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.     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

Ein neues Surjektivitätskriterium im Hilbertraum

LetA, B andC be linearm-accretive operators in a Hilbert space. Suppose further thatC is bounded, thatb:=inf {Re (C y, y)| y=1}>0, thatA ^–1 exists as a bounded operator and that Re (B ^* x, A ^–1 x)+a x²0 holds for allxD (B ^*) and a constanta with 0a<b. ThenCA+B is surjective, (CA+B)^–1 exists and C Ax+Bx (b–a) A x holds for allxD (A) D (B). This criterion can be applied to evolution equations of the formdu/dt+C(t)A(t)u=f(t) whereB:=d/dt.     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

On boundary controllability of a vibrating plate

A vibrating plate is here taken to satisfy the model equation:u _tt + ²u = 0 (where ²u:= (u); = Laplacian) with boundary conditions of the form:u _v = 0 and(u) _v = = control. Thus, the state is the pair [u, u _t] and controllability means existence of on := (0,T)× transfering any[u, u _t]₀ to any[u, u _t]_T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T ^–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u _tt = u) with log = O(T ^–1) asT0. Some related results on minimum time control are also included.This research was partially supported under the grant AFOSR-82-0271.     

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.     

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.     

On the minimum time map

Santo Si studiano alcune proprietà della funzione di tempo minimo per un'equazione differenziale multivoca su una varietà C. Si mette in relazione la funzione di tempo minimo per una famiglia di campi vettoriali con guella per l'equazione multivoca associata. Inoltre si prova che da una estensione alle varietà riemanniane complete della condizione classica F(t, x)(t)+v(t)x, si hanno le stesse conseguenze che in Rⁿ. Si prova, infine, sotto la stessa condizione, che la funzione che ad ogni t associa l'insieme raggiungibile al tempo t è localmente lipschitziana (rispetto alla metrica di Hausdorff).

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This work was performed under the auspices of the National Research Council of Italy (C.N.R.).     

Fehlerabschätzungen für Gauß-Quadraturformeln

Summary We consider Gauss quadrature formulaeQ _n ,n, approximating the integral ,w an even weight function. Let be analytic inK _r :={z:|z|<r},r>1, and . The error functionalR _n :=I-Q _n is continuous with respect to |·|_r and the relation , q_2k (x):=x ^2k holds.In this paper estimates for R _n are given. To this end we first derive two new representations of R _n which are essential for our further investigations. The R _n =r ² R _n (), with (x):=1/(r ²-x ²), is estimated in various ways by using the best uniform approximation of in P_2n-1, and also the expansion of with respect to Chebyshe polynomials of the first and second kind. Forw(x)=(1-x ²), =±1/2, R _n is calculated. The asymptotic behaviour, forr1+, of R _n and of the derived error bounds is also discussed. Finally, we compare different error bounds and give numerical examples.

     

Ergodic properties of semi-Markovian operators on the Z1-part

Summary In this note we consider a semi-Markovian operator, that is a positive linear mapping T: L ¹ L ¹ such that sup T ⁿ <. We study the behavior of T ⁿ on the Z ¹-part of the space (the disappearing part in Sucheston's terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z ¹, then the ratio theorem must fail.Research supported by the U.S.Army Research Office (Durham) under contract DA-31-124-ARO(D)-288.     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Numerical computation of least constants for the Sobolev inequality

Summary Least constantsc for the well-known Sobolev inequality fcf _{m, G} ,fH ^m (G) are obtained in closed form by a reproducing kernel technique, where the Sobolev spaceH ^m (G) for a domainG in ⁿ is defined as the completion ofC ^m (G) with respect to the Sobolev norm given by , where is the norm ofL ₂ (G) and is the supremum norm onG. Numerical values for the case whereG is the ⁿ are given.     

Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

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.     

Upper bounds for the best one-sided approximation by splines of the classes WrL1

In the present note we will investigate the problem of the one-sided approximation of functions by n-dimensional subspaces. In particular, we will find the exact value of the best one-sided approximation of the class W^rL₁ (r=1, 2, ...) of all periodic functions f(x) of period 2 for which f^(r–1)(x) (f⁽⁰⁾(x)=f(x)) is absolutely continuous and f^(r)L₁1 by periodic spline functions S_2n ( = 0, 1, ..., n=1, 2, ...) of period 2, order ,and deficiency 1.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 11–17, January, 1976.     

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.     

Index of exponential growth for a class of stochastic semigroups

We consider finite-dimensional homogeneous stochastic semigroups X _s ^t , 0 s t < assuming values in the space of real square matrices. For stochastic semigroups assuming values in the class of upper triangular matrices we compute the index of exponential growth , where · is the operator norm of a matrix. The answer is given in terms of the characteristic Y_t of the generating process Y_t of the semigroup X_s ^t:x=–(1/2), where is the smallest eigenvalue of the matrix B which defines the characteristic Y_t=Bt.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 78–84, 1988.     

A property of a system of functions close to exponential functions

We consider the system {f _n=xⁿ[l+_n]} in the interval [a,b] (0 a n > 0 and _n(x) such as the condition, we obtain a bound for the coefficients of the polynomial P(x)=#x2211;c_n f _n(x) in terms of P(x)L_p[a,b]. It is found that this bound is not valid without this condition (assuming the other conditions to remain the same).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 29–36, July, 1972.