Specker's theorem for Nöbeling's group

Specker proved that the group of integer-valued sequences is far from free; all its homomorphisms to factor through finite subproducts. Nöbeling proved that the subgroup consisting of the bounded sequences is free and therefore has many homomorphisms to . We prove that all ``reasonable' homomorphisms factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on . In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to as well as to .

     

Dual operator norms and the spectra of matrices

Let ∥·∥ be an operator norm and ∥·∥^D its dual. Then it is shown that ∥A∥^D? ∑|λ_i(A)|, where λ_i(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.     

From Nörlund matrices to Laplace representations

The existence of Laplace representations for functions in weighted Hardy spaces on the right half plane is established. The method uses an extension of an inequality involving Nörlund matrices and corresponding convolution operators on the line. Analogous inequalities are proved for power series representations of functions in weighted Hardy spaces on the disc.

     

一类缺项算子矩阵正补的谱刻画

对 于 具 有 离 散 谱 的 正 算 子 Ａ, Ｂ, 我 们 给 出 了 一 个 闭 集 成 为 ２ × ２ 缺 项 算 子 矩 阵 Ａ ？？ Ｂ 的某个正 补的谱的一些 判别条件     

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.     

Beurling-Nevanlinna inequality for subfunctions of the stationary Schrödinger operator

The classical Beurling-Nevanlinna upper bound for subharmonic functions is extended to subsolutions of the stationary Schrödinger equation.

     

Bounds for iterated normal matrices

Given a normal matrix A, asymptotic bounds are obtained for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ in terms of the spectral radius of A, the number of eigenvalues of A with modulus equal to the spectral radius of A, and the order of A. These results are extended to provide bounds for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ for all m ? 1.     

无界分块算子矩阵的谱分布

本文首先给出次对角元有界的2×2阶无界算子矩阵的Gershgorin定理,然后利用主对角元算子的谱和数值域刻画整个算子矩阵的谱分布.特别地,当次对角元算子互为共轭(反共轭)算子时,结合二次数值域和Gershgorin定理对谱分布给出更精细的描述.     

Asymptotic Dirichlet problem for the Schrödinger operator via rough isometry

We pose and solve the asymptotic Dirichlet problem for the Schrödinger operator via rough isometries on a certain class of Riemannian manifolds. With suitable potentials, we give the solvability of the problem for a naturally defined class of data functions.

     

Schur multiplier norms for Loewner matrices

We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten q  -norms of commutators _{‖[A,f(B)]‖q}/_‖[A,B]‖q${\Vert [A,f(B)]\Vert }_q/{\Vert [A,B]\Vert }_q$. We also consider operator monotone functions, for which sharper bounds are obtained.     

Weyl's theorem for operator matrices

Weyl's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison Browder's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl's theorem and Browder's theorem are liable to fail for 2×2 operator matrices. In this paper we explore how Weyl's theorem and Browder's theorem survive for 2×2 operator matrices on the Hilbert space.Supported in part by BSRI-97-1420 and KOSEF 94-0701-02-01-3.     

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite-Padé approximations.     

Bounds for eigenvalues of certain stochastic matrices

We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦?1–n^-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.