{p/2} if 2<p<infinity.
|
math
|
1,055 |
On vector-valued inequalities for Sidon sets and sets of interpolation
|
math.FA
|
Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space.
Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave
like Rademacher sums with respect to $L_p-$norms. We consider the situation
when $X$ is a quasi-Banach space. For general quasi-Banach spaces we show that
a similar result holds if and only if $E$ is a set of interpolation
($I_0$-set). However for certain special classes of quasi-Banach spaces we are
able to prove such a result for larger sets. Thus if $X$ is restricted to be
``natural'' then the result holds for all Sidon sets. We also consider spaces
with plurisubharmonic norms and introduce the class of analytic Sidon sets.
|
math
|
1,056 |
Calderón couples of re-arrangement invariant spaces
|
math.FA
|
We examine conditions under which a pair of re-arrangement invariant function
spaces on $[0,1]$ or $[0,\infty)$ form a Calder\'on couple. A very general
criterion is developed to determine whether such a pair is a Calder\'on couple,
with numerous applications. We give, for example, a complete classification of
those spaces $X$ which form a Calder\'on couple with $L_{\infty}.$ We
specialize our results to Orlicz spaces and are able to give necessary and
sufficient conditions on an Orlicz function $F$ so that the pair
$(L_F,L_{\infty})$ forms a Calder\'on pair.
|
math
|
1,057 |
A characterization of Banach spaces containing $c_0$
|
math.FA
|
A subsequence principle is obtained, characterizing Banach spaces containing
$c_0$, in the spirit of the author's 1974 characterization of Banach spaces
containing $\ell^1$.
Definition: A sequence $(b_j)$ in a Banach space is called {\it strongly
summing\/} (s.s.) if $(b_j)$ is a weak-Cauchy basic sequence so that whenever
scalars $(c_j)$ satisfy $\sup_n \|\sum_{j=1}^n c_j b_j\| <\infty$, then $\sum
c_j$ converges.
A simple permanence property: if $(b_j)$ is an (s.s.) basis for a Banach
space $B$ and $(b_j^*)$ are its biorthogonal functionals in $B^*$, then
$(\sum_{j=1}^n b_j^*)_{n=1}^ \infty$ is a non-trivial weak-Cauchy sequence in
$B^*$; hence $B^*$ fails to be weakly sequentially complete. (A weak-Cauchy
sequence is called {\it non-trivial\/} if it is {\it non-weakly convergent\/}.)
Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach
space has either an {\rm (s.s.)} subsequence, or a convex block basis
equivalent to the summing basis.
Remark : The two alternatives of the Theorem are easily seen to be mutually
exclusive.
Corollary 1. A Banach space $B$ contains no isomorph of $c_0$ if and only if
every non-trivial weak-Cauchy sequence in $B$ has an {\rm (s.s.)} subsequence.
Combining the $c_0$ and $\ell^1$ Theorems, we obtain
Corollary 2. If $B$ is a non-reflexive Banach space such that $X^*$ is weakly
sequentially complete for all linear subspaces $X$ of $B$, then $c_0$ embeds in
$X$; in fact, $B$ has property~$(u)$.
|
math
|
1,058 |
Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre
|
math.FA
|
Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to
Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the
question whether the operator $T:[X_0,X_1]_{\theta}\to [Y_0,Y_1]_{\theta}$ is
compact and show a positive answer under a variety of conditions. For example
it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a
Banach space so that $X_0=[W,X_1]_{\alpha}$ for some $0<\alpha<1.$
|
math
|
1,059 |
Schoenberg's Problem on Positive Definite Functions
|
math.FA
|
If $n \ge 3$, $q>2$ and $\beta > 0$ then the function
$\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{\beta/q})$\ is not positive definite.
This result gives an answer to a question posed by I.J.~Schoenberg in 1938.
This text is an authorized English translation of the paper published in
Russian in Algebra and Analysis 3(1991), \#3, p.78--85.
|
math
|
1,060 |
Mean Convergence of Vector--valued Walsh Series
|
math.FA
|
Given any Banach space $X$, let $L_2^X$ denote the Banach space of all
measurable functions $f:[0,1]\to X$ for which
||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2}
is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only
if
\lim_n||f-S_n(f)||_2=0 for all $f\in L_2^X$, where
S_n(f):=sum_{i=0}^{n-1} (f,w_i)w_i
is the $n$--th partial sum associated with the Walsh system $(w_i)$.
|
math
|
1,061 |
Two Remarks on Marcinkiewicz decompositions by Holomorphic Martingales
|
math.FA
|
The real part of $H^\infty(\bT)$ is not dense in $L^\infty_{\tR}(\bT)$. The
John-Nirenberg theorem in combination with the Helson-Szeg\"o theorem and the
Hunt Muckenhaupt Wheeden theorem has been used to determine whether $f\in
L^\infty_{\tR}(\bT)$ can be approximated by $\Re H^\infty(\bT)$ or not:
$\dist(f,\Re H^\infty)=0$ if and only if for every $\e>0$ there exists $\l_0>0$
so that for $\l>\l_0$ and any interval $I\sbe \bT$. $$|\{x\in I:|\tilde
f-(\tilde f)_I|>\l\}|\le |I|e^{-\l/ \e},$$ where $\tilde f$ denotes the Hilbert
transform of $f$. See [G] p. 259. This result is contrasted by the following
\begin{theor} Let $f\in L^\infty_{\tR}$ and $\e>0$. Then there is a function
$g\in H^\infty(\bT)$ and a set $E\sb \bT$ so that $|\bT\sm E|<\e$ and $$f=\Re
g\quad\mbox{ on } E.$$ \end{theor}
This theorem is best regarded as a corollary to Men'shov's correction
theorem. For the classical proof of Men'shov's theorem see [Ba, Ch VI \S
1-\S4].
Simple proofs of Men'shov's theorem -- together with significant extensions
-- have been obtained by S.V. Khruschev in [Kh] and S.V. Kislyakov in [K1],
[K2] and [K3].
In [S] C. Sundberg used $\bar\pa$-techniques (in particular [G, Theorem
VIII.1. gave a proof of Theorem 1 that does not mention Men'shov's theorem.
The purpose of this paper is to use a Marcinkiewicz decomposition on
Holomorphic Martingales to give another proof of Theorem 1. In this way we
avoid uniformly convergent Fourier series as well as $\bar\pa$-techniques.
|
math
|
1,062 |
Weakly Lindelof determined Banach spaces not containing $\ell^1(N)$
|
math.FA
|
The class of countably intersected families of sets is defined. For any such
family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain
counterexamples to certain questions related to the heredity problem for W.C.G.
Banach spaces. Among them we give a subspace of a W.C.G. Banach space not
containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
|
math
|
1,063 |
Unrestricted products of contractions in Banach spaces
|
math.FA
|
Let $X$ be a reflexive Banach space such that for any $x \ne 0$ the set $$
\{x^* \in X^*: \text {$\|x^*\|=1$ and $x^*(x)=\|x\|$}\} $$ is compact. We prove
that any unrestricted product of of a finite number of $(W)$ contractions on
$X$ converges weakly.
|
math
|
1,064 |
Factorization theorems for quasi-normed spaces
|
math.FA
|
We extend Pisier's abstract version of Grothendieck's theorem to general
non-locally convex quasi-Banach spaces. We also prove a related result on
factoring operators through a Banach space and apply our results to the study
of vector-valued inequalities for Sidon sets. We also develop the local theory
of (non-locally convex) spaces with duals of weak cotype 2.
|
math
|
1,065 |
Surjective isometries on rearrangement-invariant spaces
|
math.FA
|
We prove that if $X$ is a real rearrangement-invariant function space on
$[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective
isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel
function $a$ and an invertible Borel map $\sigma:[0,1] \to [0,1].$ If $X$ is
not equal to $L_p$, up to renorming, for some $1\le p\le \infty$ then in
addition $|a|=1$ a.e. and $\sigma$ must be measure-preserving.
|
math
|
1,066 |
Common subspaces of $L_{p}$-spaces
|
math.FA
|
For $n\geq 2, p<2$ and $q>2,$ does there exist an $n$-dimensional Banach
space different from Hilbert spaces which is isometric to subspaces of both
$L_{p}$ and $L_{q}$? Generalizing the construction from the paper "Zonoids
whose polars are zonoids" by R.Schneider we give examples of such spaces.
Moreover, for any compact subset $Q$ of $(0,\infty)\setminus \{2k, k\in N\},$
we can construct a space isometric to subspaces of $L_{q}$ for all $q\in Q$
simultaneously.
This paper requires vanilla.sty
|
math
|
1,067 |
Polynomial Schur and Polynomial Dunford-Pettis Properties
|
math.FA
|
A Banach space is {\it polynomially Schur} if sequential convergence against
analytic polynomials implies norm convergence. Carne, Cole and Gamelin show
that a space has this property and the Dunford-Pettis property if and only if
it is Schur. Herein is defined a reasonable generalization of the
Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown,
for example, that a Banach space will has the $P_N$ Dunford--Pettis property if
and only if every weakly compact $N-$homogeneous polynomial (in the sense of
Ryan) on the space is completely continuous. A certain geometric condition,
involving estimates on spreading models and implied by nontrivial type, is
shown to be sufficient to imply that a space is polynomially Schur.
|
math
|
1,068 |
Norms of Minimal Projections
|
math.FA
|
It is proved that the projection constants of two- and three-dimensional
spaces are bounded by $4/3$ and $(1+\sqrt 5)/2$, respectively. These bounds are
attained precisely by the spaces whose unit balls are the regular hexagon and
dodecahedron. In fact, a general inequality for the projection constant of a
real or complex $n$-dimensional space is obtained and the question of equality
therein is discussed.
|
math
|
1,069 |
Infinite order decoupling of random chaoses in Banach space
|
math.FA
|
We prove a number of decoupling inequalities for nonhomogeneous random
polynomials with coefficients in Banach space. Degrees of homogeneous
components enter into comparison as exponents of multipliers of terms of
certain Poincar\'e-type polynomials. It turns out that the fulfillment of most
of types of decoupling inequalities may depend on the geometry of Banach space.
|
math
|
1,070 |
Operators preserving orthogonality are isometries
|
math.FA
|
Let $E$ be a real Banach space. For $x,y \in E,$ we follow R.James in saying
that $x$ is orthogonal to $y$ if $\|x+\alpha y\|\geq \|x\|$ for every $\alpha
\in R$. We prove that every operator from $E$ into itself preserving
orthogonality is an isometry multiplied by a constant.
|
math
|
1,071 |
Interpolation Between $H^p$ Spaces and Non-Commutative Generalizations II
|
math.FA
|
We continue an investigation started in a preceding paper. We discuss the
classical results of Carleson connecting Carleson measures with the
$\d$-equation in a slightly more abstract framework than usual. We also
consider a more recent result of Peter Jones which shows the existence of a
solution of the $\d$-equation, which satisfies simultaneously a good $L_\i$
estimate and a good $L_1$ estimate. This appears as a special case of our main
result which can be stated as follows: Let $(\Omega,\cal{A},\mu)$ be any
measure space. Consider a bounded operator $u:H^1\ra L_1(\mu)$. Assume that on
one hand $u$ admits an extension $u_1:L^1\ra L_1(\mu)$ bounded with norm $C_1$,
and on the other hand that $u$ admits an extension $u_\i:L^\i\ra L_\i(\mu)$
bounded with norm $C_\i$. Then $u$ admits an extension $\w{u}$ which is bounded
simultaneously from $L^1$ into $ L_1(\mu)$ and from $L^\i$ into $ L_\i(\mu)$
and satisfies $$\eqalign{&\|\tilde u\colon \ L_\infty \to L_\infty(\mu)\|\le
CC_\infty\cr &\|\tilde u\colon \ L_1\to L_1(\mu)\|\le CC_1}$$ where $C$ is a
numerical constant.
|
math
|
1,072 |
On the ``local theory'' of operator spaces
|
math.FA
|
In Banach space theory, the ``local theory'' refers to the collection of
finite dimensional methods and ideas which are used to study infinite
dimensional spaces (see e.g. [P4,TJ]). It is natural to try to develop an
analogous theory in the recently developed category of operator spaces
[BP,B1-2,BS,ER1-7,Ru]. The object of this paper is to start such a theory. We
plan to present a more thorough discussion of the associated tensor norms in a
future publication.
|
math
|
1,073 |
Sur les opérateurs factorisables par $OH$
|
math.FA
|
Let $H,K$ be Hilbert spaces. Let $E \subset B(H)$ and $F \subset B(K)$ be
operator spaces in the sense of [1,2]. We study the operators $u : E \to F$
which admit a factorization $E \to OH \to F$ with completely bounded maps
through the operator Hilbert space $OH$ which we have introduced and studied in
a recent note. We give a characterization of these operators which allows to
develop a theory entirely analogous to that of operators between Banach spaces
which can be factored through a Hilbert space.
|
math
|
1,074 |
Multipliers and lacunary sets in non-amenable groups
|
math.FA
|
Let $G$ be a discrete group.
Let $\lambda : G \to B(\ell_2(G),\ell_2(G))$ be the left regular
representation. A function $\ph : G \to \comp$ is called a completely bounded
multiplier (= Herz-Schur multiplier) if the transformation defined on the
linear span $K(G)$ of $\{\lambda(x),x \in G\}$ by $$\sum_{x \in G} f(x)
\lambda(x) \to \sum_{x \in G} f(x) \ph(x) \lambda(x)$$ is completely bounded
(in short c.b.) on the $C^*$-algebra $C_\lambda^*(G)$ which is generated by
$\lambda$ ($C_\lambda^*(G)$ is the closure of $K(G)$ in
$B(\ell_2(G),\ell_2(G))$.) One of our main results gives a simple
characterization of the functions $\ph$ such that $\eps \ph$ is a c.b.
multiplier on $C_\lambda^*(G)$ for any bounded function $\eps$, or equivalently
for any choice of signs $\eps(x) = \pm 1$. We also consider the case when this
holds for ``almost all" choices of signs.
|
math
|
1,075 |
Espace de Hilbert d'opérateurs et Interpolation complexe
|
math.FA
|
Let $H$ be an infinite dimensional Hilbert space. We show that there exists a
subspace of $B(H)$ which is isometric to $\ell_2$ and completely isometric to
its antidual in the sense of the theory of operator spaces recently developed
by Blecher-Paulsen and Effros-Ruan. Moreover this space is unique up to a
complete isometry. We denote it by $OH$. This space has several remarkable
properties in particular with respect to the complex interpolation method.
|
math
|
1,076 |
A Uniform Kadec-klee Property For Symmetric Operator Spaces
|
math.FA
|
We show that if a rearrangement invariant Banach function space $E$ on the
positive semi-axis satisfies a non-trivial lower $q-$ estimate with constant
$1$ then the corresponding space $E(\nm)$ of $\tau-$measurable operators,
affiliated with an arbitrary semi-finite von Neumann algebra $\nm$ equipped
with a distinguished faithful, normal, semi-finite trace $\tau $, has the
uniform Kadec-Klee property for the topology of local convergence in measure.
In particular, the Lorentz function spaces $L_{q,p}$ and the Lorentz-Schatten
classes ${\cal C}_{q,p}$ have the UKK property for convergence locally in
measure and for the weak-operator topology, respectively. As a partial converse
, we show that if $E$ has the UKK property with respect to local convergence in
measure then $E$ must satisfy some non-trivial lower $q$-estimate. We also
prove a uniform Kadec-Klee result for local convergence in any Banach lattice
satisfying a lower $q$-estimate.
|
math
|
1,077 |
The Complete Continuity Property and Finite Dimensional Decompositions
|
math.FA
|
A Banach space $\X$ has the complete continuity property (CCP) if each
bounded linear operator from $L_1$ into $\X$ is completely continuous (i.e.,
maps weakly convergent sequences to norm convergent sequences). The main
theorem shows that a Banach space failing the CCP (resp., failing the CCP and
failing cotype) has a subspace with a finite dimensional decomposition (resp.,
basis) which fails the CCP.
|
math
|
1,078 |
Twisted sums and a problem of Klee
|
math.FA
|
Let F be a quasi-linear map on a separable normed space X, and assume that F
splits on an infinite-dimensional subspace of X. Then the twisted sum topology
induced by F on the direct sum of X and the real line can be written as the
supremum of a nearly convex topology and a trivial dual topology. (This
partially answers a question of Klee.) The result applies when X is \ell_1 and
F is the Ribe function or when X is James's space.
|
math
|
1,079 |
Comparing gaussian and Rademacher cotype for operators on the space of continous functions
|
math.FA
|
We will prove an abstract comparision principle which translates gaussian
cotype in Rademacher cotype conditions and vice versa. More precisely, let
$2\!<\!q\!<\!\infty$ and $T:\,C(K)\,\to\,F$ a linear, continous operator.
T is of gaussian cotype q if and only if
( \summ_1^n (\frac{|| Tx_k||_F}{\sqrt{\log(k+1)}})^q )^{1/q} \, \le c ||
\summ_1^n \varepsilon_k x_k ||_{L_2(C(K))} ,
for all sequences with $(|| Tx_k ||)_1^n$ decreasing.
T is of Rademacher cotype q if and only if
(\summ_1^n (|| Tx_k||_F \,\sqrt{\log(k+1)})^q )^{1/q} \, \le c || \summ_1^n
g_k x_k ||_{L_2(C(K))} ,
for all sequences with $(||Tx_k ||)_1^n$ decreasing.
Our methods allows a restriction to a fixed number of vectors and complements
the corresponding results of Talagrand.
|
math
|
1,080 |
How many vectors are needed to compute (p,q)-summing norms?
|
math.FA
|
We will show that for $q<p$ there exists an $\al < \infty$ such that \[
\pi_{pq}(T) \pl \le c_{pq} \pi_{pq}^{[n^{\alpha}]}(T) \mbox{for all $T$ of rank
$n$.}\] Such a polynomial number is only possible if $q=2$ or $q<p$.
Furthermore, the growth rate is linear if $q=2$ or
$\frac{1}{q}-\frac{1}{p}>\frac{1}{2}$. Unless
$\frac{1}{q}-\frac{1}{p}=\frac{1}{2}$ this is also a necessary condition .
|
math
|
1,081 |
Every nonreflexive subspace of L_1[0,1] fails the fixed point property
|
math.FA
|
The main result of this paper is that every non-reflexive subspace $Y$ of
$L_1[0,1]$ fails the fixed point property for closed, bounded, convex subsets
$C$ of $Y$ and nonexpansive (or contractive) mappings on $C$. Combined with a
theorem of Maurey we get that for subspaces $Y$ of $L_1[0,1]$, $Y$ is reflexive
if and only if $Y$ has the fixed point property. For general Banach spaces the
question as to whether reflexivity implies the fixed point property and the
converse question are both still open.
|
math
|
1,082 |
Lectures on maximal monotone operators
|
math.FA
|
This is a 30 page set of lecture notes, in Plain TeX, which were prepared for
and presented as a series of lectures (10 1/2 hours over two weeks) at the 2nd
Summer School on Banach Spaces, Related Areas and Applications in Prague and
Paseky, Czech Republic, during August, 1993. They consist of a largely
self-contained exposition of both classical and recent basic facts about
maximal monotone operators on Banach spaces, motivated in part by the goal of
highlighting several fundamental properties of such operators which remain open
questions in nonreflexive Banach spaces.
|
math
|
1,083 |
Locally Lipschitz Functions and Bornological Derivatives
|
math.FA
|
We study the relationships between Gateaux, weak Hadamard and Frechet
differentiability and their bornologies for Lipschitz and for convex functions.
In particular, Frechet and weak Hadamard differentiabily coincide for all
Lipschitz functions if and only if the space is reflexive (an earlier paper of
the first two authors shows that these two notions of differentiability
coincide for continuous convex functions if and only if the space does not
contain a copy of $\ell_1$). We also examine when Gateaux and weak Hadamard
differentiability coincide for continuous convex functions. For instance,
spaces with the Dunford-Pettis (Schur) property can be characterized by the
coincidence of Gateaux and weak Hadamard (Frechet) differentiabilty for dual
norms.
|
math
|
1,084 |
Dual Kadec-Klee norms and the relationships between Wijsman, slice and Mosco convergence
|
math.FA
|
In this paper, we completely settle several of the open questions regarding
the relationships between the three most fundamental forms of set convergence.
In particular, it is shown that Wijsman and slice convergence coincide
precisely when the weak star and norm topologies agree on the dual sphere.
Consequently, a weakly compactly generated Banach space admits a dense set of
norms for which Wijsman and slice convergence coincide if and only if it is an
Asplund space. We also show that Wijsman convergence implies Mosco convergence
precisely when the weak star and Mackey topologies coincide on the dual sphere.
A corollary of these results is that given a fixed norm on an Asplund space,
Wijsman and slice convergence coincide if and only if Wijsman convergence
implies Mosco convergence.
|
math
|
1,085 |
A factorization constant for $l^n_p
|
math.FA
|
We prove that if PT is a factorization of the identity operator on \ell_p^n
through \ell_{\infty}^k, then ||P|| ||T|| \geq Cn^{1/p-1/2}(log n)^{-1/2}. This
is a corollary of a more general result on factoring the identity operator on a
quasi-normed space through \ell_{\infty}^k.
|
math
|
1,086 |
Bounded linear operators between C^*-algebras
|
math.FA
|
Let $u:A\to B$ be a bounded linear operator between two $C^*$-algebras $A,B$.
The following result was proved by the second author.
Theorem 0.1. There is a numerical constant $K_1$ such that for all finite
sequences $x_1,\ldots, x_n$ in $A$ we have
$$\leqalignno{&\max\left\{\left\|\left(\sum u(x_i)^*
u(x_i)\right)^{1/2}\right\|_B, \left\|\left(\sum u(x_i)
u(x_i)^*\right)^{1/2}\right\|_B\right\}&(0.1)_1\cr \le &K_1\|u\|
\max\left\{\left\|\left(\sum x^*_ix_i\right)^{1/2}\right\|_A, \left\|\left(\sum
x_ix^*_i\right)^{1/2}\right\|_A\right\}.}$$
A simpler proof was given in [H1]. More recently an other alternate proof
appeared in [LPP]. In this paper we give a sequence of generalizations of this
inequality.
|
math
|
1,087 |
Projections from a von~Neumann algebra onto a subalgebra
|
math.FA
|
This paper is mainly devoted to the following question:\ Let $M,N$ be
von~Neumann algebras with $M\subset N$, if there is a completely bounded
projection $P\colon \ N\to M$, is there automatically a contractive projection
$\widetilde P\colon \ N\to M$?
We give an affirmative answer with the only restriction that $M$ is assumed
semi-finite.
|
math
|
1,088 |
Spaces Of Lipschitz Functions On Banach Spaces
|
math.FA
|
A remarkable theorem of R. C. James is the following: suppose that $X$ is a
Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such
that every linear functional $x^* \in X^*$ attains its supremum on $C$; then
$C$ is a weakly compact set. Actually, this result is significantly stronger
than this statement; indeed, the proof can be used to obtain other surprising
results. For example, suppose that $X$ is a separable Banach space and $S$ is a
norm separable subset of the unit ball of $X^*$ such that for each $x \in X$
there exists $x^* \in S$ such that $x^*(x) = \|x\|$ then $X^*$ is itself norm
separable . If we call $S$ a support set, in this case, with respect to the
entire space $X$, one can ask questions about the size and structure of a
support set, a support set not only with respect to $X$ itself but perhaps with
respect to some other subset of $X$@. We analyze one particular case of this as
well as give some applications.
|
math
|
1,089 |
Topologies on the set of all subspaces of a banach space and related questions of banach space geometry
|
math.FA
|
For a Banach space $X$ we shall denote the set of all closed subspaces of $X$
by $G(X)$. In some kinds of problems it turned out to be useful to endow $G(X)$
with a topology. The main purpose of the present paper is to survey results on
two the most common topologies on $G(X)$.
|
math
|
1,090 |
W^*-derived sets of transfinite order of subspaces of dual Banach spaces
|
math.FA
|
It is an English translation of the paper originally published in Russian and
Ukrainian in 1987. In the appendix of his book S.Banach introduced the
following definition Let $X$ be a Banach space and $\Gamma$ be a subspace of
the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in
$\Gamma $ is called the $w^*${\it -derived set} of $\Gamma $ and is denoted by
$\Gamma _{(1)}$. For an ordinal $\alpha$ the $w^{*}$-{\it derived set of order}
$\alpha $ is defined inductively by the equality: $$ \Gamma _{(\alpha
)}=\bigcup _{\beta <\alpha }((\Gamma _{(\beta )})_{(1)}. $$
|
math
|
1,091 |
Total subspaces in dual Banach spaces which are not norming
|
math.FA
|
The main result: the dual of separable Banach space $X$ contains a total
subspace which is not norming over any infinite dimensional subspace of $X$ if
and only if $X$ has a nonquasireflexive quotient space with the strictly
singular quotient mapping.
|
math
|
1,092 |
A note on analytical representability of mappings inverse to integral operators
|
math.FA
|
The condition onto pair ($F,G$) of function Banach spaces under which there
exists a integral operator $T:F\to G$ with analytic kernel such that the
inverse mapping $T^{-1}:$im$T\to F$ does not belong to arbitrary a priori given
Borel (or Baire) class is found.
|
math
|
1,093 |
Total subspaces with long chains of nowhere norming weak$^*$ sequential closures
|
math.FA
|
If a separable Banach space $X$ is such that for some nonquasireflexive
Banach space $Y$ there exists a surjective strictly singular operator $T:X\to
Y$ then for every countable ordinal $\alpha $ the dual of $X$ contains a
subspace whose weak$^*$ sequential closures of orders less than $\alpha $ are
not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$
sequential closure of order $\alpha +1$ coincides with $X^*$
|
math
|
1,094 |
Some isomorphically polyhedral Orlicz sequence spaces
|
math.FA
|
A Banach space is polyhedral if the unit ball of each of its finite
dimensional subspaces is a polyhedron. It is known that a polyhedral Banach
space has a separable dual and is $c_0$-saturated, i.e., each closed infinite
dimensional subspace contains an isomorph of $c_0$. In this paper, we show that
the Orlicz sequence space $h_M$ is isomorphic to a polyhedral Banach space if
$\lim_{t\to 0}M(Kt)/M(t) = \infty$ for some $K < \infty$. We also construct an
Orlicz sequence space $h_M$ which is $c_0$-saturated, but which is not
isomorphic to any polyhedral Banach space. This shows that being
$c_0$-saturated and having a separable dual are not sufficient for a Banach
space to be isomorphic to a polyhedral Banach space.
|
math
|
1,095 |
Random Banach spaces. The limitations of the method
|
math.FA
|
We study the properties of "generic", in the sense of the Haar measure on the
corresponding Grassmann manifold, subspaces of l^N_infinity of given dimension.
We prove that every "well bounded" operator on such a subspace, say E, is a
"small" perturbation of a multiple of identity, where "smallness" is defined
intrinsically in terms of the geometry of E. In the opposite direction, we
prove that such "generic subspaces of l^N_infinity" do admit "nontrivial well
bounded" projections, which shows the "near optimality" of the first mentioned
result, and proves the so called "Pisier's dichotomy conjecture" in the
"generic" case.
|
math
|
1,096 |
Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
|
math.FA
|
Let $E$ be an operator space in the sense of the theory recently developed by
Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non
commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting
operator space satisfies the natural properties to be expected with respect to
e.g. duality and interpolation. This notion leads to the definition of a
``completely p-summing" map which is the operator space analogue of the
$p$-absolutely summing maps in the sense of Pietsch-Kwapie\'n. These notions
extend the particular case $p=1$ which was previously studied by Effros-Ruan.
|
math
|
1,097 |
Complex Interpolation and Regular Operators Between Banach
|
math.FA
|
We study certain interpolation and extension properties of the space of
regular operators between two Banach lattices. Let $R_p$ be the space of all
the regular (or equivalently order bounded) operators on $L_p$ equipped with
the regular norm. We prove the isometric identity $R_p = (R_\infty,R_1)^\theta$
if $\theta = 1/p$, which shows that the spaces $(R_p)$ form an interpolation
scale relative to Calder\'on's interpolation method. We also prove that if
$S\subset L_p$ is a subspace, every regular operator $u : S \to L_p$ admits a
regular extension $\tilde u : L_p \to L_p$ with the same regular norm. This
extends a result due to Mireille L\'evy in the case $p = 1$. Finally, we apply
these ideas to the Hardy space $H^p$ viewed as a subspace of $L_p$ on the
circle. We show that the space of regular operators from $H^p$ to $L_p$
possesses a similar interpolation property as the spaces $R_p$ defined above.
|
math
|
1,098 |
Isometric stability property of certain Banach spaces
|
math.FA
|
Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach
space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that
$E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if
$E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$
into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow
R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from
$E$ to $F.$
|
math
|
1,099 |
The k_t--functional for the interpolation couple L^\infty(dμ;L^1(dν)), L^\infty(dν;L^1(dμ))
|
math.FA
|
Let $(M,\mu)$ and $(N,\nu)$ be measure spaces. In this paper, we study the
$K_t$--\,functional for the couple $$A_0=L^\infty(d\mu\,;
L^1(d\nu))\,,~~A_1=L^\infty(d\nu\,; L^1(d\mu))\,. $$
Here, and in what follows the vector valued $L^p$--\,spaces $L^p(d\mu\,;
L^q(d\nu))$ are meant in Bochner's sense.
One of our main results is the following, which can be viewed as a refinement
of a lemma due to Varopoulos [V].
\proclaim Theorem 0.1. Let $(A_0,A_1)$ be as above. Then for all $f$ in
$A_0+A_1$ we have $${1\over 2}\,K_t(f;\,A_0\,,A_1)\leq \sup\,\bigg\{
\Big(\mu(E)\vee t^{-1}\nu(F)\Big)^{-1} \int_{E\times F} \vert
f\vert\,d\mu\,d\nu\,\bigg\} \leq K_t(f;\,A_0\,,A_1)\,,$$ where the supremum
runs over all measurable subsets $E\subset M\,,~ F\subset N$ with positive and
finite measure and $u\!\vee\!v$ denotes the maximum of the reals $u$ and $v$.
|
math
|
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