Linear pencils and quadratic programming problems with a quadratic constraint

Characteristic function of pencils. Model representations of a quadratic operator pencil

Notion of an open system of second order is introduced. Characteristic function for such an open system is obtained. Model representations of a quadratic non-self-adjoint operator pencil are found.

A factorization of a quadratic pencils of accretive operators and applications

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. The paper is concluded with some questions left open from the preceding discussions.

On spectra of quadratic operator pencils with rank one gyroscopic linear part

The spectrum of a selfadjoint quadratic operator pencil of the form $λ^2M-λG-A$ is investigated where $M\geq 0$, $G\geq 0$ are bounded operators and $A$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $G$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $G$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $A$ are given.

The Moore-Penrose inverse of accretive operators with application to quadratic operator pencils

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

线性矩阵束与非交换凸性

Linear matrix pencils and noncommutative convexity

本综述从自由实代数几何视角阐述自由谱体的凸性本质，介绍其检测与线性矩阵束表示方法，并说明其正置定理可用于构建非交换约束下凸矩阵集特征值优化的半定规划形式。

Quadratic programming with one quadratic constraint in Hilbert spaces

A quadratically constrained quadratic programming problem is considered in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are provided to guarantee that the problem admits solutions for every initial data (in an adequate set).