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.