Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let Mn be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A Mn is defined by Wk(A) = (1/k)∑jk=1xj*Axj : x1, …, xk is an orthonormal set in n]. It is known that tr A/n = Wn(A) Wn−1(A) W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.