Two Sided Ideals of Operators
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چکیده
1. 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 instance, when X = l, l ^ p < « > , L(X) is a chain (i.e., totally ordered) with three elements: {o}, 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, Kp < oo, the lattice L(X) is not a chain. A treatment for X a Hubert space of arbitrary dimension can be found in [4]. We aim to exhibit here a Banach space X such that L(X) is both "long" and "wide." Precisely, we have
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