Strong Convergence of Generalized Resolvents

Abstract

Let M and fLng be a linear, closed, densely defined operator and a sequence of linear, closed, densely defined operators in a Banach Space X respectively. We consider a sequence of generalized resolvents fRn(¸)g, where Rn(¸) = (Ln¡¸M)¡1M. In this paper, we will prove that the sequence fRn(¸)g is uniformly bounded in n and ¸ in any compact subset of a certain open set. Then we will concern with consideration on strong convergence of fRn(¸)g. Finally we will give a criterion for the sequence fRn(¸)g converges strongly.

Keywords : generalized resolvent, uniformly bounded, strong convergence