A characterization and sum decomposition for operator ideals

Abstract

Let L ( H ) L(H) be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in L ( H ) L(H) every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of L ( H ) L(H) . This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.