Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients

Answering a question raised by S. Friedland, we show that the possible eige nvalues of Hermitian matrices (or compact operators) A, B, and C with C les s than or equal to A + B are given by the same inequalities as in Klyachko' s theorem for the case where C = A + B, except that the equality correspond ing to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C) less than or equal to tr(A) + tr(B). The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such t hat there is an exact sequence B --> C --> A are characterized by the same inequalities. (C) 2000 Published by Elsevier Science Inc. All rights reserv ed. AMS classification: 15A42; 22E46; 14M15; 05E15; 13F10; 47B07.