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 frac(1, 2) < γ ≤ 1, then we show that there exists a rank-1 operator B ∈ L (D ((ω - A)...