Invariant Convex Sets in Operator Lie Algebras

We study the closed convex subsets of Lie algebras of bounded linear operators on a Hilbert space that are invariant under the corresponding group of unitary operators. We will give a family f_(j) of convex functions such, that for each closed convex invariant set C there are real numbers c_(j) satisfying C = { X: f_(j)(X) ≤ c_(j) for all j }