Showing papers in "Israel Journal of Mathematics in 1977"

Quelques propriétés des opérateurs angle-bornés et n -cycliquement monotones

Abstract: The paper deals with properties of σ-angle bounded linear and nonlinear operators. In particular it is proved that ifA is a σ-angle bounded linear maximal monotone operator in a real Hilbert space then −A is the infinitesimal generator of an analytic semi-group.

Non-ergodic interval exchange transformations

Abstract: We construct interval exchange transformations on four intervals satisfying a strong irrationality condition and having exactly two ergodic invariant probability measures. This shows that although Kronecker’s theorem remains true for interval exchange transformations, the Weyl equidistribution theorem is false even under the strongest irrationality assumptions.

Equivalence of topological Markov shifts

Abstract: We show that any two topological Markov shifts both of whose topological entropy equals logn (for somen) are equivalent by a finitistic coding.

On dentability and the Bishop-Phelps property

Abstract: It is shown that for Banach spaces the Radon-Nikodym property and the Bishop-Phelps property are equivalent. Using similar techniques, we prove that ifC is a bounded, closed and convex subset of a Banach space such that every nonempty subset ofC is dentable, then the strongly exposing functionals ofC form a denseGδ-subset of the dual.

On the singular cardinals problem i

Abstract: We show how to get a model of set theory in which ℵω is a strong limit cardinal which violates the generalized continuum hypothesis. Generalizations to other cardinals are also given.

Unique ergodicity for horocycle foliations

Abstract: The notion of unique ergodicity is extended to multidimensional foliations, and it is shown that ifg is the strong stable or strong unstable foliation of a topologically mixing basic set Ωκ for an Axiom A diffeomorphism or flow theng is uniquely ergodic

Proof of Springer’s hypothesis

Abstract: LetG be a reductive group over a finite fieldk of a characteristicp. Π:G k → AutU is an irreducible representation ofG in “a general position”. Springer formulated a conjecture about values of the character of Π on unipotent elements. This conjecture is proved in the article.

Nonlinear evolution equations in an arbitrary Banach space

Abstract: This paper proves the existence of an evolution operatorU(t, s)x 0 corresponding to a weak or generalized solution of the differential equation:du (t)/dt +A (t)u(t) ∋ f(t), u(s) =x 0,t ≧ s; the operatorsA(t) are eachm-accretive in a Banach spaceX and, loosely speaking, have an “L1 modulus of continuity” int. The continuity and differentiability properties ofU(t, s)x0 are investigated, and some simple examples are presented.

On uncomplemented subspaces of L p, 1 <p <2

Abstract: It is shown tat, for 1

The separable extension problem

Abstract: It is proved that every infinite dimensional separable Banach space having the separable extension property is isomorphic to c0. It is also proved that every Banach space with a separable dual is “close” to a space of continuous functions on a countable compact space.

On Banach spaces whose dual balls are not weak∗ sequentially compact

Abstract: Theorem 1. LetX be a Banach space. (a) IfX ∗ has a closed subspace in which no normalized sequence converges weak∗ to zero, thenl 1 is isomorphic to a subspace ofX. (b) IfX ∗ contains a bounded sequence which has no weak∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable.

Whitehead groups may be not free, even assuming CH, I

Abstract: We prove the consistency with ZFC+G.C.H. of an assertion, which implies several consequences of\(MA + 2^{\aleph _0 } > \aleph _1 \), which\(\diamondsuit \aleph _1 \) implies their negation.

The structure of skew products with ergodic group automorphisms

Abstract: We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.

Every planar graph has an acyclic 7-coloring

Abstract: A proper coloring of the vertices of a graph is said to be acyclic provided that no cycle is two colored. We prove that every planar graph has an acyclic seven coloring.

A class of finitary codes

Abstract: It is shown that for any two Bernoulli schemes with a finite number of states and unequal entropies, there exists a finitary homomorphism from the scheme with larger entropy to the one with smaller entropy.

Bernoulli maps of the interval

Abstract: The ergodic properties of expanding piecewiseC2 maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness).

Rational ergodicity and a metric invariant for Markov shifts

Abstract: The concept of rational ergodicity is introduced and used to construct a metric invariant for the class of rationally ergodic transformations (which includes all ergodic Markov shifts).

Power and taxes in a multi-commodity economy

Abstract: Taxation and redistribution in a democratic majority-rule society are analyzed, using the Harsanyi-Shapley non-transferable utility value. The context is that of a multi-commodity pure exchange economy. Two approaches are treated: one in which taxes are in kind and exchange takes the form of barter; and one in which taxes are in money, exchange takes the form of sale and purchase, and prices are determined by a process of supply and demand. It is shown that in the presence of a non-atomic continuum of agents, the two approaches are equivalent, but that this is not so when there are only finitely many agents. It is also shown that the value exists under both approaches, and a characterization is found in the non-atomic case.

On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space

Abstract: LetT be a nonexpansive mapping on a closed convex subsetC of a real Hilbert spaceH. In the present note we deal with the weak convergence of the sequenceT n x and the sequenceS n x of the arithmetical means of the sequenceT n x, asn → ∞. We also give some results concerning the strong convergence ofS n x.

On the ergodic theory of non-integrable functions and infinite measure spaces

Abstract: We prove a Chow-Robbins type result for an ergodic, non-negative SSSP, and a similar result for transformations preserving infinite measure, which implies that for these transformations, no “absolute” version of Hopf's theorem can hold.

A generalization of Quillen’s lemma and its application to the Weyl algebras

Abstract: Quillen’s lemma [17] is generalized to modules of arbitrary Krull dimension. This leads to some generalizations of the results of [5] and [12] for the Weyl algebras of index > 1.

On banach spaces which containl 1(τ) and types of measures on compact spaces

Abstract: Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczynski's about Banach spaces containingL 1. Let τ be a regular cardinal satisfying the hypothesis that κω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]τ. 2) A Banach spaceX has a subspace isomorphic tol 1(τ) if and only ifX ∗ has a subspace isomorphic toL 1({0, 1}τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl 1 does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl 1 (ω1), while the unit ball ofX ∗ is not weakly∗ sequentially compact.

Mixed width-integrals of convex bodies

Abstract: The mixed width-integrals are defined and shown to have properties similar to those of the mixed volumes of Minkowski. An inequality is established for the mixed width-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed width-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened versions of this general inequality are obtained by introducing indexed mixed width-integrals. This leads to an isoperimetric inequality similar to Busemann’s inequality involving concurrent cross-sections of convex bodies.

On the range of accretive operators

Abstract: The solvability of the nonlinear operator equationw=x+Bx, whereB is accretive in a general Banach spaceX is studied by means of discrete approximations. In particular, ifB is continuous and everywhere defined an algorithm is given for solving the equation.

Compact and strictly singular operators on orlicz spaces

Abstract: If 0 < p < 1 andT: Lp(0,1) →E is a continuous linear operator into a topological vector space, there is an infinite-dimensional subspaceX ofL p on whichT is an isomorphism; thus there are no compact operators onL p . Results of this type are proved for general non-locally convex Orlicz spaces.

Singular cohomology inL

Abstract: We prove, that in the world of constructible sets, there does not exist a spaceX withH″ (X,Z) isomorphic to the rational numbers. The proof requires a result about the growth of ExtzEmphasis>/i(-, Z) inside of Godel’s constructible universeL.

Random walks on compact groups and the existence of cocycles

Abstract: We show that certain skew products in ergodic theory are isomorphic to the shifts defined by random walks. We conclude the existence of cocycles for any finite measure preserving ergodic automorphism or flow, taking values in an arbitrary compact group, which determine ergodic skew products.

Invariant sets for a class of perturbed differential equations of retarded type

Abstract: LetX be a Banach space and leta, b, q be real numbers such thata 0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b 1],X),a0,f: [a, b)×C([−q,0],D λ)→X is continuous and ϕ∈C([−q,0],D λ) with ϕ(0)∈D. D λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.

Sous-espaces de L

Abstract: A typical result of the paper isTheorem. LetE be a reflexive subspace ofL1(Ω, A, P) [(Ω,A, P) a probability space]. IfE contains a subspace isomorphic to lp then for every e > 0,E contains a subspace (1 + e) isomorphic to lp.

Primariness of spaces of continuous functions on ordinals

Abstract: Necessary and sufficient conditions on an ordinal α are given, such thatC(α) is primary, and that the general linear group ofC(α) is contractible. In particularC(α) possesses both of these properties if α is countable.