分块矩阵的块数值值域等于谱的条件

doi:10.3976/j.issn.1002-4026.2012.01.001

Abstract

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

CLC Number:

O177

ZHAI Fa-Hui, HAN Jie. The condition of the block numerical range of a block operator matrix equal to its spectrum[J].J4, 2012, 25(1): 1-4.

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

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