SOLUTION PROPERTIES OF SOME CLASSES OF OPERATOR EQUATIONS IN HILBERT SPACES

We study properties of solutions of the operator equation Lu = f, u, f is an element of 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(A) := E(A)H, i.e. when for every f is an element of H(A) there exists a unique (mild) solution u in H(A) of this equation. We introduce the notion of equation spectrum E associated with Eq. (*), and prove that if A subset of R is a compact subset such that A boolean AND Sigma = empty set, then H(A) is regularly admissible. If A subset of R is an arbitrary Borel subset such that A boolean AND Sigma = empty set, then, in general, H(A) needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of H(A). Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Priiss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schiller [10] and Vu [11], [12].