A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

New associative and nonassociative Gelfand-Naimark theorems

We prove that, ifA is a complex Banach algebra with a unit 1 and a conjugate-linear vector space involution^* such that 1^*=1 and‖a ^*a‖=‖a^*‖ ‖a‖ for alla inA, and ifdim(A)≥3, thenA is a C^*-algebra. The two-dimensional case is also considered and described.     

Orthogonal almost locally minimal projections on ℓ 1 n

A projectionP on a Banach spaceX with ‖P‖≤λ₀ is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα²), whereD=D(λ₀) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL _p(G) is almost locally minimal. Orthogonal projections on ℓ ₁ ⁿ are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Shape preserving widths of Sobolev-type classes ofs-monotone functions on a finite interval

LetI be a finite interval andr ∈ ℕ. Denote by △ ₊ ^s L _q the subset of all functionsy ∈L _q such that thes-difference △ _T ^s y(·) is nonnegative onI, ∀τ>0. Further, denote by △ ₊ ^s W _p ^r the class of functionsx onI with the seminorm ‖x ^(r) ‖L _p ≤1, such that △ _T ^s x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths , whereM ⁿ is the set of all linear manifoldsM ⁿ inL _q , dimM ⁿ ≤n, such thatM ⁿ ⋂△ ₊ ^s L _q ≠ 0. Part of this work was done while the first author visited Tel Aviv University in 2001 and part of it while the second author was a member of the Industrial Mathematics Institute (IMI), University of South Carolina.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

The bounds for matrices with monotonic rows

The bounds LX,Y (A) and ‖A‖ _X,Y of an operator A = (a _n,k ) _{n, k} ≥0 with monotonic rows are evaluated, where X and Y are quasi-normed real valued sequence spaces. In particular, in the case where X = ℓ _p and Y = ℓ _q , our results give the results when part 0 < p ≤ 1 and 0 < q < ∞ to complement the other results with range p ≥ 1 and 0 < q < ∞. Moreover, we give a partial answer to a problem [1, Problem 7.23] which was posed by Bennett.     

Averaging distances in real quasihypermetric Banach spaces of finite dimension

The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x ₁,x ₂, …,x _n inE with ‖x ₁‖=‖x ₂‖=…=‖x _n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL ¹-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ ^N equipped with the usual 1-norm.     

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)⁻¹‖≤C|z−1|⁻¹, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup _n≥0‖T ⁿ‖. A similar estimate is shown for lim sup _n ‖T ⁿ‖ in terms of the Ritt constantM=lim sup _z→1‖(1−z)(zI−T)⁻¹‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C _q‖f‖ _Mult , where ‖·‖ _Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1. Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H ^∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖_∞.     

Limit theorems for convolutions of probabilities on non abelian semigroups

Let S be a locally compact semigroup. We study the sequence (λⁿ) of the convolution powers of a probability measure λ on S and their shifts by a probability measure η on S. We shall give sufficient conditions for lim ‖λⁿ−η*λⁿ‖ = 0 (where ‖.‖ denotes the norm). In particular we consider the case the η is a point measure and we study the subsemigroup L_O(λ) = {x ∈ S : lim ‖λⁿ−δ_X*λⁿ‖ = 0}. We shall give necessary and sufficient conditions for L_o(λ)=S. In this case we want to treat the problem of the convergence of the sequence (λⁿ).     

Carathéodory balls and norm balls inH n ={z ε ℂ n : ‖z‖1<1}

It is shown that the only balls in the Carathéodory distance onH _n ={z ε ℂ ⁿ :‖z‖₁<1},n≥2, which are balls with respect to the ℓ₁ norm in ℂ ⁿ are those centered at the origin. In memory of Albert Pfluger The research was partially supported by the Fund for the Promotion of Research at the Technion.     

Non-isogenous superelliptic Jacobians

Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set of roots of f and that Gal(h) acts doubly transitively on as well. Let J(C_f,ℓ) and J(C_h,ℓ) be the Jacobians of the superelliptic curves C_f,ℓ:y^ℓ=f(x) and C_h,ℓ:y^ℓ=h(x) respectively. We prove that J(C_f,ℓ) and J(C_h,ℓ) are not isogenous over K_a if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables

LetX ₁,X ₂, …,X _n be a sequence of independent random variables, letM be a rearrangement invariant space on the underlying probability space, and letN be a symmetric sequence space. This paper gives an approximate formula for the quantity ‖‖(X _i )‖ _N ‖ _M wheneverL _q embeds intoM for some 1≤q<∞. This extends work of Johnson and Schechtman who tackled the case whenN=ℓ _p , and recent work of Gordon, Litvak, Schütt and Werner who obtained similar results for Orlicz spaces. The author was partially supported by NSF grant DMS 9870026, and a grant from the Research Office of the University of Missouri.     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

Best monotone approximation and the Hardy-Littlewood maximal function

If f∈L₂[0, 1] and g^*∈L₂[0, 1] is the best non-decreasing approximation to f, then it's shown that ‖f−g^*‖₂=‖f−θ(f)‖₂, where θ(f) denotes the Hardy-Littlewood maximal function of f.     

Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let