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.     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Riemann surfaces in fibered polynomial hulls

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and letA(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. LetK be a compact set in C² such that Π(K)=Γ, and letK _λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,K _λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))²-S(λ) quadratic in w withR,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ intK _λ, whereS has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component ofK _λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull ofK. Furthermore, we show that БК/K is the disjoint union of such disks.     

Certaines formes linéaires pathologiques sur un espace de Banach dual

We construct a Banach spaceE such thatE′ isw ^*-separable, andf∈E″/E, which isw ^*-continuous on every set ofE′ which is thew ^*-closure of a countablebounded set ofE′.      

Absolute upper semicontinuity

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ _t′ ^t″ g (s) ds. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

Elementary equivalence of Chevalley groups over fields

It is proved that (elementary) Chevalley groups G _π(Φ,K) and G _π′(Φ′,K′) (or E _π(Φ,K) and E _π′(Φ′,K′)) over infinite fields K and K′ of characteristic different from 2, with weight lattices Λ and Λ′, respectively, are elementarily equivalent if and only if the root systems Φ and Φ′ are isomorphic, the fields K and K′ are elementarily equivalent, and the lattices Λ and Λ′ coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 29–77, 2006.     

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.     

On the length of commutators in Chevalley groups

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A _l , with a better bound, was earlier obtained by Sivatski and Stepanov.     

BMO-scale of distribution on ℝ n

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J ^−α ƒ the Bessel potential of ƒ of order α. We prove that if J ^−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J ^−α (f)_λ ∈ BMO, where (f)_λ = λ⁻ⁿ f(φ(λ⁻¹)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

Counterexample to Peano's theorem in infinite-dimensionalF′-spaces

LetE be a nonnormable Fréchet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt ₀∈ℝ,x ₀∈E′, the Cauchy problemx=f(x), x(t ₀ )=x ₀ has no solutions is constructed. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 128–137, July, 1997. Translated by V. N. Dubrovsky     

Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.     

Relationship between the Jung constant and acertain projection constant in Banach spaces

Summary For a Banach spaceX the parameter λ₁(X) is the infimum of the numbers a with the following property: for everyZ )X with dimZ/X=1 there exists a projectionP:Z→X with |P|≤a. We give an upper bound for λ₁(X) in terms of the Jung constantJ(X). Some known and some new results are then deduced. A characteristic property ofJ ₁-spaces is also proved.

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Riassunto Scopo di questa nota è mostrare la stretta relazione che intercorre tra la costante di JungJ(X) di uno spazio di BanachX e la costante λ₁(X). Si stabilisce una limitazione superiore di λ₁, come funzione diJ, dalla quale si possono dedurre alcuni nuovi risultati e altri conosciuti. è inoltre dimostrata una caratterizzazione deiJ ₁-spazi.

     

On the peripheral spectrum of positive operators

We prove that ifE is a Banach lattice andS, T ∈ ℒ (E) are such that 0≦s≦T,r(s)=r(T) andr(T) is a Riesz point ofσ(T) thenr(S) is a Riesz point ofσ(S). We prove also some results on compact positive perturbations of positive irreducible operators and lattice homomorphisms.     

Sous-Potentiels D’un Operateur Nonlineaire

LetA be a (nonlinear) operator in an ordered linear spaceX with resolvantJ _λ=(I+λA)^-1 well-defined onX and non-decreasing for any smallλ>0, andν ∈X. We define sub-potential ofν with respect toA, as anyu ∈X satisfyingu≧J _λ(u+λv) for smallλ>0, and show that this coincides with the notion of sub-solution of the equationAu∋ν in some abstract cases where such notion is defined in a natural way. At last, we give some general properties of sub-potentials, in particular an extension of the Kato inequality whenX is a lattice, and, for good set of constraintsU, existence of a largest solution for the control problem:u ∈U andu is a sub-potential ofν with respect toA.