Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Functions of a second order elliptic operator in rearrangement invariant spaces

The functional calculus of positive operators is applied to second-order elliptic operatorsP. For any absolutely concave (t), the corresponding operators (P ^–1) are represented as integral operators, their kernels are estimated, and these estimates are used for studying (P ^–1) in Lorentz, Marcinkiewicz and Orlicz spaces. Most of results obtained are sharp.     

Equivalent composition operators

If is an analytic function taking the unit disk into itself then the composition operatorC can be defined on the Hardy spaceH ^p(D) byC (f)=fo. In this work, the question of when two of these operators are equivalent in some sense is studied. In some cases, it turns out that the only equivalences are those that are induced by an invertible composition operator. However, other cases are exhibited in which there are equivalences that are not induced by an invertible composition operator.Much of this work comes from the author's thesis, written under the direction of Carl C. Cowen at Purdue University. Final preparations were accomplished with the help of a faculty development grant from the University of Tampa.     

Spectral properties of certain operators on Hardy spaces of planar regions

If G is any region in the complex plane and H^P (G), 1p, denotes the Hardy space of analytic functions on G, then the principal aim of this paper is to present the spectral properties of the multiplication operators M defined on H^P(G) by Mf=f, where is any bounded analytic function on G. In order to do this, the speccial case in which G is bounded and (z)=z is studied in detail. In addition an operator called the truncation operator that is not a multilication operator and acts on the Hardy spaces of regions of the form /K for a compact set K is studied.Dedicated to the memory of my brotherThis research was partially supported by National Science Foundation grant MCS 83-204-26.     

A Class of Operators on L h 2 . II

Let D be the open unit disk in C, and L _h ² the space of quadratic integrable harmonic functions defined on D. Let be a function in L(D) with the property that (b) = lim _x b,xD (x) for all b D. Define the operator C on L _h ² as follows: C(f) = Q( · f), where Q is the orthogonal projection of L²(D) onto L _h ² . In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then |D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L _h ² generated by , and K(D) is the ideal of compact operators on L _h ² . Finally, the spectrum of classes of operators defined on L _h ² is characterized.     

On the stability of the transport equation

We consider boundary value problems associated with the equation T=–A in a Hilbert Space, where T and A are bounded, self adjoint, injective, and A has a bounded inverse. We discuss the stability of the solution when A is perturbed by a self adjoint operator.     

On the reduction of a set of numerical matrices to quasi-diagonal form with one similarity transformation

We consider finite sets of numerical matrices and the polynomial matrices corresponding to them that have the Smith form diag (1, (x), ..., (x)). We solve the problem of reducing such sets to canonical form with one similarity transformation assuming that all the roots of the invariant polynomial (x) are simple.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 101–105.     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

L ∞ — Decay estimations of the spherical mean value on symmetric spaces

LetM _t[](x) be the spherical mean value operator applied to a function on a symmetric Riemannian space of the non-compact type.L —decay estimations forM _t [](x) as well as for its derivatives with respect to (t, x) are given, provided that belongs to a Banach space with suitable weighted supremum norm. This leads to estimates of the solutions to the wave equation in certain cases in which Huygens' principle is valid.     

On the Existence and Regularity of Mass-Minimizing Currents with an Elastic Boundary

We study the following variational problem. For a compact manifold S₀ embedded in the Euclidean space we consider deformations of S₀. They are represented by Lipschitz continuous homeomorphisms of S₀ whose images are embedded manifolds. We introduce an energy of a deformation which depends on the first derivative of the curvature of (S₀) and the mass of a mass minimizing current which is bounded by (S₀). In this paper it is shown that an energy minimizing deformation of (S₀) exists. Moreover, in the case that S₀ has codimension 1, (S₀) is an embedded C^3a -submanifold, if is of the class C^2,1.     

Inclusion statements for operator equations by a continuity principle

The paper is concerned with Range-Domain Implications MvCvK, where M is a given operator and C,K denote given sets. Sufficient conditions are derived by a very general continuity principle. Various special cases are considered such as inverse-positivity, MvMwvw, and a generalization H(,[,])MvH(,[,]) v, where Mu=H(u,u) and [,] denotes an order interval. These results are applied to differential operators related to boundary or initial value problems. The goal is to furnish a simple uniform approach, to explain its application, and to provide a kind of survey on what problems have been treated in this way.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

Holomorphic solutions of an inhomogeneous Cauchy equation

Summary We consider the functional equation(x + y) – (x) – (y) = f(x)f(y)h(x + y) and we find all its homomorphic solutionsf, h, defined in a neighbourhood of the origin.     

Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities

We consider an extension of the auxiliary problem principle for solving a general variational inequality problem. This problem consists in finding a zero of the sum of two operators defined on a real Hilbert space H: the first is a monotone single-valued operator; the second is the subdifferential of a lower semicontinuous proper convex function . To make the subproblems easier to solve, we consider two kinds of lower approximations for the function : a smooth approximation and a piecewise linear convex approximation. We explain how to construct these approximations and we prove the weak convergence and the strong convergence of the sequence generated by the corresponding algorithms under a pseudo Dunn condition on the single-valued operator. Finally, we report some numerical experiences to illustrate the behavior of the two algorithms.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.