Inertia of Kraus matrices II

For positive real numbers \(r, p_0,\) and \(p_1< \cdots < p_n,\) let \(K_r\) be the Kraus matrix whose (i, j) entry is equal to

$$\begin{aligned} \frac{1}{p_i - p_j} \Bigl ( \frac{p_i^r - p_0^r}{p_i -p_0} - \frac{p_j^r - p_0^r}{p_j -p_0} \Bigr ). \end{aligned}$$

In this article, we give a supplemental result to Sano and Takeuchi (J. Spectr. Theory, 2022) about the Kraus matrices \(K_r\): the simplicity of non-zero eigenvalues. Our proof is accomplished by arguments similar to those for Loewner matrices given by Bhatia, Friedland and Jain (Indiana Univ. Math. J., 2016).