关键词: 块数值值域, 谱, 分块算子矩阵

Abstract:

Let H_i（i=1,2,…,n） be unitary space. Let A be a linear operator on H=H₁⊕H₂⊕…⊕H_n with block matrix expression (［A_ij］_n×n,n≤dim H_i<∞). 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,…,λ_m,（m≤n） exist and A_ii=μ_iI（i=1,2,…,n）. Moreover, the set of all μ_i is equal to {λ₁,λ₂,…,λ_m}, and can be transformed into an upper triangular block matrix by some elementary row operations and the corresponding elementary column operations.

Key words: block numerical range, spectrum, block operator matrix