Prescribed Szlenk index of separable Banach spaces

In a previous work, the first named author described the set P of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer n and any ordinal alpha in P, there exists a separable Banach space X such that the Szlenk index of the dual of order k of X is equal to the first infinite ordinal omega for all k in {0, ..., n - 1} and equal to alpha for k = n. One of the ingredients is to show that the Lindenstrauss space and its dual both have Szlenk index equal to omega. We also show that any element of P can be realized as the Szlenk index of a reflexive Banach space with an unconditional basis.