Quasi-Banach operator ideals with a very strange trace

Abstract

As shown by S. Lord, F. Sukochev, and D. Zanin (see [7]), the theory of singular traces is well understood for operators on the Hilbert space. The situation turns out to be completely different in the Banach space setting. Indeed, quite strange phenomena may occur. We will construct quasi-Banach operator ideals ${\mathfrak A}$ with seemingly contradictory properties: On the one hand, ${\mathfrak A}$ supports a continuous trace τ that vanishes at all finite rank operators, which means that τ is singular. On the other hand, ${\mathfrak A}$ contains the identity map I _Z of an infinite-dimensional Banach space Z and τ (I _Z ) =  1. This implies that there exist operators ${T \in \mathfrak A (Z)}$ such that ${\tau (T^n) = 1}$ for ${n = 1,2,{\dots} \;}$ , which is impossible for singular traces in the case of a Hilbert space. As most counterexamples, the new operator ideals have no useful application. They provide, however, a deeper insight into the philosophy of traces.