Two sided ideals of operators

Abstract

Let X be a Banach space, and B(X) the Banach algebra of all bounded linear operators in X. The closed two sided ideals of B(X) (actually, of any Banach algebra) form a complete lattice L(X). Aside from very concrete cases, L(X) has not yet been determined; for inst- ance, when X = l<SUP>p</SUP>, l ≦ p < ∞, L(X) is a chain (i.e., totally ordered) with three elements: (0), B(X) and the ideal C(X) of compact operators (see (3)). On the other hand, it is known ((2), 5.23) that for X = L<SUP>p</SUP>, 1 < p < ∞, the lattice L(X) is not a chain. A treatment for X a Hilbert space of arbitrary dimensión can be found in (4). We aim to exhibit here a Banach space X such that L(X) is both "long" and "wide".