Quasi-similarp-hyponormal operators

IfA _i i=1, 2 are quasi-similarp-hyponormal operators such thatUi is unitary in the polar decompositionA _i =U _i |A _i |, then (A ₁)=(A ₂) and _c(A₁) = _e(A₂). Also a Putnam-Fuglede type commutativity theorem holds for p-hyponomral operators.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.     

PAT – a Reliable Path-Following Algorithm

This paper presents a new technique for the reliable computation of the -pseudospectrum defined by (A)={zC : _min(A–zI)} where _min is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve (A)={zC : _min(A–zI)=} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to . The method is guaranteed to terminate, even in the presence of round-off errors.     

Analytic structures for subnormal operators

For a compactly supported measure on , we construct a mutually absolutely continuous measure so thatP ²() has analytic bounded point evaluations, and the operator of multiplication byz onP ²() has every invariant subspace hyperinvariant. We also construct an equivalent measure so thatR ²(K, ) has as analytic bounded point evaluations precisely the interior of the set of weak-star continuous point evaluations ofR (K, ). In the course of the proof, we classify weak-star closed super-algebras ofR (K, ) whenR(K) is hypo-Dirichlet.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

The Connes–Moscovici Approach to Cyclic Cohomology for Compact Quantum Groups

Consider the Hopf algebra (A, ) of regular functions on a compact quantum group. Let (A ^o ,) denote its maximal dual Hopf algebra. We show that the tensor product Hopf algebra (H ₂,₂) of (A ^o ,) and its opposite Hopf algebra is endowed with a modular pair (,) in involution; a notion introduced by A. Connes and J. Moscovici, who associate canonically a cocyclic object to such Hopf algebras. Denote the Hopf cyclic cohomology thus obtained by HC ^* _(,)(H ₂). Next we define an action of H ₂),₂ on A and show that the Haar state of (A, ) is a -invariant -trace on A with respect to this action. This gives us a canonical map from HC ^* _(,)(H ₂) to the ordinary cyclic cohomology of A.     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Diffusions réfléchies réversibles dégénérées

On a domain D in ^d, for a smooth enough probability density and a diffusion matrix which can degenerate, we construct the law Q ^s of a (x)d -symmetric reflecting process in D with matrix . Therefore, we use the associated Dirichlet form and a sequence of approximating processes already used by Pardoux and R. Williams in [23]. Under mild conditions on the boundary ofD (finite Minkowski content), we prove that Q ^s is the law of a semi-martingale and provide its decomposition. Comparing with the decomposition in additive functionals, we conclude that the process is reflected in the conormal direction ^* n where n denotes Chen's normal (cf [10]), that is, the reflection direction of the Brownian motion in Kuramochi compactification.     

On the On-line Number of Snacks Problem

In the number of snacks problem (NSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as _: C_A()/C_OPT(), where denotes a sequence of numbers of customers, C _OPT() is the cost of satisfying by an optimal off-line algorithm, and C _A() is the cost of satisfying by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm(ENHA), with competitive ratio 1+p(M-m)/(M+m), where M and m are two extreme numbers of customers over the total period of the game, and p is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Onp-hyponormal operators for 0<p<1

The distance formula Tt-I)^–1=[Dist(, (T)]^–1, (T), for hyponormal operators, is generalized top-hyponormal operators for 0<p<1. Several other results involving eigenspaces ofU and |T|, the joint point spectrum, and the spectral radius are also otained, where |T|=(T ^* T)^1/2 andU is the unitary operator in the polar decomposition of thep-hyponormal operatorT=U|T|.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

On the convergence order of accelerated root iterations

Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (kN) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+ _n (k), where _n (k)>1 is the unique positive root of the equation ⁿ --k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.     

Estimations de l'erreur d'approximation sur un domaine borné deR n parD m -splines d'interpolation et d'ajustement discrètes

Résumé Considérant un espace discretV _h associé àH ^m (), une fonctionfH ^m+1 () et laD ^m -spline d'interpolation discrète _h ^d def dansV _h (cf. [1]), on établit des estimations de l'erreurf– _h ^d en fonction de la distance de Hausdorffd de et de l'ensembleA ^d des points de données, du type |– _h ^d | _l, =o(d ^m–l ), en utilisant des résultats de Duchon [5].De la même façon, on établit des estimations de l'erreurf– _h ^d , oùfH ^m (),m entier >m, et _h ^d désigne laD ^m -spline d'ajustement discrète def dansV _h de paramètre >0 (cf. [1]), du type |– _h ^d | _l, =o(d ^m–l )+O(d ^n/2–1/2). La méthode suivie est applicable auxD ^m -splines d'ajustement surR ⁿ de Duchon [4].

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Approximation error estimates on a bounded domain inR ⁿ for interpolating and smoothing discreteD ^m -splines

Summary Considering a discrete spaceV _h associated withH ^m (), a functionfH ^m+1 () and the interpolation discreteD ^m -spline _h ^d off inV _h (cf. [1]), and using Duchon's results [5], we establish estimates of the errorf– _h ^d . These estimates are of the type |– _h ^d | _l, =o(d ^m–l ), whered is the Hausdorffs distance between and the setA ^d of data points.In the same way, we establish estimates of the errorf– _h ^d , wherefH ^m (), andm>m, and _h ^d is the smoothing discreteD ^m -spline off inV _h associated to the parameter >0 (cf. [1]). These estimates are of the type |– _h ^d | _l, =o(d ^m–l )+O(d ^n/2–1/2). The proposed method can be applied to the smoothingD ^m -splines in #x211D; ⁿ of Duchon [4].

     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.