Generalized Convergence and Uniform Bounds for Semigroups of Restrictions of Nonselfadjoint Operators

Abstract

In Pellicer (Nonlinear Anal. 69:3110–3127, 2008) we showed the existence of exponentially attracting invariant manifolds for a nonlinear strongly damped viscoelastic problem. In that paper we used an special result, that we did not prove there, concerning certain uniform bounds for families of semigroups generated by restrictions of nonselfadjoint linear operators. The fact of applying this to nonselfadjoint operators represented the main novelty and difficulty of Pellicer (2008). In the present paper we prove this special result that was used in Pellicer (2008). This is done using the notion of generalized convergence between operators.