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 .