Gideon Schechtman

Let $X$ be a quasinormed rearrangement invariant function space on (0, 1) which contains $L_q(0, 1)$ for some finite $q$. There is an extension of $X$ to a quasinormed rearrangement … Let $X$ be a quasinormed rearrangement invariant function space on (0, 1) which contains $L_q(0, 1)$ for some finite $q$. There is an extension of $X$ to a quasinormed rearrangement invariant function space $Y$ on $(0, \infty)$ so that for any sequence $(f_i)^\infty_{i = 1}$ of symmetric random variables on (0,1), the quasinorm of $\sum f_i$ in $X$ is equivalent to the quasinorm of $\sum\mathbf{f}_i$ in $Y$, where $(\mathbf{f}_i)^\infty_{i = 1}$ is a sequence of disjoint functions on $(0, \infty)$ such that for each $i, \mathbf{f}_i$ has the same decreasing rearrangement as $f_i$. When specialized to the case $X = L_q(0, 1)$, this result gives new information on the quantitative local structure of $L_q$.

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Planar Earthmover Is Not in $L_1$

2007-01-01

Assaf Naor , Gideon Schechtman

We show that any $L_1$ embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid $\{0,1,\ldots,n\}^2 \subseteq \mathbb{R}^2$ incurs distortion $\Omega \left(\sqrt{\log n}\right)$. We also … We show that any $L_1$ embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid $\{0,1,\ldots,n\}^2 \subseteq \mathbb{R}^2$ incurs distortion $\Omega \left(\sqrt{\log n}\right)$. We also use Fourier analytic techniques to construct a simple $L_1$ embedding of this space which has distortion $O(\log n)$.

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Affine Approximation of Lipschitz Functions and Nonlinear Quotients

1999-12-01

S. M. Bates , William B. Johnson , Joram Lindenstrauss +2 more

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Two observations regarding embedding subsets of Euclidean spaces in normed spaces

2004-12-31

Gideon Schechtman

Remarks on non linear type and Pisiers inequality

2002-01-06

Assaf Naor , Gideon Schechtman

On the Gaussian measure of the intersection

1998-01-01

Gideon Schechtman , Th. Schlumprecht , Joel Zinn

The Gaussian correlation conjecture states that for any two symmetric, convex sets in $n$-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is … The Gaussian correlation conjecture states that for any two symmetric, convex sets in $n$-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, $c$ , such that the conjecture is true if the two sets are in the Euclidean ball of radius $c \sqrt{n}$. Further we show that if for every $n$ the conjecture is true when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.

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Averages of norms and quasi-norms

1998-09-01

Alexander E. Litvak , Vitali Milman , Gideon Schechtman

Banach spaces determined by their uniform structures

1996-05-01

W. B. Johson , Joram Lindenstrauss , Gideon Schechtman

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ALMOST FRÉCHET DIFFERENTIABILITY OF LIPSCHITZ MAPPINGS BETWEEN INFINITE-DIMENSIONAL BANACH SPACES

2002-04-29

William B. Johnson , Joram Lindenstrauss , David Preiss +1 more

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon … We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results. 2000 Mathematical Subject Classification: 46G05, 46T20.

On the Volume of the Intersection of Two L n p Balls

1990-09-01

Gideon Schechtman , Joel Zinn

Martingale inequalities in rearrangement invariant function spaces

1988-12-01

William B. Johnson , Gideon Schechtman

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

1997-01-01

T. Figiel , Paweł Hitczenko , W. Johnson +2 more

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moment of a … The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moment of a sum of independent symmetric random variables to that of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moments of the individual variables, are computed in the range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 greater-than p less-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2>p\le 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This complements the work of Utev who has done the same for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The qualitative nature of the extreme cases turns out to be different for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> than for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The method developed yields results in some more general and other related moment inequalities.

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Finite Dimensional Subspaces of Lp

2001-01-01

William B. Johnson , Gideon Schechtman

Geometric Aspects of Functional Analysis

2000-01-01

Albrecht Dold , Floris Takens , Bernard Teissier +2 more

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Global versus local asymptotic theories of finite-dimensional normed spaces

1997-10-15

Vitali Milman , Gideon Schechtman

Embedding l ${}_{p}^{m}$ into l ${}_{1}^{n}$

1982-01-01

William B. Johnson , Gideon Schechtman

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An Ordinal L p -Index for Banach Spaces, with Application to Complemented Subspaces of L p

1981-09-01

Jean Bourgain , Haskell P. Rosenthal , Gideon Schechtman

One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < … One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five simple examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an EP-space, that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns large countable ordinals to any

DIAMOND GRAPHS AND SUPER-REFLEXIVITY

2009-06-01

William B. Johnson , Gideon Schechtman

The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs D n Lipschitz embed into X with distortions independent of n. … The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs D n Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a la Johnson–Lindenstrauss fails in any non-super-reflexive space with nontrivial type. We also introduce the concept of Lipschitz (p,r)-summing map and prove that every Lipschitz mapping is Lipschitz (p,r)-summing for every 1 ≤ r < p.

Interpolation of weighted Banach lattices. A characterization of relatively decomposable Banach lattices

2003-01-01

Michael Ćwikel , Per G. Nilsson , Gideon Schechtman

Interpolation of weighted Banach lattices, by Michael Cwikel and Per G. Nilsson: Introduction Definitions, terminology and preliminary results The main results A uniqueness theorem Two properties of the $K$-functional for … Interpolation of weighted Banach lattices, by Michael Cwikel and Per G. Nilsson: Introduction Definitions, terminology and preliminary results The main results A uniqueness theorem Two properties of the $K$-functional for a couple of Banach lattices Characterizations of couples which are uniformly Calderon-Mityagin for all weights Some uniform boundedness principles for interpolation of Banach lattices Appendix: Lozanovskii's formula for general Banach lattices of measurable functions References A characterization of relatively decomposable Banach lattices, by Michael Cwikel, Per G. Nilsson and Gideon Schechtman: Introduction Equal norm upper and lower $p$-estimates and some other preliminary results Completion of the proof of the main theorem Application to the problem of characterizing interpolation spaces References.

Concentration, Results and Applications

2003-01-01

Gideon Schechtman

Geometric Aspects of Functional Analysis

2003-01-01

Vitali Milman , Gideon Schechtman

Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers

2004-01-01

Uriel Feige , Michael Langberg , Gideon Schechtman

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also … Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 < \delta < 1$. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ colors). For $k = O(\log n/\log\log n)$ we show vector k-colorable graphs that do not have independent sets of size (log n)c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylog n. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653--750] for this problem.

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Another remark on the volume of the intersection of two L p n balls

1991-01-01

Gideon Schechtman , Michael Schmuckenschläger

More on embedding subspaces of $L_p$ in $l^n_r$

1987-01-01

Gideon Schechtman

On the relation between several notions of unconditional structure

1980-03-01

William B. Johnson , Joram Lindenstrauss , Gideon Schechtman

Random embeddings of Euclidean spaces in sequence spaces

1981-06-01

Gideon Schechtman

Geometric Aspects of Functional Analysis

2004-01-01

Vitali Milman , Gideon Schechtman

Lipschitz Quotients from Metric Trees and from Banach Spaces Containing ℓ1

2002-10-01

William B. Johnson , Joram Lindenstrauss , David Preiss +1 more

A tree-like Tsirelson space

1979-08-01

Gideon Schechtman

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On the volume of the intersection of two $L_p^n$ balls

1992-01-01

Gideon Schechtman , Joel Zinn

This note deals with the following problem, the case $p=1$, $q=2$ of which was introduced to us by Vitali Milman: What is the volume left in the $L_p^n$ ball after … This note deals with the following problem, the case $p=1$, $q=2$ of which was introduced to us by Vitali Milman: What is the volume left in the $L_p^n$ ball after removing a t-multiple of the $L_q^n$ ball? Recall that the $L_r^n$ ball is the set $\{(t_1,t_2,\dots,t_n);\ t_i\in{\bf R},\ n^{-1}\sum_{i=1}^n|t_i|^r\le 1\}$ and note that for $0

Remarks on Talagrand’s deviation inequality for Rademacher functions

1991-01-01

William B. Johnson , Gideon Schechtman

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Concentration on the ℓ p n ball

2000-01-01

Gideon Schechtman , Joel Zinn

On the volume of the intersection of two $L\sp n\sb p$ balls

1990-01-01

Gideon Schechtman , Joel Zinn

In this note we prove that the proportion of the volume left in the $L_p^n$ ball after removing a $t$-multiple of the $L_q^n$ ball is of order ${e^{ - c{t^p}{n^{p/q}}}}.$ In this note we prove that the proportion of the volume left in the $L_p^n$ ball after removing a $t$-multiple of the $L_q^n$ ball is of order ${e^{ - c{t^p}{n^{p/q}}}}.$

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A Concentration Inequality for Harmonic Measures on the Sphere

1995-01-01

Gideon Schechtman , Michael Schmuckenschläger

Let μ be the normalized Lebesgue measure on S n −1. For x = (x 1,…,x n ) with ||x||2 < 1 we denote by μ x the probability measure … Let μ be the normalized Lebesgue measure on S n −1. For x = (x 1,…,x n ) with ||x||2 < 1 we denote by μ x the probability measure on S n −1 given by $$ \frac{{1 - {{\left\| x \right\|}^2}}} {{{{\left\| {y - x} \right\|}^n}}}d\mu \left( y \right) $$ . We recall that if f is an integrable function on S n -1 then u(x) = $$ u(x) = {\int_{{S^{n - 1}}} {f(y)d\mu }^x}(y) $$ is a harmonic function whose radial limits are equal μ-almost everywhere to f.

Bourgain’s discretization theorem

2012-10-23

Ohad Giladi , Assaf Naor , Gideon Schechtman

Bourgain's discretization theorem asserts that there exists a universal constant C∈(0,∞) with the following property. Let X,Y be Banach spaces with dimX=n. Fix D∈(1,∞) and set δ=e -n Cn . … Bourgain's discretization theorem asserts that there exists a universal constant C∈(0,∞) with the following property. Let X,Y be Banach spaces with dimX=n. Fix D∈(1,∞) and set δ=e -n Cn . Assume that 𝒩 is a δ-net in the unit ball of X and that 𝒩 admits a bi-Lipschitz embedding into Y with distortion at most D. Then the entire space X admits a bi-Lipschitz embedding into Y with distortion at most CD. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem.

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Lévy type inequality for a class of finite metric spaces

1982-01-01

Gideon Schechtman

Uniform quotient mappings of the plane.

2000-05-01

William B. Johnson , Joram Lindenstrauss , David Preiss +1 more

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Hypercontractivity and Comparison of Moments of Iterated Maxima and Minima of Independent Random Variables

1998-01-01

Paweł Hitczenko , Stanisław Kwapień , Wenbo Li +3 more

We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such … We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.

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On the Gaussian measure of the intersection of symmetric, convex sets

1996-01-01

Gideon Schechtman , Joel Zinn

The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is … The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, c, such that the conjecture is true if the two sets are in the Euclidean ball of radius $c\sqrt{n}$. Further we show that if for every n the conjecture is true when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.

Special orthogonal splittings ofL 1 2k

2004-12-01

Gideon Schechtman

A remark concerning the dependence on ɛ in dvoretzky's theorem

2006-11-24

Gideon Schechtman

On lipschitz embedding of finite metric spaces in low dimensional normed spaces

1987-01-01

William B. Johnson , Joram Lindenstrauss , Gideon Schechtman

On pelczynski’s paper “universal bases”

1975-12-01

Gideon Schechtman

Weakly null sequences in 𝐿₁

2006-09-19

William B. Johnson , B. Maurey , Gideon Schechtman

We construct a weakly null normalized sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> … We construct a weakly null normalized sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{f_i\}_{i=1}^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">L_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Haar basis is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 plus epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1+\varepsilon )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalent to a block basis of every subsequence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{f_i\}_{i=1}^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{f_i\}_{i=1}^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces.

Almost Isometric L p Subspaces of L p (0, 1)

1979-12-01

Gideon Schechtman

Factorizations of natural embeddings oflpnintoLr. II

1991-10-01

Tadeusz Figiel , William B. Johnson , Gideon Schechtman

This is a continuation of the paper by Figiel, Johnson and Schechtman with a similar title.Several results from there are strengthened, in particular: 1.If T is a "natural" embedding of … This is a continuation of the paper by Figiel, Johnson and Schechtman with a similar title.Several results from there are strengthened, in particular: 1.If T is a "natural" embedding of lζ into L\ then, for any well-bounded factorization of T through an L\ space in the form T -uv with υ of norm one, u well-preserves a copy of /f with k exponential in n .2. Any norm one operator from a C(K) space which well-preserves a copy of /£ also well-preserves a copy of /£, with k exponential in n .As an application of these and other results we show the existence, for any n , of an π-dimensional space which well-embeds into a space with an unconditional basis only if the latter contains a copy of /£> with k exponential in n .

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Approximate identities for ideals in L(Lp)$L(L^p)$

2024-09-28

William B. Johnson , Gideon Schechtman

Abstract The main result is that the only non‐trivial closed ideal in the Banach algebra of bounded linear operators on , , that has a left approximate identity is the … Abstract The main result is that the only non‐trivial closed ideal in the Banach algebra of bounded linear operators on , , that has a left approximate identity is the ideal of compact operators. The algebra has at least one non‐trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

Approximate identities for Ideals in $L(L^p))$

2024-02-20

William B. Johnson , Gideon Schechtman

The main result is that the only non trivial closed ideal in the Banach algebra $L(L^p)$ of bounded linear operators on $L^p(0,1)$, $1\le p < \infty$, that has a left … The main result is that the only non trivial closed ideal in the Banach algebra $L(L^p)$ of bounded linear operators on $L^p(0,1)$, $1\le p < \infty$, that has a left approximate identity is the ideal of compact operators. The algebra $L(L^1)$ has at least one non trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

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The number of closed ideals in the algebra of bounded operators on Lebesgue spaces

2023-12-15

Gideon Schechtman

We survey some of the recent progress in determining the number of two-sided closed ideals in the Banach algebras of bounded linear operators on Lebesgue spaces, $L\_{p}\[0,1]$. In particular, we … We survey some of the recent progress in determining the number of two-sided closed ideals in the Banach algebras of bounded linear operators on Lebesgue spaces, $L\_{p}\[0,1]$. In particular, we discuss two recent results: the first of Johnson, Pisier, and the author, showing that there are a continuum of such ideal in the case of $p=1$; the second a result of Johnson and the author, showing that in the case $1\<p<\infty$, $p\not =2$, there are exactly 2 to the continuum such ideals.

The SHAI property for the operators on L

2021-11-30

William B. Johnson , N. Christopher Phillips , Gideon Schechtman

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The SHAI property for the operators on L^p

2021-01-01

William B. Johnson , N. Christopher Phillips , Gideon Schechtman

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on … A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < \infty.

The number of closed ideals in $L(L_p)$

2021-01-01

William B. Johnson , Gideon Schechtman

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. This solves a problem in A. Pietsch's 1978 book Operator Ideals. The proof is … We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. This solves a problem in A. Pietsch's 1978 book Operator Ideals. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${\frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.

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Polylog Dimensional Subspaces of $$\ell _\infty ^N$$

2020-01-01

Gideon Schechtman , Nicole Tomczak-Jaegermann

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The number of closed ideals in $L(L_p)$

2020-01-01

William B. Johnson , Gideon Schechtman

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1 We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1

Entropy versus influence for complex functions of modulus one

2020-01-01

Gideon Schechtman

We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$. … We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$. We also present an example of a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$ with $L_2$ norm $1$, $L_\infty$ norm bounded by $\sqrt{2}$, influence bounded by $1$ and entropy of $\hat f^2$ larger than $\frac12\log n$.

Impossibility of Dimension Reduction in the Nuclear Norm

2019-12-16

Assaf Naor , Gilles Pisier , Gideon Schechtman

Ideals in $$L(L_1)$$

2019-11-27

William B. Johnson , Gilles Pisier , Gideon Schechtman

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Ideals in $L(L_1)$.

2018-11-15

William B. Johnson , Gilles Pisier , Gideon Schechtman

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a … The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book Operator Ideals. The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals.

Impossibility of dimension reduction in the nuclear norm

2018-01-01

Assaf Naor , Gilles Pisier , Gideon Schechtman

Let S1 (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators T : ℓ2 → ℓ2 whose nuclear norm ||T||S1 = Σj=1∞ σj(T) is finite, … Let S1 (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators T : ℓ2 → ℓ2 whose nuclear norm ||T||S1 = Σj=1∞ σj(T) is finite, where {σj(T)}j=1∞ are the singular values of T. We prove that for arbitrarily large n ∊ ℕ there exists a subset with that cannot be embedded with bi-Lipschitz distortion O(1) into any no(1)-dimensional linear subspace of S1. is not even a O(1)-Lipschitz quotient of any subset of any no(1)-dimensional linear subspace of S1. Thus, S1 does not admit a dimension reduction result á la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with S1 replaced by the Banach space ℓ1 of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in S1. The challenge is to demonstrate that cannot be faithfully realized in an arbitrary low-dimensional subspace of S1, while Brinkman and Charikar obtained such an assertion only for subspaces of S1 that consist of diagonal operators (i.e., subspaces of ℓ1). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of S1 is at most a universal constant multiple of .

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Polylog dimensional subspaces of $\ell_\infty^N$

2018-01-01

Gideon Schechtman , Nicole Tomczak-Jaegermann

We show that a subspace of of $\ell_\infty^N$ of dimension $n>(\log N\log \log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$ where $k$ tends to infinity with $n/(\log N\log \log N)^2$. More … We show that a subspace of of $\ell_\infty^N$ of dimension $n>(\log N\log \log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$ where $k$ tends to infinity with $n/(\log N\log \log N)^2$. More precisely, for every $\eta>0$, we show that any subspace of $\ell_\infty^N$ of dimension $n$ contains a subspace of dimension $m=c(\eta)\sqrt{n}/(\log N\log \log N)$ of distance at most $1+\eta$ from $\ell_\infty^m$.

Ideals in $L(L_1)$

2018-01-01

William B. Johnson , Gilles Pisier , Gideon Schechtman

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a … The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book "Operator Ideals". The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals.

Impossibility of dimension reduction in the nuclear norm

2017-10-24

Assaf Naor , Gilles Pisier , Gideon Schechtman

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values … Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $\mathsf{S}_1$ replaced by the Banach space $\ell_1$ of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set $\mathcal{C}$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $\mathsf{S}_1$. The challenge is to demonstrate that $\mathcal{C}$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $\mathsf{S}_1$, while Brinkman and Charikar obtained such an assertion only for subspaces of $\mathsf{S}_1$ that consist of diagonal operators (i.e., subspaces of $\ell_1$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace $X$ of $\mathsf{S}_1$ is at most a universal constant multiple of $\sqrt{\log \mathrm{dim}(X)}$.

Impossibility of dimension reduction in the nuclear norm

2017-01-01

Assaf Naor , Gilles Pisier , Gideon Schechtman

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Book Review: Asymptotic geometric analysis, Part I

2016-09-14

Gideon Schechtman

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Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support

2016-01-20

William B. Johnson , Tomasz Kania , Gideon Schechtman

Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that assume at most countably many non-zero values. We classify all … Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(\lambda)$, proving that they are isomorphic to $\ell_\infty^c(\kappa)$ for some cardinal number $\kappa$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(\lambda)$ or $\ell_\infty(\lambda)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws' approach description of the lattice of all closed ideals of $\mathscr{B}(X)$, where $X = c_0(\lambda)$ or $X=\ell_p(\lambda)$ for some $p\in [1,\infty)$, and we classify the closed ideals of $\mathscr{B}(\ell_\infty^c(\lambda))$ that contains the ideal of weakly compact operators.

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Three observations regarding Schatten $p$ classes

2016-01-01

Gideon Schechtman

The paper contains three results, the common feature of which is that they deal with the Schatten p class.The first is a presentation of a new complemented subspace of C … The paper contains three results, the common feature of which is that they deal with the Schatten p class.The first is a presentation of a new complemented subspace of C p in the reflexive range (and p = 2).This construction answers a question of Arazy and Lindestrauss from 1975.The second result relates to tight embeddings of finite dimensional subspaces of C p in C n p with small n and shows that ℓ k p nicely embeds into C n p only if n is at least proportional to k (and then of course the dimension of C n p is at least of order k 2 ).The third result concerns single elements of C n p and shows that for p > 2 any n × n matrix of C p norm one and zero diagonal admits, for every ε > 0, a k-paving of C p norm at most ε with k depending on ε and p only.

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Pythagorean powers of hypercubes

2016-01-01

Assaf Naor , Gideon Schechtman

On montre que pour tout n∈ℕ, tout plongement dans L 1 de la puissance pythagoricienne n-ième du cube de Hamming de dimension n admet une distortion qui est au moins … On montre que pour tout n∈ℕ, tout plongement dans L 1 de la puissance pythagoricienne n-ième du cube de Hamming de dimension n admet une distortion qui est au moins un multiple de n par une constante. Pour cela on introduit un nouvel invariant bi-Lipschitz des espaces métriques, inspiré par une inégalité linéaire de Kwapień et Schütt (1989). C’est en évaluant ce nouvel invariant sur L 1 que l’on obtient l’énoncé ci-dessus. On explique le rapport avec le programme de Ribe, et on discute des questions ouvertes.

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METRIC INEQUALITIES

2016-01-01

Assaf Naor , Gideon Schechtman

For every $p\in (0,\infty )$ we associate to every metric space $(X,d_{X})$ a numerical invariant $\mathfrak{X}_{p}(X)\in [0,\infty ]$ such that if $\mathfrak{X}_{p}(X)<\infty$ and a metric space $(Y,d_{Y})$ admits a bi-Lipschitz … For every $p\in (0,\infty )$ we associate to every metric space $(X,d_{X})$ a numerical invariant $\mathfrak{X}_{p}(X)\in [0,\infty ]$ such that if $\mathfrak{X}_{p}(X)<\infty$ and a metric space $(Y,d_{Y})$ admits a bi-Lipschitz embedding into $X$ then also $\mathfrak{X}_{p}(Y)<\infty$ . We prove that if $p,q\in (2,\infty )$ satisfy $q<p$ then $\mathfrak{X}_{p}(L_{p})<\infty$ yet $\mathfrak{X}_{p}(L_{q})=\infty$ . Thus, our new bi-Lipschitz invariant certifies that $L_{q}$ does not admit a bi-Lipschitz embedding into $L_{p}$ when $2<q<p<\infty$ . This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_{p}$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of $L_{q}$ into $L_{p}$ when $2<q<p<\infty$ . Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into $L_{p}$ of snowflakes of $L_{q}$ and integer grids in $\ell _{q}^{n}$ , for $2<q<p<\infty$ . As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_{p}$ that are new even in the linear setting.

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The Hilbert–Schmidt version of the commutator theorem for zero trace matrices

2015-06-21

Omer Angel , Gideon Schechtman

Let A be an m × m complex matrix with zero trace. Then there are m × m matrices B and C such that A = [ B , C … Let A be an m × m complex matrix with zero trace. Then there are m × m matrices B and C such that A = [ B , C ] and ∥ B ∥ ∥ C ∥ 2 ⩽ ( log m + O ( 1 ) ) 1 / 2 ∥ A ∥ 2 where ∥ D ∥ is the norm of D as an operator on ℓ 2 m and ∥ D ∥ 2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely, there is a zero trace m × m matrix A such that whenever A = [ B , C ] , ∥ B ∥ ∥ C ∥ 2 ⩾ | log m − O ( 1 ) | 1 / 2 ∥ A ∥ 2 for some absolute constant c > 0 .

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A Schauder basis for $L_1(0,\infty)$ consisting of non-negative functions

2015-02-26

William B. Johnson , Gideon Schechtman

We construct a Schauder basis for $L_1$ consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in $L_p$, $1\le p < \infty$. We construct a Schauder basis for $L_1$ consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in $L_p$, $1\le p < \infty$.

Pythagorean powers of hypercubes

2015-01-21

Assaf Naor , Gideon Schechtman

For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with … For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced by the $\ell_1^n$ norm when one identifies $\mathbb{F}_2^n$ with $\{0,1\}^n\subseteq \mathbb{R}^n$. Denote by $\ell_2^n(\mathbb{F}_2^n)$ the $n$-fold Pythagorean product of $\mathbb{F}_2^n$, i.e., the space of all $x=(x_1,\ldots,x_n)\in \prod_{j=1}^n \mathbb{F}_2^n$, equipped with the metric $$ \forall\, x,y\in \prod_{j=1}^n \mathbb{F}_2^n,\qquad d_{\ell_2^n(\mathbb{F}_2^n)}(x,y)= \sqrt{ \|x_1-y_1\|_1^2+\ldots+\|x_n-y_n\|_1^2}. $$ It is shown here that the bi-Lipschitz distortion of any embedding of $\ell_2^n(\mathbb{F}_2^n)$ into $L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapie{\'n} and Sch\utt (1989). Letting $\{e_{jk}\}_{j,k\in \{1,\ldots,n\}}$ denote the standard basis of the space of all $n$ by $n$ matrices $M_n(\mathbb{F}_2)$, say that a metric space $(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in 2\mathbb{N}$, every mapping $f:M_n(\mathbb{F}_2)\to X$ satisfies \begin{equation*}\label{eq:metric KS abstract} \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right], \end{equation*} where the expectations above are with respect to $x\in M_n(\mathbb{F}_2)$ and $k=(k_1,\ldots,k_n)\in \{1,\ldots,n\}^n$ chosen uniformly at random. It is shown here that $L_1$ is a KS space (with $C= 2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

A Schauder basis for $L_{1}(0,\infty)$ consisting of non-negative functions

2015-01-01

William B. Johnson , Gideon Schechtman

We construct a Schauder basis for L 1 consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in L p , 1 ≤ p < … We construct a Schauder basis for L 1 consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in L p , 1 ≤ p < ∞.

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Pythagorean powers of hypercubes

2015-01-01

Assaf Naor , Gideon Schechtman

For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with … For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced by the $\ell_1^n$ norm when one identifies $\mathbb{F}_2^n$ with $\{0,1\}^n\subseteq \mathbb{R}^n$. Denote by $\ell_2^n(\mathbb{F}_2^n)$ the $n$-fold Pythagorean product of $\mathbb{F}_2^n$, i.e., the space of all $x=(x_1,\ldots,x_n)\in \prod_{j=1}^n \mathbb{F}_2^n$, equipped with the metric $$ \forall\, x,y\in \prod_{j=1}^n \mathbb{F}_2^n,\qquad d_{\ell_2^n(\mathbb{F}_2^n)}(x,y)= \sqrt{ \|x_1-y_1\|_1^2+\ldots+\|x_n-y_n\|_1^2}. $$ It is shown here that the bi-Lipschitz distortion of any embedding of $\ell_2^n(\mathbb{F}_2^n)$ into $L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapie{\'n} and Sch\"utt (1989). Letting $\{e_{jk}\}_{j,k\in \{1,\ldots,n\}}$ denote the standard basis of the space of all $n$ by $n$ matrices $M_n(\mathbb{F}_2)$, say that a metric space $(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in 2\mathbb{N}$, every mapping $f:M_n(\mathbb{F}_2)\to X$ satisfies \begin{equation*}\label{eq:metric KS abstract} \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right], \end{equation*} where the expectations above are with respect to $x\in M_n(\mathbb{F}_2)$ and $k=(k_1,\ldots,k_n)\in \{1,\ldots,n\}^n$ chosen uniformly at random. It is shown here that $L_1$ is a KS space (with $C= 2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

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A Schauder basis for $L_1(0,\infty)$ consisting of non-negative functions

2015-01-01

William B. Johnson , Gideon Schechtman

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Three observations regarding Schatten p classes

2014-11-17

Gideon Schechtman

The paper contains three results, the common feature of which is that they deal with the Schatten $p$ class. The first is a presentation of a new complemented subspace of … The paper contains three results, the common feature of which is that they deal with the Schatten $p$ class. The first is a presentation of a new complemented subspace of $C_p$ in the reflexive range (and $p\not= 2$). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of $C_p$ in $C_p^n$ with small $n$ and shows that $\ell_p^k$ nicely embeds into $C_p^n$ only if $n$ is at least proportional to $k$ (and then of course the dimension of $C_p^n$ is at least of order $k^2$). The third result concerns single element of $C_p^n$ and shows that for $p>2$ any $n\times n$ matrix of $C_p$ norm one and zero diagonal admits, for every $\varepsilon>0$, a $k$-paving of $C_p$ norm at most $\varepsilon$ with $k$ depending on $\varepsilon$ and $p$ only.

Injective Tauberian Operators on L1 and Operators with Dense Range on ℓ∞

2014-10-06

William B. Johnson , Amir Bahman Nasseri , Gideon Schechtman +1 more

Abstract. There exist injective Tauberian operators on L 1 (0, 1) that have dense, nonclosed range. This gives injective nonsurjective operators on ℓ ∞ that have dense range. Consequently, there … Abstract. There exist injective Tauberian operators on L 1 (0, 1) that have dense, nonclosed range. This gives injective nonsurjective operators on ℓ ∞ that have dense range. Consequently, there are two quasi-complementary noncomplementary subspaces of ℓ ∞ that are isometric to ℓ ∞ .

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On the mathematical contributions of Joram Lindenstrauss

2014-10-01

Assaf Naor , Gideon Schechtman

Injective Tauberian operators on $L_1$ and operators with dense range on $\ell_\infty$

2014-08-06

William B. Johnson , Amir Bahman Nasseri , Gideon Schechtman +1 more

There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, … There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, non complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$.

Subspaces of L p that embed into L p (µ) with µ finite

2014-06-24

William B. Johnson , Gideon Schechtman

No greedy bases for matrix spaces with mixed<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>norms

2014-05-13

Gideon Schechtman

We show that none of the spaces (⨁n=1∞ℓp)ℓq, 1≤p≠q<∞ have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprecht. Similarly, the spaces (⨁n=1∞ℓp)c0, 1≤p<∞, and … We show that none of the spaces (⨁n=1∞ℓp)ℓq, 1≤p≠q<∞ have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprecht. Similarly, the spaces (⨁n=1∞ℓp)c0, 1≤p<∞, and (⨁n=1∞co)ℓq, 1≤q<∞, do not have greedy bases. It follows from that and known results that a class of Besov spaces on Rn lack greedy bases as well.

Narrow and ℓ2-strictly singular operators from L p

2014-02-07

Volodymyr Mykhaylyuk , M. M. Popov , Beata Randrianantoanina +1 more

Metric ${X}_p$ inequalities

2014-01-01

Assaf Naor , Gideon Schechtman

For every $p\in (0,\infty)$ we associate to every metric space $(X,d_X)$ a numerical invariant $\mathfrak{X}_p(X)\in [0,\infty]$ such that if $\mathfrak{X}_p(X) For every $p\in (0,\infty)$ we associate to every metric space $(X,d_X)$ a numerical invariant $\mathfrak{X}_p(X)\in [0,\infty]$ such that if $\mathfrak{X}_p(X)

Injective Tauberian operators on $L_1$ and operators with dense range on $\ell_\infty$

2014-01-01

William B. Johnson , Amir Bahman Nasseri , Gideon Schechtman +1 more

Three observations regarding Schatten p classes

2014-01-01

Gideon Schechtman

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No greedy bases for matrix spaces with mixed $\ell_p$ and $\ell_q$ norms

2013-10-09

Gideon Schechtman

We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, … We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, $1\le p<\infty$, and $(\bigoplus_{n=1}^\infty c_o)_{\ell_q}$, $1\le q<\infty$, do not have greedy bases. It follows from that and known results that a class of Besov spaces on $\R^n$ lack greedy bases as well.

Matrix Subspaces of $L_1$

2013-03-19

Gideon Schechtman

If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r<p\le 2$, then the space of matrices $\{a_{i,j}\}$ with norm … If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r<p\le 2$, then the space of matrices $\{a_{i,j}\}$ with norm $\|\{a_{i,j}\}\|_{E(F)}=\big\|\sum_k \|\sum_l a_{k,l}f_l\|e_k\big\|$ embeds into $L_1$. This generalizes a recent result of Prochno and Sch\utt.

Subspaces of $L_p$ that embed into $L_p(\mu)$ with $\mu$ finite

2013-01-17

William B. Johnson , Gideon Schechtman

Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ a finite measure. We prove that if $X$ is a subspace … Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ a finite measure. We prove that if $X$ is a subspace of an $L_p$ space, $1< p < 2$, and $\ell_p(\aleph_1)$ does not embed into $X$, then $X$ embeds into $L_p(\mu)$ for some finite measure $\mu$.

Euclidean Sections of Convex Bodies

2013-01-01

Gideon Schechtman

Matrix subspaces of L1

2013-01-01

Gideon Schechtman

If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r< p\le 2$, then the space of matrices $\{a_{i,j}\}$ with … If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r< p\le 2$, then the space of matrices $\{a_{i,j}\}$ with norm $ \|\{a_{i,j}\}\|_{E(F)}=\left\|\sum_k \|\sum_l a_{k,l}f

No greedy bases for matrix spaces with mixed $\ell_p$ and $\ell_q$ norms

2013-01-01

Gideon Schechtman

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Subspaces of $L_p$ that embed into $L_p(μ)$ with $μ$ finite

2013-01-01

William B. Johnson , Gideon Schechtman

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Matrix Subspaces of $L_1$

2013-01-01

Gideon Schechtman

If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r

Narrow and $\ell_2$-strictly singular operators from $L_p$

2012-11-20

Volodymyr Mykhaylyuk , M. M. Popov , Beata Randrianantoanina +1 more

In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of … In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and function Lebesgue spaces $X$ with the property that every operator $T:L_p \to X$ is narrow. Next, using similar methods we prove that every $\ell_2$-strictly singular operator from $L_p$, $1<p<\infty$, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator $T$ on $L_1[0,1]$ satisfies the assumption that for each measurable set $A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic embedding, then $T$ is narrow. (Here $L_1(A) = \{x \in L_1: {\rm supp} \, x \subseteq A\}$.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being $\ell_2$-strictly singular, for operators on $L_p[0,1]$, $1<p<2$, to be narrow. We define a notion of a gentle growth of a function and we prove that for $1 < p < 2$ every operator $T$ on $L_p$ which, for every $A\subseteq[0,1]$, sends a function of gentle growth supported on $A$ to a function of arbitrarily small norm is narrow.

Bourgain’s discretization theorem

2012-10-23

Ohad Giladi , Assaf Naor , Gideon Schechtman

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Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

2012-08-21

Detelin Dosev , William B. Johnson , Gideon Schechtman

The operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p Baseline equals upper L Subscript p Baseline left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>=</mml:mo> … The operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p Baseline equals upper L Subscript p Baseline left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_p=L_p[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1\leq p>\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which are not commutators are those of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda upper I plus upper S"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda I + S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda not-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to the largest ideal in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L left-parenthesis upper L Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}(L_p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof involves new structural results for operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are of independent interest.

A quantitative version of the commutator theorem for zero trace matrices

2012-08-06

William B. Johnson , Narutaka Ozawa , Gideon Schechtman

Let A be an m × m complex matrix with zero trace and let ε > 0. Then there are m × m matrices B and C such that A … Let A be an m × m complex matrix with zero trace and let ε > 0. Then there are m × m matrices B and C such that A = [ B , C ] and ‖ B ‖‖ C ‖ ≤ K ε m ε ‖ A ‖ where K ε depends only on ε . Moreover, the matrix B can be taken to be normal.

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Coauthor	Papers Together
William B. Johnson	58
Assaf Naor	17
Joram Lindenstrauss	12
Joel Zinn	9
Vitali Milman	7
Gilles Pisier	7
T. Figiel	7
David Preiss	6
S. M. Bates	3
Amir Bahman Nasseri	3
Paweł Hitczenko	3
Detelin Dosev	3
Ohad Giladi	3
Tomasz Tkocz	3
B. Maurey	3
Nicole Tomczak-Jaegermann	3
Paul F. X. Müller	3
Alvaro Arias	3
Michael Langberg	2
Shahar Mendelson	2
Uriel Feige	2
M. M. Popov	2
Adi Shraibman	2
Thomas Schlumprecht	2
Edna Schechtman	2
Volodymyr Mykhaylyuk	2
Michael Schmuckenschläger	2
N. Christopher Phillips	2
Beata Randrianantoanina	2
Stanisław Kwapień	2
W. Johnson	2
Alexander E. Litvak	2
Albrecht Dold	1
Artem Zvavitch	1
Guy Kindler	1
Michael Ćwikel	1
Floris Takens	1
Jesús Bastero	1
Wenbo Li	1
Michel Ledoux	1
Nati Linial	1
Dvir Kleper	1
Ana Peña	1
Bernard Teissier	1
Lior Tzafriri	1
Tomasz Kania	1
Jean Bourgain	1
Haskell P. Rosenthal	1
Wenbo V. Li	1
Ran Levy	1

The Classical Banach Spaces

2000-07-01

Joram Lindenstrauss , Lior Tzafriri

Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A … Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later a new series Ergebnisse der Mathematik und ihrer Grenzgebiete, survey reports of recent mathematical research, was added. Of over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissuing a selected few of these highly successful books in a new, inexpensive sofcover edition to make them easily accessible to younger generations of students and researchers.

Asymptotic Theory of Finite Dimensional Normed Spaces

1986-01-01

Vitali Milman , Gideon Schechtman

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Symmetric structures in Banach spaces

1979-01-01

William B. Johnson , B. Maurey , Gideon Schechtman +1 more

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

1970-09-01

Haskell P. Rosenthal

Geometric Nonlinear Functional Analysis

1999-12-14

Yoav Benyamini , Joram Lindenstrauss

Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces … Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classification of Banach spaces Nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local isometries Perturbations of global isometries Twisted sums Group structure on Banach spaces Appendices Bibliography Index.

The dimension of almost spherical sections of convex bodies

1977-01-01

T. Figiel , Joram Lindenstrauss , Vitali Milman

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Approximation of zonoids by zonotopes

1989-01-01

Jean Bourgain , Joram Lindenstrauss , Vitali Milman

On etudie une propriete d'approximation des corps convexes de R n qui sont des limites de sommes de segments On etudie une propriete d'approximation des corps convexes de R n qui sont des limites de sommes de segments

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Classical Banach Spaces I

1977-01-01

Joram Lindenstrauss , Lior Tzafriri

On uncomplemented subspaces ofL p , 1 <p <2

1977-06-01

Grahame Bennett , Leonard E. Dor , Victor Goodman +2 more

Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$

1962-01-01

M. Kadec , A. Pełczyński

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Topics in Banach Space Theory

2006-01-01

Fernando Albiac , N. J. Kalton

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On uniformly homeomorphic normed spaces II

1978-12-01

M. Ribe

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Classical Banach Spaces II

1979-01-01

Joram Lindenstrauss , Lior Tzafriri

Extensions of Lipschitz mappings into a Hilbert space

1984-01-01

William B. Johnson , Joram Lindenstrauss

An Ordinal L p -Index for Banach Spaces, with Application to Complemented Subspaces of L p

1981-09-01

Jean Bourgain , Haskell P. Rosenthal , Gideon Schechtman

Best Constants in Moment Inequalities for Linear Combinations of Independent and Exchangeable Random Variables

1985-02-01

William B. Johnson

In [4] Rosenthal proved the following generalization of Khintchine's inequality: \begin{equation*} \tag{B} \begin{cases} \max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i=1} \| X_i \|^p_p)^{1/p}\} \\ \leq \| \sigma^n_{i=1} X_i \|_p \leq … In [4] Rosenthal proved the following generalization of Khintchine's inequality: \begin{equation*} \tag{B} \begin{cases} \max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i=1} \| X_i \|^p_p)^{1/p}\} \\ \leq \| \sigma^n_{i=1} X_i \|_p \leq B_p\max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i-i} \| X_i \|^p_p)^{1/p}\} \\ \text{for all independent symmetric random variables} X_1, X_2,\cdots, \text{with finite} pth \text{moment}, 2 < p < \infty.\end{cases}\end{equation*} Rosenthal's proof of (B) as well as later proofs of more general results by Burkholder [1] yielded only exponential of $p$ estimates for the growth rate of $B_p$ as $p \rightarrow \infty$. The main result of this paper is that the actual growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\operatorname{Log} p$, as compared with a growth rate of $\sqrt p$ in Khintchine's inequality.

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Absolutely Summing Operators

2006-05-27

Joseph Diestel , Hans Jarchow , Andrew Tonge

Ultraproducts in Banach space theory.

1980-01-01

Stefan Heinrich

Distribution Function Inequalities for Martingales

1973-02-01

D. L. Burkholder

This is a guide to some recent work in the theory of martingale inequalities. Methods are simplified; some new proofs are given. A number of new results are also included. This is a guide to some recent work in the theory of martingale inequalities. Methods are simplified; some new proofs are given. A number of new results are also included.

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Probabilistic methods in the geometry of Banach spaces

1986-01-01

Gilles Pisier

Finite dimensional subspaces of $L_{p}$

1978-01-01

Debra Lewis

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Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces LP

1974-01-01

B. Maurey

On uniformly homeomorphic normed spaces

1976-12-01

M. Ribe

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Uniformly Non-Square Banach Spaces

1964-11-01

Robert James

Markov chains, Riesz transforms and Lipschitz maps

1992-06-01

Keith Ball

An introduction to the Ribe program

2012-11-01

Assaf Naor

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The ℒ p spaces

1969-03-01

Joram Lindenstrauss , Haskell P. Rosenthal

Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces

1982-01-01

Stefan Heinrich , Piotr Mankiewicz

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On type of metric spaces

1986-01-01

Jean Bourgain , Vitali Milman , Haim J. Wolfson

Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of … Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cubes into the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cube (Hamming cube) are discussed.

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Absolutely summing operators in $ℒ_{p}$-spaces and their applications

1968-01-01

Joram Lindenstrauss , A. Pełczyński

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Some results on Banach spaces without local unconditional structure

1978-01-01

Gilles Pisier

Asymptotic Theory of Finite Dimensional Normed Spaces

1986-01-01

Sharp uniform convexity and smoothness inequalities for trace norms

1994-12-01

Keith Ball , Eric A. Carlen , Élliott H. Lieb

Metric cotype

2008-07-01

Manor Mendel , Assaf Naor

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype.This yields a concrete version of … We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype.This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces.We apply our results to several problems in metric geometry.Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L p coarsely or uniformly embeds into L q .We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

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Ellipsoids defined by Banach ideal norms

1979-06-01

D. R. Lewis

A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E … A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E of Lp and a λ > 1 such that 1E has no λ-unconditional decomposition in terms of rank one operators.

Examples of ℒp spaces (1<p≠2<∞)

1975-06-01

Gideon Schechtman

Some linear topological properties of the spaces $C_p$ of operators on Hilbert space

1975-01-01

Jonathan Arazy , Joram Lindenstrauss

The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities

1955-04-01

T. W. Anderson

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Bounded orthogonal systems and the Λ(p)-set problem

1989-01-01

Jean Bourgain

Observe that the size n ~p is optimal.Indeed, if one considers for instance a finite Cantor group G={I,-1} k and let ~=G*, the space LPs(G) is a Hilbertian subspace of Observe that the size n ~p is optimal.Indeed, if one considers for instance a finite Cantor group G={I,-1} k and let ~=G*, the space LPs(G) is a Hilbertian subspace of

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Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis

1987-06-01

Jean Bourgain , Lior Tzafriri

Affine Approximation of Lipschitz Functions and Nonlinear Quotients

1999-12-01

S. M. Bates , William B. Johnson , Joram Lindenstrauss +2 more

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On the nonexistence of uniform homeomorphisms between L~-spacesBy P~. On the nonexistence of uniform homeomorphisms between L~-spacesBy P~.

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