Estimate of the decay exponent of an operator semigroup associated with a second-order linear differential equation

Abstract

Since |(x, y)| = |(A−s/2x,As/2y)| ≤ ‖x‖−s · ‖y‖s for all x ∈ H and y ∈ Hs, s > 0, it follows that the sesquilinear form (x, y) extends from H ×Hs by continuity to a sesquilinear form (x, y)−s,s onH−s ×Hs. Consequently,H−s can be treated as the dual space of Hs (H−s = H∗ s ); the duality pairing is given by the sesquilinear form (x, y)−s,s (duality with respect toH). ByL (X,Y ) we denote the set of bounded linear operators from a spaceX to a space Y . The operator A can be viewed as a bounded operator in a scale of Hilbert spaces, A ∈ L (Hs,Hs−2) for s ∈ R. Just as in [3], we make the following assumption about the operatorD.