Boundedness of the numerical range (Q922618)

Let A be an \(n\times n\) complex matrix. Define \(W(A)=\{x^*Ax:\) \(x^*x=1\), \(x\in {\mathbb{C}}^ n\}\). Given an \(n\times n\) Hermitian matrix H, denote the minimum (maximum) eigenvalue of H by \(\lambda_ 1(H)(\lambda_ n(H)\) respectively). By identifying the complex plane with \(R^ 2\), it is well-known and very easy to show that \(W(A)\subseteq [\lambda_ 1(H_ 1),\lambda_ n(H_ 1)]\times [\lambda_ 1(H_ 2),\lambda_ n(H_ 2)]\), where \(H_ 1=(A+A^*)/2\) and \(H_ 2=(A- A^*)/2i\). The author proves the above result for real A by a lengthy computation. However, in the theorem, the author incorrectly claims that \([\lambda_ 1(H_ 1),\lambda_ n(H_ 1)]\times [\lambda_ 1(H_ 2),\lambda_ n(H_ 2)]\) is the smallest rectangle containing W(A). A counterexample can be obtained by taking \(A=\left[\begin{matrix} 1&0 \\ 0&-1 \end{matrix}\right]\oplus \left[\begin{matrix} 0&1 \\ -1&0 \end{matrix}\right]\). Also, the author raises a conjecture which can easily be shown to be incorrect when \(n=4\). (Note: the matrix K used in the paper is unitarily similar to \(2H_ 2\oplus 2H_ 2\).)