Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space
Under the assumptions that \((V,\|.\|)\) is a Hilbert space densely and continuously imbedded in a Hilbert space H, the imbedding \(V\subset H\) is compact, \(-A_ 0\) is the operator associated with a bounded sesquilinear form a(u,v) defined in \(V\times V\) and satisfying Gårding's inequality Re a(u,v)\(\geq c\| u\|^ 2\), \(\gamma\in R\) and the real function a(s) is Hölder continuous, the spectral properties of the infinitesimal generator A of the solution semigroup S(t) for \[ \frac{du}{dt}=A_ 0u(t)+\gamma A_ 0u(t-h)+\int^{0}_{- h}a(s)A_ 0u(t+s)ds \] are investigated. If \(m(\lambda)=1+\gamma e^{- \lambda h}+\int^{0}_{-h}e^{\lambda s}a(s)ds\) and \(m(0)=0\), then 0 is an eigenvalue of A with infinite multiplicity. 0 is an isolated point of \(\sigma\) (A) if it is a simple zero of m(\(\lambda)\), and it is a cluster point of \(\sigma\) (A) if it is a multiple zero of m(\(\lambda)\). If m(0)\(\neq 0\), \(\gamma\neq 0\) and the generalized eigenvectors of \(A_ 0\) are complete in H, then the generalized eigenvectors of A are complete in \(M=H\times L^ 2(-h,0;V)\).
scientific article; zbMATH DE number 4146483