ABSTRACT
In this chapter, the authors investigate the ideal structure of B(X). They discuss the norm closed ideals, and give some attention to the ideal of finite dimensional operators, F(X). It is to be understood throughout that ideal means two sided ideal. The authors look at the question of containment of various of the well known ideals of B(X). They prove that the ideal of finite dimensional operators is contained in every nonzero ideal of B(X). Most of the results on ideal structure deal with the well known closed ideals which have arisen from applied work with operators or from natural topological considerations. The authors define many ideals in B(X) and it is natural to ask what kind of containment relationships exist between these ideals.
