Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B ∈ L(D((ω−A)1/2),X). If A is unbounded and 1/2 < γ 1, then we show that there exists a rank-1 operator B ∈ L(D((ω − A)γ ),X) such that A + B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A + B generates a distribution semigroup for each operator B ∈ L(D(A),X) of rank-1, then A generates a holomorphic C0-semigroup. If A + B generates a C0-semigroup for each operator B ∈ L(D((ω − A)γ ),X) of rank-1 where 0 < γ <1, then the semigroup T generated by A is differentiable and T (t) = O(t−α) as t ↓ 0 for any α >1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups. © 2006 Elsevier Inc. All rights reserved.