Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say yes to answer the following question raised by D. A. Herrero.Given an operatorT with connected spectrum (T) and a positive number , can we find a compact operatorK with K < such thatT+K is strongly irreducible?Supported by National Natural Science Foundation of China(19901011), Mathematical Center of State Education Commission of China and 973 Project of China     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

Cellular-indecomposable subnormal operators III

A bounded operatorT is called cellular-indecomposable ifL M {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH . This completes our previous work [5], [6].     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

Matrix functions with arbitrarily prescribed left and right partial indices

We prove that ifn2 and , are two given vectors inZ ⁿ, then there exists a matrix function inL ^n×n (T) which has a right Wiener-Hopf factorization inL ² with the partial indices and a left Wiener-Hopf factorization inL ² with the partial indices .     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

On the cauchy problem for some non-Kowalewskian equations inD {σ} L2 (*)

A pseudo-differential operator is considered, which generalizes some peculiar non-Kowalewskian operators of 2-evolution type. A result is proved about the well-posedness of the Cauchy problem inD _{} ^L2 , where 1 is Gevrey index.     

Quadrature method for singular integral equations on closed curves

Summary A method which combines quadrature with trigonometric interpolation is proposed for singular integral equations on closed curves. For the case of the circle, the present method is shown to be equivalent to the trigonometric -collocation method together with numerical quadrature for the compact term, and is shown to be stable inL ² provided the operatorA is invertible inL ². The results are extended to arbitraryC curves, to give a complete error analysis in the scale of Sobolev spacesH ^s . In the final section the case of a non-invertible operatorA is considered.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Geometry and the Jones projection of a state

LetA be a von Neumann algebra and a faithful normal state. ThenO = { ºAd(g ¹) :g G _A }andU = { ºAd(u ^*) :u U _A are homogeneous reductive spaces. IfA is aC ^* algebra,e the Jones projection of the faithful state viewed as a conditional expectation, then we prove that the similarity orbit ofe by invertible elements ofA can be imbedded inAA in such a way thate is carried to 1 1 and the orbit ofe to a homogeneous reductive space and an analytic submanifold ofAA.     

Cardinal hermite interpolation with box splines II

Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L ² together with its directional derivatives mentioned above. Moreover, for data sequences inl ^p ( ^d ), 1p2, there is a spline function inL ^p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst     

Lototsky-Schnabl operators associated with a strictly elliptic differential operator and their corresponding Feller semigroup

Given an open bounded convex subset of ^p , a strictly elliptic differential operatorL and a continuous function , and denoted withT _L the Dirichlet operator associated withL, the Lototsky-Schnabl operators associated withT _L and are investigated. In particular, conditions are established which ensure the existence of a Feller semigroup represented by limit of powers of these operators. Then the analytic expression of the infinitesimal generator is determined and some properties of the semigroup are deduced. Finally, the saturation class of Lototsky-Schnabl operators is determined.Work supported by a C.N.R. Research Grant (n. 201.19.1, November 30, 1994)     

A class of algebrasA with SubA anti-isomorphic to ConA

It is proved that for a finite non-empty latticeL satisfying (L, ) (L, ), Sub (L, , 0) is anti-isomorphic to Con (L, , 0).Presented by I. Rosenberg.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

On power series with positive coefficients

M. . , . , ^p () (). , , .