On isometries of symmetric products of metric spaces

By Fn(X), n≥1, we denote the n-th symmetric product of a metric space (X,d) as the space of the nonempty finite subsets of X with at most n elements endowed with the Hausdorff metric dH. By Iso(X) we denote the group of all isometries from X onto itself with the topology of pointwise convergence. In this paper, we show that, under the certain hypothesis, Iso(Fn(X)) is topologically isomorphic to the semidirect product group Iso(Fn(X),F1(X))⋊Iso(X). We apply those results to ℓpq, (p,q)∈[1,∞]×N≥2⁎, as particular spaces and prove the following statements: