On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

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.     

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

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.     

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.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

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.     

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.     

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.     

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:

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 ‖.‖_∞.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.     

Norm conditions for real-algebra isomorphisms between uniform algebras

Let A and B be uniform algebras. Suppose that α ≠ 0 and A ₁ ⊂ A. Let ρ, τ: A ₁ → A and S, T: A ₁ → B be mappings. Suppose that ρ(A ₁), τ(A ₁) and S(A ₁), T(A ₁) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖_∞ = ‖ρ(f)τ(g) − α‖_∞ for all f, g ∈ A ₁, S(e ₁)⁻¹ ∈ S(A ₁) and S(e ₁) ∈ T(A ₁) for some e ₁ ∈ A ₁ with ρ(e ₁) = 1, then there exists a real-algebra isomorphism $ \tilde S $ \tilde S : A → B such that $ \tilde S $ \tilde S (ρ(f)) = S(e ₁)⁻¹ S(f) for every f ∈ A ₁. We also give some applications of this result.     

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.