  We investigate the properties of bounded operators which satisfy a certain
spectral additivity condition, and use our results to study Lie and Jordan
algebras of compact operators. We prove that these algebras have nontrivial
invariant subspaces when their elements have sublinear or submultiplicative
spectrum, and when they satisfy simple trace conditions. In certain cases we
show that these conditions imply that the algebra is (simultaneously)
triangularizable.
