Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C 0 semigroup on a Hilbert space and σ ( A ) = σ 1 ( A ) ∪ σ 2 ( A ) with σ 2 ( A ) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ 2 ( A ) is separated and for each λ ∈ σ 2 ( A ) , the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ 2 ( A ) form an L -basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.