Equality of higher-rank numerical ranges of matrices

Abstract

Let denote the rank- numerical range of an -by- complex matrix . We give a characterization for , where , via the compressions and the principal submatrices of . As an application, the matrix satisfying , where is the classical numerical range of and , is under consideration. We show that if for some , then is unitarily similar to , where is a 2-by-2 matrix, is a -by- matrix and .