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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... | 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 wri... | 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.... | 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... | 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 ... | 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 H... | 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... | 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 opera... | 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 ... | 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 ... | 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\| ... | 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 w... | 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 prov... | 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 factori... | 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 seque... | 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 ba... | 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 >... | 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 ... | 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)... | 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 ... | 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 dep... | 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 in... | 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 is... | 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 ... | 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
dimensio... | 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 tha... | 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 spac... | 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-Ben... | 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 ... | 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 c... | 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... | 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 th... | 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 Se... | 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 t... | 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 stabiliza... | 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 ... | 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 sp... | 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 wh... | 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 suffice... | 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)... | 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} ... | 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^\inft... | 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
containin... | 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 ... | 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 $L_p$,... | 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 comp... | 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 propert... | 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 comp... | 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 ineq... | 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,... | 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 objec... | 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. W... | 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)
... | 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 iso... | 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 distinguish... | 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... | 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 answer... | 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... | 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
$\... | 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 onl... | 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 ... | 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 aut... | 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 wea... | 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... | 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$ ... | 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 s... | 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 ca... | 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 an... | 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 sequenc... | 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
intrinsic... | 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. duali... | 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... | 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(... | 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... | math |
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