2013/04/01 by Gau, Hwa-Long, Wu, Pei Yuan
#15A60 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1304.0295
A KMS matrix is one of the form Jn(a)=[arrayccccc 0 amp; a amp; a2 amp;... amp; an-1 amp; 0 amp; a amp; \ddots amp; ⋮ amp; amp; \ddots amp; \ddots amp; a2 amp; amp; amp; \ddots amp; a 0 amp; amp; amp; amp; 0array] for n≥ 1 and a in ℂ. Among other things, we prove the following properties of its numerical range: (1) W(Jn(a)) is a circular disc if and only if n=2 and a≠ 0, (2) its boundary ∂ W(Jn(a)) contains a line segment if and only if n≥ 3 and |a|=1, and (3) the intersection of the boundaries ∂ W(Jn(a)) and ∂ W(Jn(a)[j]) is either the singleton \minσ(\re Jn(a))\ if n is odd, j=(n+1)/2 and |a|>1, or the empty set ∅ if otherwise, where, for any n-by-n matrix A, A[j] denotes its jth principal submatrix obtained by deleting its jth row and jth column (1≤ j≤ n), \re A its real part (A+A^*)/2, and σ(A) its spectrum.