Asymptotic Behaviour of C0–Semigroups with Bounded Local Resolvents

✍ Scribed by Charles J. K. Batty; Ralph Chill; Jan van Neerven

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 255 KB
• Volume: 219
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/1522-2616(200011)219:1<65::aid-mana65>3.0.co;2-o

✦ Synopsis

Let {T (t)} t≥0 be a C 0 -semigroup on a Banach space X with generator A, and let

T be the space of all x ∈ X such that the local resolvent λ → R(λ, A)x 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 → T φ , as operators on H ∞ T . We show that each orbit T ( • )T φ x is bounded and uniformly continuous, and T (t)T φ x → 0 weakly as t → ∞, and we give a new proof that T (t)R(µ, A)x = O(t). We also show that T (t)T φ x → 0 when T is sun -reflexive, and that T (t)R(µ, A)x = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H ∞ T .