SOLUTION PROPERTIES OF SOME CLASSES OF OPERATOR EQUATIONS IN HILBERT SPACES

Abstract

We study properties of solutions of the operator equation [Formula: see text], [Formula: see text], where [Formula: see text] a closable linear operator on a Hilbert space [Formula: see text], such that there exists a self-adjoint operator [Formula: see text] on [Formula: see text], with the resolution of identity E(·), which commutes with [Formula: see text]. We are interested in the question of regular admissibility of the subspace [Formula: see text], i.e. when for every [Formula: see text] there exists a unique (mild) solution u in [Formula: see text] of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ ℝ is a compact subset such that Λ ⋂ Σ = ∅, then [Formula: see text] is regularly admissible. If Λ ⊂ ℝ is an arbitrary Borel subset such that Λ ⋂ Σ = ∅, then, in general, [Formula: see text] needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of [Formula: see text]. 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].