Definition 7

Let $\varrho$ be a unitary representation of a topological group G in a Hilbert space E. One says that $\varrho$ is semisimple if there exists a family $(F_i)_{i\in I}$ of irreducible subrepresentations of $\varrho$ such that E is the Hilbert sum of the subspaces $F_i$.

One sometimes also says that a semisimple unitary representation admits a discrete decomposition or is discretely decomposable.

If $(\varrho_i)_{i\in I}$ is a family of semisimple unitary representations of a topological group G, then the Hilbert sum of the representations $\varrho_i$ is semisimple.