Asymptotic behaviour of C-0-semigroups with bounded local resolvents

Journal article

Let {Τ(t)}t≥0 be a C0-semigroup on a Banach space Χ with generator Α, and let H∞Τ be the space of all x ∈ Χ such that the local resolvent λ → R(λ, Α)cursive Greek chi has a bounded holomorphic extension to the right half-plane. For the class of integrable functions φ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus φ → Τφ, as operators on H∞Τ. We show that each orbit Τ(·)Τφcursive Greek chi is bounded and uniformly continuous, and Τ(t)Τφ → 0 weakly as t → ∞, and we give a new proof that ∥Τ(t)R(μ, Α)cursive Greek chi∥ = O(t). We also show that ∥Τ(t)Τφ,cursive Greek chi∥ → 0 when Τ is sun-reflexive, and that ∥Τ(t)R(μ, Α)cursive Greek chi∥ = O(ln t) when Τ is a positive semigroup on a normal ordered space Χ and cursive Greek chi is a positive vector in H∞Τ.

Authors: Batty, C; Chill, R; van Neerven, J

• Publication status: Published
• Journal: MATHEMATISCHE NACHRICHTEN
• Volume: 219
• Issue: 1
• Pages: 65-83
• Publication date: 2000-01-01
• DOI: 10.1002/1522-2616(200011)219:1<65::AID-MANA65>3.0.CO;2-O
• EISSN: 1522-2616
• ISSN: 0025-584X
• Language: English
• Keywords: local resolvent; functional calculus; Tauberian theorem; Laplace transform; C-0-semigroup; individual stability; H-infinity(C+); circle dot-reflexive; fractional power