An extremum property of sums of eigenvalues

We present in this note a maximum-minimum characterization of sums like a3+a7+asj where a1> . * * a. are the eigenvalues of a hermitian nXn matrix. The result contains the classic characterization of am as well as the maximum property of a1+a2+ * * * +a. given recently by Fan [4]. Though the result is valid also for a completely continuous hermitian operator in Hilbert space, we shall for the sake of simplicity assume the dimension to be finite. As an application we obtain linear inequalities relating the eigenvalues of the sum of two hermitian matrices to the eigenvalues of the summands.