Rank-1 perturbations of cosine functions and semigroups

Journal article

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)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 {norm of matrix} T′ (t) {norm of matrix} = 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.

Authors: Batty, C; Arendt, W

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Functional Analysis
• Volume: 238
• Issue: 1
• Pages: 340-352
• Publication date: 2006-03-22
• DOI: 10.1016/j.jfa.2006.02.003
• EISSN: 1096-0783
• ISSN: 0022-1236
• Language: English
• Keywords: C-0-semigroup; cosine function; fractional power; rank-one; distribution semigroup; perturbation