The expansion of a semigroup and a Riesz basis criterion

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 C0 semigroup on a Hilbert space and σ(A)=σ1(A) σ2(A) with σ2(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ2(A) is separated and for each λ σ2(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ2(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.