The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let and suppose that f : K ⁿ →K ⁿ is nonexpansive with respect to the l ₁-norm, , and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.     

Lipschitz algebras and peripherally-multiplicative maps

Let X be a compact metric space and let Lip（X） be the Banach algebra of all scalar- valued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip（X）, σπ（f） denotes the peripheral spectrum of f. We state that any map Φ from Lip（X） onto Lip（Y） which preserves multiplicatively the peripheral spectrum：
σπ（Φ（f）Φ（g）） = σπ（fg）, A↓f, g ∈ Lip（X）
is a weighted composition operator of the form Φ（f） = τ· （f °φ） for all f ∈ Lip（X）, where τ ： Y → {-1, 1} is a Lipschitz function and φ ： Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip（X）-algebras is of the form above.     

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC ¹-mapX→ℝ for which {c∈X: T′c=0} is finite. Set . Fix ann ∈ ℕ. For ε>0, theC ¹-map is called an ε-perturbation ofT if is a piecewise monotonic map with at mostn intervals of monotonicity and is ε-close toT in theC ¹-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation ofT has a unique measure of maximal entropy, and the map is continuous atT in the weak star-topology.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖ _X and ‖.‖ _Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖ _Y = ‖fg‖ _X , for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖ _X = ‖Tf Tg + α‖ _Y , f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,

Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

For metric spaces (X, d _x) and (Y, d _y) we consider the Hausdorff metric topology on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ _uc on the set F(X, 2^Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),) of closed subsets of Y metrized by the Hausdorff metric . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank.ubica Holá for suggestions and comments.     

A joint norm control Nehari type theorem forN-tuples of Hardy spaces

If N ∈ ℕ, 0 < p ≤ 1, and(X_k) _k=1 ^N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ _k=1 ^N X_k, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ², it is proved that the N-tuple (H(X₁),…, H(X_n)) is K_⊓-closed with respect to (X₁,…, X_N) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.     

A general Hobby-Rice Theorem and cake cutting

LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L ₁ (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x ₀<x ₁<x ₂<…<x _n<x _n+1=∞with n≤m, such that for all g ∈ G,      

Absolute continuity of the distribution of some Markov geometric series

Let (∈n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift space ∑A defined by Fibonacci incident matrix A.We consider the measure μλ of the probability distribution of the random series ∑∞n=0 εnλn (0 ＜λ＜ 1).It is proved that μλ is singular if λ∈ (0,√5-1/2) and that μλ is absolutely continuous for almost all λ∈ (√5-1/2,0.739).     

Extensions of lipschitz maps into Banach spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map fromX intoZ so that wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX. Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. Supported in part by NSF MCS-8102714.     

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.     

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB ^m ×C ⁿ be a compact set over the unit sphere ϱB ^m such that for eachz∈ϱB ^m the fiberX _z ={ω∈C ⁿ ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC ⁿ . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z ₀,w ₀)∈Int there exists a smooth up to the boundary analytic discF:Δ→B ^m ×C ⁿ with the boundary inX such thatF(0)=(z ₀,w ₀). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005     

Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.     

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.