The condition of the block numerical range of a block operator matrix equal to its spectrum
Jie Han · Shandong kexue · 2012
Let Hi(i=1,2,…,n) be unitary space.Let A be a linear operator on H=H1H2…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.