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 $C = \{X : (\forall j)\, f_j(X) \leq c_j\}$