Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part II)

Abstract

The theory of operator ideals on the infinite-dimensional separable Hilbert space is equivalent to the theory of shift-monotone sequence ideals. In particular, traces correspond to $${\frac {1}{2} S_+}$$ -invariant linear forms, where S + is the forward shift. Important consequences of these facts are presented. We concentrate on the study of sequence ideals, which are easier to handle. So the reader should not be surprised that operator ideals play almost no role. All results, however, can be translated into the context of operators. In this way, algebraic and topological classifications of operator ideals are obtained.