Showing papers in "Israel Journal of Mathematics in 1972"
TL;DR: In this paper, the authors extend Simon's theorem to a more general case, where the Schrodinger operator is essentially self-adjoint onC(R)m), if 0≦q ∈L>>\s 2(R>>\s m), andq>>\s 1(x)≧−q*(|x|) withq *(r) monotone nondecreasing inr ando(r petertodd 2) asr → ∞.
Abstract: Recently B. Simon proved a remarkable theorem to the effect that the Schrodinger operatorT=−Δ+q(x) is essentially selfadjoint onC
0
∞
(R
m
if 0≦q ∈L
2(R
m). Here we extend the theorem to a more general case,T=−Σ
=1/
(∂/∂x
j −ib
j(x))2 +q
1(x) +q
2(x), whereb
j, q1,q
2 are real-valued,b
j ∈C(R
m),q
1 ∈L
loc
2
(R
m),q
1(x)≧−q*(|x|) withq*(r) monotone nondecreasing inr ando(r
2) asr → ∞, andq
2 satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq
1 is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.
677 citations
TL;DR: In this article, the evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where the A(T) are nonlinear operators acting in a Banach space, is studied.
Abstract: The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases.
347 citations
TL;DR: In this paper, the concept of zero set for the Haar measure can be generalized to abelian Polish groups which are not necessarily locally compact, and it turns out that these groups, in many respects, behave like locally compact groups.
Abstract: It is shown that the concept of zero set for the Haar measure can be generalized to abelian Polish groups which are not necessarily locally compact. It turns out that these groups, in many respects, behave like locally compact groups. Suitably modified, many theorems from harmonic analysis carry over to this case. A few applications are given and some open problems are mentioned.
286 citations
TL;DR: In this article, the following theorem is proved: if A and B are twoPI-algebras over a field F, then A⊗ B is a PI-algebra.
Abstract: The following theorem is proved: ifA andB are twoPI-algebras over a fieldF, thenA⊗ F B is aPI-algebra.
277 citations
TL;DR: In this paper, necessary and sufficient conditions for a Banach space to have an equivalent uniformly convex norm are given, where the norm is defined in terms of the convexity of the norm.
Abstract: Necessary and sufficient conditions for a Banach space to have an equivalent uniformly convex norm are given.
225 citations
TL;DR: In this paper, the Cauchy problem for ut + Σi = 1n (φi(u))xi = 0 is treated via the theory of semigroups of nonlinear transformations.
Abstract: The Cauchy problem for ut + Σi = 1n (φi(u))xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σi = 1n (φi(u))xi = h for h∈L1(Rn), which in turn is studied via the regularized equation
$$ u + \sum
olimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$
.
175 citations
TL;DR: In this paper, the union of the arcs covers C with probability one if and only if the arcs are uniformly thrown independently and uniformly on a circumference C of unit length, and the union coversC with probability only if
Abstract: Arcs of lengthsl
n, 0
136 citations
TL;DR: In this article, a generalisation of the equations d'evolution non lineaires monotones a entrees stochastiques du type suivant is presented. But this generalisation is only applicable to the case of the stochastic model.
Abstract: Le but de cet article est d’etudier les equations d’evolution non lineaires monotones a entrees stochastiques du type suivant:
$$\frac{{dy}}{{dt}}\left( {t; } \right) + A\left( t \right)y\left( {; } \right) = g \left( {t; } \right) + \frac{{df}}{{dt}}\left( {t; } \right)$$
ouA(t) est une famille d’operateurs monotones d’un espace de BanachV dans son dualV′. Il s’agit d’une generalisation a des equations aux derivees partielles des travaux de Ito sur les equations differentielles stochastiques.
135 citations
TL;DR: For any integern such that 8|n or for which there exists an odd primeq such thatq 2|n, there is a central division algebra of dimensionn 2 over its center which is not a crossed product as mentioned in this paper.
Abstract: For any integern such that 8|n or for which there exists an odd primeq such thatq 2|n, there is a central division algebra of dimensionn 2 over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X 1,…,X)m, the algebra generated over the rationalQ bym(≧2) generic matrices.
128 citations
TL;DR: In this article, the Kakutani-Rokhlin tower theorem is proved in a refined form for nonperiodic groups Φ, and the Shannon-McMillan theorem is extended to ergodic groups.
Abstract: Let φ1, ... ,φd be commuting measure-preserving transformations, \( \phi ^l \equiv \phi _1^{l_1 } \phi _2^{l_2 } \cdot \cdot \cdot \phi _d^{l_d } ,\Phi = \left\{ {\phi ^l } \right\} \). The Kakutani-Rokhlin tower theorem is proved in a refined form for non-periodic groups Φ, and the Shannon-McMillan theorem is extended to ergodic groups. These results are used to extend recent isomorphism results to groups of transformations.
122 citations
TL;DR: In this paper, it was shown that every Banach space has the metric approximation property (m] a.a.p., in short) iff (ΣG�Ω(G� *� )� (G� *)ⓘ m� Ω mⓘ does not, and if there is a space failing the m. a.p, then the space can be equivalently normed to fail m. p.
Abstract: LetC 1=(ΣG
n
)
l
1, where (G
n
) is a sequence which is dense (in the Banach-Mazur sense) in the class of all finite dimensional Banach spaces. IfX is a separable Banach space, thenX
* is isometric to a subspace ofC
1
*
=(ΣG
*
)
m
which is the range of a contractive projection onC
1
*
. Separable Banach spaces whose conjugates are isomorphic toC
1
*
are classified as those spaces which contain complemented copies of C1. Applications are that every Banach space has the [metric] approximation property ([m.] a.p., in short) iff (ΣG
*
)
m
does, and if there is a space failing the m.a.p., thenC
1 can be equivalently normed to fail the m.a.p.
TL;DR: In this paper, the convex functional J(u) = ∫Ωj(u)/dx on the space W0s,p (Θ) is considered and a description of its conjugate J* on W−s, p′(Ω) and its subdifferential∂J are given.
Abstract: The convex functionalJ(u) = ∫Ωj(u)dx on the spaceW0s,p (Θ) is considered. A description of its conjugateJ* onW−s, p′(Ω) and its subdifferential∂J are given.
TL;DR: In this article, an example of a group of class 3 with a non-trivial fourth dimension subgroup is given, where the subgroup can be represented by a 4-dimensional subgroup.
Abstract: An example of a group of class 3 with a non-trivial fourth integer dimension subgroup is given.
TL;DR: In this article, a relatively easy proof for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball is given.
Abstract: A relatively easy proof is given for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball. This proof simplifies considerably for separable spaces and can be extended to give a proof that a boundedw-closed subsetX of a complete locally convex linear topological space isw-compact if and only if each continuous linear functional attains its sup onX.
TL;DR: In this paper, it was shown that the set of p's such thatlp is isomorphic to a subspace of a given Orlicz spacelF forms an interval.
Abstract: It is proved that the set ofp's such thatlp is isomorphic to a subspace of a given Orlicz spacelFforms an interval. Some examples and properties of minimal Orlicz sequence spaces are presented. It is proved that an Orlicz function space (different froml2) is not isomorphic to a subspace of an Orlicz sequence space. Finally it is shown (under a certain restriction) that if two Orlicz function spaces are isomorphic, then they are identical (i.e. consist of the same functions).
TL;DR: In this paper, it was shown that every complemented subspace of C with non-separable dual is isomorphic (linearly homeomorphic) to C. An immediate consequence of this and a result of A. Pelczynski, is that every subspace Y of C, isometric to C, such that T|Y is an isomorphism.
Abstract: LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT
*
X
* non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC.
TL;DR: In this article, the authors show that every subspace of a quotient space is isomorphic to a subspace in the space ℒ¯¯¯¯ p� (resp.p=∞).
Abstract: Let 1
TL;DR: In this paper, a variational inequality for a second order uniformly elliptic operator on a bounded domain is studied, the solution of which is required to lie above a given obstacle and to assume assigned values on a part of the boundary of the domain.
Abstract: In this paper we study a variational inequality for a second order uniformly elliptic operator on a bounded domain, the solution of which is required to lie above a given obstacle and to assume assigned values on a part of the boundary of the domain. We are mainly concerned with the regularity of the solution in relation to the regularity of the data.
TL;DR: In this article, the Cauchy problem in a separable Hilbert space setting was studied, where A is a maximal monotone operator, and B is a multivalued operator which is measurable with respect to the time variable.
Abstract: The Cauchy problemdu/dt+Au+B(t,u)∋0,u(0)=u
0 is studied in a separable Hilbert space setting, whenA is a multivalued maximal monotone operator, andB is a multivalued operator which is measurable with respect to the time variable and upper semi-continuous with respect to the space variable. Under some boundedness conditions onB, an existence theorem is proved, with the extra assumption, in the infinite dimensional case thatA is the subdifferential of a proper lower semi-continuous inf-compact convex function. A theorem of dependence upon the initial condition is also given.
TL;DR: In this article, it was shown that there exists an ann×n matrix all of whose eigenvalues and off-diagonal elements are prescribed, and the number of such matrices is finite.
Abstract: It is shown that there exists ann×n matrix all of whose eigenvalues and off-diagonal elements are prescribed. The number of such matrices is finite.
TL;DR: In this article, the existence of disjoint sets has been studied in the context of set-theoretic axioms and existence theorems, and the authors solve some problems arising from a work by Hechler.
Abstract: We solve here some problems arising from a work by Hechler [3]. We eliminate extra set-theoretic axioms (MA, in fact) from existence theorems and deal with the existence of disjoint sets.
TL;DR: In this paper, the minimax solution for a game in which player I chooses a real number and player II seeks it by choosing a trajectory represented by a positive function was found.
Abstract: The minimax solution is found for a game in which player I chooses a real number and player II seeks it by choosing a trajectory represented by a positive function.
TL;DR: In this paper, the existence and uniqueness of monosplines satisfying certain boundary conditions with a maximal prescribed number of zeros is established, which is of value in characterizing optimal quadrature formulas and in problems of best approximations involving free knots.
Abstract: The existence and uniqueness of monosplines satisfying certain boundary conditions with a maximal prescribed number of zeros is established. This results is of value in characterizing optimal quadrature formulas and in problems of best approximations involving free knots.
TL;DR: In this article, the authors give an example of a Banach space X such that X * is isometric to a subspace of C(ωθ) and X is not isomorphic to a complemented subspace in any C(K) space.
Abstract: We give an example of a Banach spaceX such that (i)X
* is isometric tol
1, (ii)X is isometric to a subspace ofC(ωθ) and (iii)X is not isomorphic to a complemented subspace of anyC(K) space.
TL;DR: In this paper, an existence and uniqueness theorem for bounded solutions of the initial value problem has been proved, where the generator of this semigroup is denoted by a semi-group of contractions on D(A).
Abstract: This paper extends some recent results of V. Barbu. It is concerned with bounded solutions of the problem:u″∈Au, u′(0)∈ϖj(u(0)−a) whereA is a maximal monotone operator in a Hilbert spaceH, a∈D(A) andj is a strictly convex l.s.c. function fromH to [0,+∞]. An existence and uniqueness theorem for this problem is proved. Takingj to be the indicator function of a pointu
0∈D(A), one obtains a bounded solutionu(t) of the initial value problem:u″∈Au, u(0)=u
0. Denotingu(t)=S
1/2(t)u0 one obtains a semi-group of contractions onD(A). The generator of this semigroup is denoted byA
1/2. Further properties ofS
1/2(t) andA
1/2 are studied.
TL;DR: If a homeomorphism possesses a non-atomic ergodic measure, then it has recurrent points as discussed by the authors and there are uncountably many inequivalent ergodically quasi-invariant measures.
Abstract: If a homeomorphism possesses a non-atomic ergodic measure it has recurrent points. If it has recurrent points then there are uncountably many inequivalent ergodic quasi-invariant measures.
TL;DR: In this article, it was shown that M-ideals coincide with the algebraic ideals generated by p-sets, which in turn yields a description of the primitive ideal space, PrimA, as the Choquet-boundary endowed with p-set topology.
Abstract: Convexity arguments are applied to characterize theM-ideals of a given complex function spaceA⊆C(X) The main result is the following: A closed subspaceJ ofA is anM-ideal if and only ifJ={a∈ A⇑ a≠ O onE}, whereE is anM-set ofX Specializing to uniform algebras it is shown thatM-ideals coincides with the algebraic ideals generated byp-sets, which in turn yields a description of the primitive ideal space, PrimA, as the Choquet-boundary endowed withp-set topology
TL;DR: In this paper, it was shown that T* maps (the natural imbedding inC(Y) of) Y into the weak*-closure of extS(E*), provided that Y is extremally disconnected, or E = C(X), where X is a dispersed compact Hausdorff space.
Abstract: LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT* maps (the natural imbedding inC(Y)* of)Y into the weak*-closure of extS(E*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.
TL;DR: It was shown in this article that Mumford's compactness theorem for Fuchsian groups remains valid for groups containing elliptic and parabolic elements, and it was shown that the compactness of Fuchsians can be maintained for groups with only two elements.
Abstract: It is shown that a recent compactness theorem for Fuchsian groups, due to Mumford, remains valid for groups containing elliptic and parabolic elements.
TL;DR: In this paper, the existence of a prime minimal model containing an infinite set of indiscernibles was shown and a sentence ofLω1ω which is categorical in ω 1 but whose model of power ω1 is not (ω 1,Lω 1ω-homogeneous).
Abstract: We show the existence of a prime minimal model containing an infinite set of indiscernibles. We then find a sentence ofLω1ω which is categorical in ω1 but whose model of power ω1 is not (ω1,Lω1ω-homogeneous. This answers a question posed by Keisler.