Solution properties of some classes of operator equations in hilbert spaces

Abstract

We study properties of solutions of the operator equation ℒu = f, u, f ⋯ H,(*), where ℒ a closable linear operator on a Hilbert space H, such that there exists a self-adjoint operator D on H, with the resolution of identity E(·), which commutes with ℒ. We are interested in the question of regular admissibility of the subspace H(Λ): =E(Λ )H, i.e. when for every f ⋯ H(Λ) there exists a unique (mild) solution u in H (Λ) of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ is a compact subset such that Λ ∩ Σ = ∅, then H (Λ) is regularly admissible. If Λ ⊂ is an arbitrary Borel subset such that Λ ∩ Σ = ∅, then, in general, H(Λ) needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of H(Λ). Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12]. © 2010 World Scientific Publishing Company.