Sums and intersections of symmetric operator spaces

Let L(H) be the algebra of all bounded operators on a separable Hilbert space H. We completely describe symmetric operator ideals E in L(H), which can be represented as a sum (or, an intersection) of two other (distinct from E) symmetric operator ideals in L(H). We also present a version of our results for symmetric operator spaces affiliated with a semifinite atomless von Neumann algebra M.