Every separable metric space is Lipschitz equivalent to a subset ofc 0 +

It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →c _o satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X. This is a part of the author's Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor J. Lindenstrauss.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

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 ‖.‖_∞.     

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.     

Invariant densities and mean ergodicity of markov operators

We prove that a Markov operatorT onL ₁ has an invariant density if and only if there exists a densityf that satisfies lim sup _n→∞‖T ⁿ f − f‖ < 2. Using this result, we show that a Frobenius-Perron operatorP is mean ergodic if and only if there exists a densityw such that lim sup _n→∞ ‖P ⁿ f − w‖<2 for every densityf. Corresponding results hold for strongly continuous semigroups.     

Asymptotic behavior of nonexpansive mappings in normed linear spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, lim_n→x f(T ⁿ x/n)=lim_n→x‖T ⁿ x/n ‖=α, where α≡inf _y∈c ‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT ⁿ x/n converges weakly for allx (inf_z∈f g(T ⁿ x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT. Supported by National Science Foundation Grant MCS-79-066.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

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.     

Fourier analysis with respect to bilinear maps

Several results about convolution and about Fourier coefficients for X-valued functions defined on t he torus satisfying the condition sup ｜｜y｜｜=1∫-π^π｜｜ B （f （e^iθ）, y）｜｜dθ/2π〈 ∞ for a bounded bilinear map B ： X × Y → Z are presented and some applications are given.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.     

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.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ⁻¹ Σ _k ⁼⁰ⁿ⁻¹ Xk for a stationary processX = (X _n ,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n |σ _k | ≥ε) asn → ∞ are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998.     

Extreme value limit laws in the nonidentically distributed case

LetX ₁, ...,X _n be independent random variables, letF _i be the distribution function ofX _i (1≦i≦n) and letX _1n ≦... ≦X _nn be the corresponding order statistics. We consider the statisticsX _kn, wherek=k(n),k/n → 1 andn−k → ∞. Under some additional restrictions concerning the behaviour of the sequences {a _n>0,b _n,k(n),F _n} we characterize the class of all distribution functionsH such that Prob{(X _kn −b _n )/a _n <x)}→H. Dedicated to the Memory of N. V. Smirnov (1900–1966)     

The range of a projection of small norm inl 1 n

It is proved that there exists a positive function Φ(∈) defined for sufficiently small ∈ 〉 0 and satisfying lim_t→0 Φ(∈) =0 such that for any integersn 〉>0, ifQ is a projection ofl ₁ ⁿ onto ak-dimensional subspaceE with ‖|Q‖|≦1+∈ then there is an integerh〉=k(1−Φ(∈)) and anh-dimensional subspaceF ofE withd(F,l ₁ ^h ) 〈= 1+Φ (∈) whered(X, Y) denotes the Banach-Mazur distance between the Banach spacesX andY. Moreover, there is a projectionP ofl ₁ ⁿ ontoF with ‖|P‖| ≦1+Φ(∈). Author was partially supported by the N.S.F. Grant MCS 79-03042.     

A condition for asymptotic finite-dimensionality of an operator semigroup

Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X ₀ = {x ∈ X ∣ T ⁿ x → 0}. Assume given a compact set K ⊂ X such that lim inf _n→∞ ρ{T ⁿ x, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If $\eta < \tfrac{1} {2} $\eta < \tfrac{1} {2} , then codim X ₀ < ∞. This is true in X reflexive for $\eta \in [\tfrac{1} {2},1) $\eta \in [\tfrac{1} {2},1) , but fails in the general case.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

Positive definite functions and stable random vectors

We say that a random vector X = (X ₁, …, X _n ) in ℝ ⁿ is an n-dimensional version of a random variable Y if, for any a ∈ ℝ ⁿ , the random variables Σa _i X _i and γ(a)Y are identically distributed, where γ: ℝ ⁿ → [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L ₀. This result is almost optimal, as the norm of any finite-dimensional subspace of L _p with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖ _K ) is positive definite on ℝ ⁿ , where K is an origin symmetric star body in ℝ ⁿ and f: ℝ → ℝ is an even continuous function, then either the space (ℝ ⁿ , ‖·‖ _K ) embeds in L ₀ or f is a constant function. Combined with known facts about embedding in L ₀, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions.     

Substitution tilings and separated nets with similarities to the integer lattice

We show that any primitive substitution tiling of ℝ² creates a separated net which is biLipschitz to ℤ². Then we show that if H is a primitive Pisot substitution in ℝ ^d , for every separated net Y, that corresponds to some tiling τ ∈ X _H , there exists a bijection Φ between Y and the integer lattice such that sup _y∈Y ∥Φ(y) − y∥ < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.