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Let $\Hn$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a compactgroup of automorphisms of $\Hn$ such that $(K\ltimes \Hn,K)$ is a Gelfand pair.We prove that the Gelfand transform is a topological isomorphism between thespace of $K$-invariant Schwartz functions on $\Hn$ and the space of Schwartzfunction on a closed subset of $\R^s$ homeomorphic to the Gelfand spectrum ofthe Banach algebra of $K$-invariant integrable functions on $\Hn$.

Gelfand pairs on the Heisenberg group and Schwartz functions