On the boundary of numerical ranges of operators

In this note, a new characterization of an attained boundary point of the numerical range of an operator on a Hilbert space is given. As an application, we point out a gap in the proof of the main result in the paper [M.T. Chien, L. Yeh, On the boundary of the numerical range of a matrix, Appl. Math. Lett. 23 (2010) 725–727].