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].     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

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     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

Variational principles for real eigenvalues of self-adjoint operator pencils

Double variational principles are established for eigenvalues of a (norm) continuous self-adjoint operator valued functionL defined on a real interval [, [.L() is not required to be definite for any . Applications are made to linear, quadratic and rational functionsL.This author acknowledges support from NSERC of Canada and the I.W. Killam Foundation.This author was supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 12176-MAT.     

A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

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     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

A Method of Solution for the One-Dimensional Heat Equation Subject to Nonlocal Conditions

The problem of solving the one-dimensional heat equation /t - ²/x² = f(x, t) subject to given initial and nonlocal conditions is considered. It is solved in the Laplace transform domain by taking the Laplace transform of the unknown function with respect to time t. The physical solution is recovered with the help of a numerical technique for inverting the Laplace transform.AMS Subject Classification (1991): 35K20.     

Normal and seminormal eigenvalues of matrix functions

LetP() be ann×n analytic matrix function andW(P) be its numerical range. In this paper classical results on the normality of matrix eigenvalues on W(P) are generalized to the context of such matrix functions. Special attention is paid to corners of W(P) and to the special case of matrix polynomialsP().     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Let be a barreled locally convex space. A continuous operator on is called anequicontinuous generator if { ⁿ /n!;n=0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series. There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators ₁, ₂ satisfy [₁,₂]=₂ for some thena₁+b₂ is also an equicontinuous generator for anya, b. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.Work supported by Alexander von Humboldt-Stiftung and Japan Society for Promotion of Sciences     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.