The expansion of a semigroup and a Riesz basis criterion
Let \(\mathcal{A}\) be the generator for a strongly continuous semigroup \(T(t)\) on a Hilbert space \(X\). Suppose that the singular set for \(\mathcal{A}\) can be split into two parts \(\sigma(\mathcal{A})=\sigma_1(\mathcal{A})\cup\sigma_2(\mathcal{A})\), where \(\sigma_2(\mathcal{A})\) consists all isolated eigenvalues. Under the assumption that the isolated eigenvalues are contained in a vertical strip, that the multiplicity of the eigenvalues are uniformly bounded and that they satisfy \(\inf_{k\neq l}| \lambda_k-\lambda_l| >0\), the authors prove their main theorem, which says that there exist two \(T(t)\)-invariant subspaces \(X_1\), \(X_2\) such that \(\sigma(\left.\mathcal{A}\right| _{X_1})=\sigma_1(\mathcal{A})\), \(\sigma(\left.\mathcal{A}\right| _{X_2})=\sigma_2(\mathcal{A})\) and \(X_1\oplus X_2\subset X\) (the topological direct sum). In addition, if the Riesz projectors for the eigenvalues satisfy a certain inequality, then \(X=X_1\oplus X_2\). As an application, the authors consider a heat exchanger system with boundary feedback. It is shown that in a certain Hilbert space, the corresponding semigroup fulfils the conditions of the main theorem.
scientific article; zbMATH DE number 2136777