On the spectra of certain matrices generated by involutory automorphisms

Let $A=P+K$ be an $n\times n$ complex matrix with $P = \frac12(A-HAH)$ and $K=\frac12(A+HAH)$, $H$ being a unitary involution. Having characterised all unitary involutions, we investigate the spectral structure of $P$ and $K$ and, in particular, characterise the eigenvalues of $K$ as zeros of a rational function and prove that, for normal $A$, $\sigma(K)$ resides in the convex hull of $\sigma(A)$. We also demonstrate that this need not be true when $A$ is not normal.