Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

Perron-frobenius theory for Banach spaces with a hyperbolic cone

If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Simultaneous approximation of algebraic irrationalities

This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers ₁ and ₂ from a totally real algebraic number field, the constant ₁₂ can be explicitly calculated, this being the upper limit of the numbers c₁₂ such that the inequality max (q₁, q₂)(qc₁₂)^–1/2 holds for infinitely many natural numbers q; likewise for the constant a₁₂ such that the inequality q₁·q₂< a₁₂(q^logq) holds for infinitely many natural numbers q. It is shown that there exist n –1 numbers ₁, ..., _n–1 in an algebraic number field of degree n and discriminant d such that the inequality holds only for finitely many natural numbers q if 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d $$ " align="middle" border="0"> . is fixed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 142–154, 1982.     

Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Eine Verallgemeinerung des Satzes vom abgeschlossenen Wertebereich in lokalkonvexen Räumen

This paper extends Kato's proof [5] of Banach's closed range theorem to locally convex spaces. Thus we consider a locally convex space (E,) and pairs (M,N) of closed subspaces. We call such a pair -open, if and only if there exists a directed, total system of seminorms generating the topology induced by a on M+N, such that the minimal gap _p(M,N)>O for each p. Our main result is a generalisation of the closed range theorem and it consists of statements on relationships between the following properties: (a) M+N -closed, (b) M+N (E,E)-closed, (c) M+N (E,E)-closed, (d) (M,N) -open, (e) (M,N) (E,E)-open, (f) (M,N) (E,E)-open, (g) (M,N) (E,E)-open, (h) M+N=(MN), (i) M+N=(MN).By specialising the space (E,) and the subspaces M,N, our generalisation includes the closed range theorems of Dieudonné and Schwartz [4], Browder [1] and Mochizuki [12]. It is shown that these theorems not only hold for closed linear operators but even for closed linear relations. We are therefore able to obtain closed domain theorems which extend Brown's examinations in Banach-spaces [2] to locally convex spaces.

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Herrn Gottfried Köthe zum 70. Geburtstag am 25.12.1975 gewidmet     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Convergence of simultaneous Padé approximants to two dilogarithms

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Joint norm of operators and an investigation of nonlinear problems

Nonlinear operator equations of the form x=Fx in a real-valued Hilbert space H are studied. If the operator F is completely continuous and admits the bound Fx< Bx+b, where B is a continuous linear operator then for B<1 the Schauder principle is applicable to the equation x=Fx and this equation possesses at least one solution x H. If the bound Fx<,B₁x+B₂x+b is valid where B₁ and B₂ are bounded linear operators then the simplest conditions for solvability of the equation x=Fx is of the form B₁+B₂<1. This condition could be relaxed. The proposed method is applied to the investigation of a two-point boundary problem (cf., e.g., [1–3]). New conditions for the existence of solutions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1605–1616, December, 1990.     

Diagonale Netze aus Schmieg- und Krümmungslinien I

In this paper we develop the theory of nets of curves in a regular C^r-2-surface Eⁿ (r1, n2) using the concept C^s-net (of curves); the term diagonal nets of curves defined by W. BPLASCHKE [2] in E² is generalized accordingly. A regular C^r-surface E³ (r2) of negative GAUSSian curvature is called a (C^r-)D_SK-surface if its asymptotic lines (S-lines) and lines of curvature (K-lines) locally form a pair of diagonal nets. For the C³-D_SK-surfaces a criterion is given and distinct categories are determined, in particular all those C³-D_SK-surfaces in which the S- and K-lines can be arranged as (curvilinear) kites, respectively parallelograms and their diagonals.

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Auszugsweise vorgetragen auf der Geometrietagung in Oberwolfach (1.10.l974).     

Über die Existenz gemeinsamer Elemente bester Approximation bezüglich zweier Normen II

The present paper deals with the possibility of existence of best approximation elements, simultaneously with respect to two norms ·_i,i=1,2, for all the elements of a class of subspaces. In case this class in any of the following: (a) All n-dimensional subspaces, (b) All ·₁-or ·||₂-closed, n-codimensional subspaces, (c) All ·₁-or ·₂-closed subspaces with infinite dimension and codimension, we prove that the two norms differ at most by a constant factor.     

Unique <Emphasis Type="Italic">a</Emphasis>-closure for some ℓ-groups of rational valued functions

Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct a-closures. Here, we find a reasonably large class with unique and perfectly describable a-closure, the class of archimedean -groups with weak unit which are -convex. ( is the group of rationals.) Any C(X, ) is -convex and its unique a-closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C(X).     

Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+(x,x), >0. For example, ifA is a bounded operator with a bounded inverse from a Hilbert spaceH into itself and the control setU is the unit ball inH, then an optimal control is bang-bang (has norm l) if 0<1/2;A ^–1*r·A ^–1–2, but is singular (an interior point ofU) if >1/2A ^–1*r·A².This work was supported by NRC Grant No. A-4047 and NSF Grant No. GP-7445.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

Approximation and regular perturbation of optimal control problems via Hamilton-Jacobi theory

We present two convergence theorems for Hamilton-Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. LetT andT _h be the minimal time functions to reach the origin of two control systemsy = f(y, a) andy = f _h (y, a), both locally controllable in the origin, and letK be any compact set of points controllable to the origin. If f _h –f Ch, then |T(x) – T _h (x)| C _K h , for all x K, where is the exponent of Hölder continuity ofT(x).     

Polynomial Wavelet-Type Expansions on the Sphere

We present a polynomial wavelet-type system on S ^d such that any continuous function can be expanded with respect to these wavelets. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a dual wavelet system. The dual wavelets system is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.