2013/10/18 by Hwa-Long Gau, Pei Yuan Wu, Gau, Hwa-Long +1
Mathematics · #15A60 #15A99 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:15A60 #msc:15A99
paper · pdf · doi:10.48550/arxiv.1310.4952
28 pages
arxiv created 2013/10/18 · arxiv updated 2013/10/21
We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A, A2, …, Ak (k≥ 1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form \scriptsize[0 · B 0 · C] with B^*B+C^*C=I. Using this model, we show that if A has ascent k and A, A2, …, Ak-1 are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any Sn-matrix A, W(A) (resp., W(A⊗ A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block Jn. Finally, examples are given to show that the conditions that W(A) and W(A⊗ A) are circular discs at 0 are independent of each other for a general matrix A.