Nuclear Cyclic Diagonal Mappings

In this note we construct nuclear operators whose eigenvalues are preassigned up to change of sign (or the n th roots of unity in the general case). These operators have a remarkably simple structure and in a special case yield a “natural” way of obtaining the modular sequence spaces of H. Rosenthal. Recall that a (Schauder) basis for a Banach space X is a sequence ( x n ) in X such that for every x ε X there is a unique sequence of scalars ( a n ) such thatconvergence in the topology of X . If X is a Banach space with a basis ( x n ) we can identify X with a sequence space by the correspondence.