On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial (Q1827504)

Let \(L(\lambda)=\lambda^2I-2\lambda A+B\) be a monic quadratic matrix polynomial, where \(A\) and \(B\) are \(n\)-by-\(n\) Hermitian complex matrices, and let \(W(L)=\{ \lambda\in\mathbb{C}:\langle L(\lambda)x,x\rangle =0\) for some \(x\in\mathbb{C}^n\), \(\|x\|=1 \}\) be its numerical range. \(W(L)\) is related to the (classical) numerical range \(W(S)\) of \(S=A+iB\) by \(W(L)=\{\lambda\in\mathbb{C}: \lambda^2-2p\lambda+q=0\) for some \(p+iq\) in \(W(S)\}\). The present paper addresses the questions on (a) the connection between the geometric structures of the boundaries \(\partial W(L)\) and \(\partial W(S)\) and the lengths of the Jordan chains of eigenvalues of \(L(\lambda)\), and (b) the possible lengths of such Jordan chains. For example, (a) is answered by the result asserting that if \(\partial W(S)\) is tangent to the parabola \(y=x^2\) of order \(t\leq m\) \((m\) a positive integer) at the point \((w,w^2)\) and \(w\) is an eigenvalue of \(L(\lambda)\), then the length of the Jordan chain for \(w\) is at most \(m+1\). Examples are provided to show the sharpness of the number \(m+1\) (at least for \(n=2\) and 3). On the other hand, since it is known that the lengths of the Jordan chains of an eigenvalue of \(L(\lambda)\) is at most \(2n\), (b) is answered by proving that every integer \(m\), \(1\leq m \leq 2n\), can be realized as such a length for some eigenvalue in \(\partial W(L)\) of some \(L\).