Traces on operator ideals and related linear forms on sequence ideals (part III)

Let S+S+ be the forward shift acting on sequences indexed by N0:={0,1,2,…}N0:={0,1,2,…}. With certain sequence ideals z(N0)z(N0) we associate an operator ideal Dzapp over the class of all Banach spaces. It will be proved that every 12S+-invariant linear form λ   on z(N0)z(N0) generates a trace on Dzapp, which is presented by an explicit expression. If the considerations are restricted to operators on the infinite-dimensional separable Hilbert space, then our elementary approach yields all traces on all operator ideals.