ACTA issues

Numerical ranges of KMS matrices

Hwa-Long Gau, Pei Yuan Wu

Acta Sci. Math. (Szeged) 79:3-4(2013), 583-610

Abstract. We prove the following properties of the numerical range of a KMS matrix $J_n(a)$: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\not=0$, (2) its boundary $\partial W(J_n(a))$ contains a line segment if and only if $n\ge3$ and $|a|=1$, and (3) the intersection of the boundaries $\partial W(J_n(a))$ and $\partial W(J_n(a)[j])$ is either the singleton $\{\min\sigma (\mathop{\rm Re}J_n(a))\} $ if $n$ is odd, $j=(n+1)/2$ and $|a|>1$, or the empty set $\emptyset $ if otherwise, where, for any $n$-by-$n$ matrix $A$, $A[j]$ denotes its $j$th principal submatrix obtained by deleting its $j$th row and $j$th column ($1\le j\le n$), $\mathop{\rm Re}A$ its real part $(A+A^*)/2$, and $\sigma(A)$ its spectrum.

AMS Subject Classification (1991): 15A60

Keyword(s): Numerical range, KMS matrix, $S_n$-matrix, $S_n^{-1}$-matrix

Received October 4, 2012, and in revised form May 11, 2013. (Registered under 79/2012.)