分块矩阵的块数值值域等于谱的条件

设H_i是酉空间,i=1,2,…,n, A是作用在H=H₁⊕H₂⊕…⊕H_n上的线性变换具有分块矩阵表示［A_ij］_n×n,n≤dim H_i<∞。本文给出了A的块数值值域等于A的谱的充要条件是存在复数λ₁,λ₂,…,λ_m,m≤n,使得A_ii=μ_iI,i=1,2,…,n, 这里μ_i组成的集合等于集合{λ₁,λ₂,…,λ_m}, 而且存在一些初等行变换和对应的初等列变换将A化为上三角分块矩阵。

The condition of the block numerical range of a block operator matrix equal to its spectrum

Let Hi(i=1,2,…,n) be unitary space.Let A be a linear operator on H=H1H2…Hn with block matrix expression(n×n,n≤dim Hi<∞).This paper presents a necessary and sufficient condition of the block numerical range of matrix A equal to its spectrum: a set of complex numbers λ1,λ2,…,λm,(m≤n) exist and Aii=μiI(i=1,2,…,n).Moreover,the set of all μi is equal to {λ1,λ2,…,λm},and can be transformed into an upper triangular block matrix by some elementary row operations and the corresponding elementary column operations.