The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing
sequence $\lambda=(\lambda_1,\dots,\lambda_n)$, with repetitions according to
multiplicity, and the diagonal of A is a point of $R^n$ that bears some
relation to $\lambda$. The Schur-Horn theorem characterizes that relation in
terms of a system of linear inequalities.
We give a new proof of the latter result for positive trace-class operators
on infinite dimensional Hilbert spaces, generalizing results of one of us on
the diagonals of projections. We also establish an appropriate counterpart of
the Schur inequalities that relate spectral properties of self-adjoint
operators in $II_1$ factors to their images under a conditional expectation
onto a maximal abelian subalgebra.