An Infinite Dimensional Version of the Schur–Horn Convexity Theorem

The Schur–Horn Convexity Theorem states that forain Rnp({U* diag(a) U: U∈U(n)})=conv(Sna),wherepdenotes the projection on the diagonal. In this paper we generalize this result to the setting of arbitrary separable Hilbert spaces. It turns out that the theorem still holds, if we take thel∞-closure on both sides. We will also give a description of the left-hand side for nondiagonalizable hermitian operators. In the last section we use this result to get an extension theorem for invariant closed convex subsets of the diagonal operators.