The expansion of a semigroup and a Riesz basis criterion

Abstract

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. © 2004 Elsevier Inc. All rights reserved.