J4 ›› 2012, Vol. 25 ›› Issue (1): 1-4.doi: 10.3976/j.issn.1002-4026.2012.01.001

• Content •     Next Articles

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

 ZHAI Fa-Hui, HAN Jie   

  1. Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China
  • Received:2011-08-09 Online:2012-02-20 Published:2012-02-20

Abstract:

Let Hi(i=1,2,…,n) be unitary space. Let A be a linear operator on H=H1H2Hn with block matrix expression ([Aijn×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 λ12,…,λm,(m≤n) exist and AiiiI(i=1,2,…,n). Moreover, the set of all μi is equal to {λ12,…,λm}, 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