THE BEHAVIOUR OF SEMIGROUP OF CONTRACTIONS ON HILBERT SPACE

Abstract

Abstract. Let be a strongly continuous, one-parameter semigroup of contractions on a complex Hilbert space 𝐻. The generator of is the linear operator with domain defined by Let be a closed densely defined operator on a Hilbert space with domain . For we define to be the set of all for which there exists a neighborhood of with analytic on having values in such that for all . This set is open and contains the resolvent set of . By definition, the local spectrum of at denoted by is the complement of , so it is a closed subset of The set is called local unitary spectrum of at We will say that 𝑻 is a multiple of the identity if there is 𝜆 ∈ ℝ such that 𝑇(𝑡) = 𝑒𝑥𝑝(𝑖𝜆𝑡) for all 𝑡 ≥ 0. Theorem. Let be a strongly continuous, one-parameter semi-group of contractions on a Hilbert space 𝐻. Assume that there exists a vector such that and is at most countable. If is not a multiple of the identity, then there exists a nonzero vector such that