The works of Klyachko and Knutson-Tao solved the Horn's conjecture about the polytope of eigenvalues of $A+B$, where $A$ and $B$ are hermitian matrices with prescribed eigenvalues. Klyachko solved also Thompson's conjecture about the singular values of $CD$, where $C$ and $D$ are nonsingular matrices with prescribed singular values. In this lecture we discuss generalizations of these results to matrices, which can be extended to certain compact operators. We will relate these results to the classical Golden-Thompson inequality. Some open problems will be mentioned.