Effective Construction of a Positive Operator which does not admit Triangular Factorization

Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

A class of non-factorable positive operators is constructed. As a result, pure existence theorems in the well-known problems by Ringrose, Kadison and Singer are substituted by concrete examples.

Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. In this paper conditions, in terms of continued fractions, for an oscillatory tetradiagonal Hessenberg matrix to have such positive bidiagonal factorization are found. Oscillatory tetradiagonal Toeplitz matrices are taken as a case study of matrix that admits a positive bidiagonal factorization. Moreover, it is proved that oscillatory banded Hessenberg matrices are organized in rays, with the origin of the ray not having the positive bidiagonal factoriza…

Invariance properties of thematic factorizations of matrix functions

We study the problem of invariance of indices of thematic factorizations. Such factorizations were introduced in [PY1] for studying superoptimal approximation by bounded analytic matrix functions. As shown in [PY1], the indices may depend on the choice of a thematic factorization. We introduce the notion of a monotone thematic factorization. The main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function it…

马尔可夫链的谱理论：具有允许正随机双对角因子分解的转移矩阵

Spectral theory for Markov chains with transition matrix admitting a stochastic bidiagonal factorization

本文研究了具有正随机双对角因子分解的马尔可夫链的谱理论，推导了有限和无限情况下的谱表示，并给出了平稳分布的显式公式以及遍历性的判定条件。

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices leads to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue-Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

Green's function of the problem of bounded solutions in the case of a block triangular coefficient

It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums…