Similarity Techniques in the Spectral Analysis of Perturbed Operator Matrices

关于扰动乘法算子的相似性

On similarity of perturbed multiplication operators

设S是L_2（0，1）中与自变量x相乘的算子，V是Volterra型积分算子.我们发现条件T：=S+V是类似于S和讨论一些概括的结果得到一个抽象的设置。

Study on Hirano Invertibility of Two New Types of Perturbed Operator Matrices

We investigate the Hirano invertibility of block-operator matrices in Banach algebras, and obtain the Hirano inverse of matrix $\begin{bmatrix} A&B\\ C&D \end{bmatrix}$ under two types of new perturbation conditions. Furthermore, we provide a new operator matrix decomposed into the sum of tripotent and nilpotent elements on Banach spaces.

On the similarity of AB and BA for normal and other matrices

It is well-known that $AB$ and $BA$ are similar when $A$ and $B$ are complex square Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if $A$ is Hermitian and $B$ is normal. Perhaps surprisingly, similarity does hold when $A$ is positive semidefinite and $B$ is normal.

Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach

The Wigner-von Neumann method, which was previously used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary $T$-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, $λ$, into the operator's absolutely continuous spectrum. Introducing a new rational function, $C(λ;T)$, related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum …

Preservers of Unitary Similarity Functions on Lie Products of Matrices

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(μUAU^*)= f(A)$ for any complex unit $μ$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \mapsto f(tX)$ is monotonic increasing on $[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given for surjective maps $ϕ$ on $M_n$ satisfying $f(AB-BA) = f(ϕ(A)ϕ(B)-ϕ(B)ϕ(A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, t…

Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

For an arbitrary Hermitian period-$T$ Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, $S$, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an $l^1$-perturbation. An analogue of the quan…