Polynomially Isometric Matrices in Low Dimensions

Operators which are polynomially isometric to a normal operator

Let $\mathcal{H}$ be a complex, separable Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators acting on $\mathcal{H}$. Given a unitarily-invariant norm $\| \cdot \|_u$ on $\mathcal{B}(\mathcal{H})$ and two linear operators $A$ and $B$ in $\mathcal{B}(\mathcal{H})$, we shall say that $A$ and $B$ are \emph{polynomially isometric relative to} $\| \cdot \|_u$ if $\| p(A) \|_u = \| p(B) \|_u$ for all polynomials $p$. In this paper, we examine to what extent an operator $A$ being polynomially isometric to a normal operator $N$ implies that $A$ is itself no…

Partially isometric matrices: a brief and selective survey

We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also discuss the unitary similarity problem and several other results.

Triangularizability of Families of Polynomially Compact Operators

A recent paper of Shemesh shows triangularizability of a pair $\{A, B\}$ of complex matrices satisfying the condition $A [A,B]=[A,B] B=0$, or equivalently, the matrices $A$ and $B$ commute with their product $A B$. In this paper we extend this result to polynomially compact operators on Banach spaces. The case when the underlying space is Hilbert and one of operators is normal is also studied. Furthermore, we consider families of polynomially compact operators whose iterated commutators of some fixed length are zero. We also obtain a structure result in the special case of a finite family of a…

Polynomial Stability of Semigroups Generated by Operator Matrices

In this paper we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretic results are applied to deriving conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equations.

谱与幂零约束下多项式变换矩阵的存在性与计数

Existence and Enumeration of Polynomially Transformed Matrices under Spectral and Nilpotent Constraints

本文研究多项式变换后矩阵何时幂零或可对角化，区分单/多互异特征值及稀疏、稠密情形，综合组合根计数与解析代数方法建立存在性和分类框架。

迹联合谱测度

Tracial joint spectral measures

该研究针对两个厄米矩阵提出平面上的迹联合谱测度，证明其存在性可推导Schatten-p类二维子空间与Lp子空间等距、迹函数导数非负性两个结论，并给出该测度的显式表达式。