Refined scales of decaying rates of operator semigroups on Hilbert spaces: Typical behavior

Abstract

We study relations between the decaying rates of operator semigroups on Hilbert spaces and some spectral properties of their respective generators; in particular, we show that the decaying rates of orbits of semigroups which are stable but not exponentially stable, typically in Baire's sense, depend on sequences of time going to infinity. * Corresponding author. Telephone +55 16 3351 9153, fax +55 16 3361 2081. applications of the theory to PDEs; namely, estimates on the norm of the resolvent of the generator are often easier to compute than the estimates on the norm of the semigroup itself. In this context, we refer to [1,3,7,8,16], among others. An important intermediate step from Gearhart-Prüss Theorem to results by Rozendaal et al. [16] was the Batty-Duyckaerts Theorem (Theorem 1.1 below) which relates the decaying rates of T (t)A −1 B(X) , iR ⊂ ̺(A), with the arbitrary growth of the norm of the resolvent of the generator. In order to properly recall such result, we need some preliminaries. For every A, the generator of a bounded C 0 -semigroup (T (t)) t≥0 on a Banach space X, with iR ⊂ ̺(A), consider a continuous non-decreasing function M (y) := max λ∈[−y,y] R(iλ, A) B(X) , y ≥ 0, and the associated function M log (y) := M (y)(log(1 + M (y)) + log(1 + y)), y ≥ 0. Denote by M −1 log : [M log (0), ∞) → R the inverse of M log .Theorem 1.1 (Theorem 1.5 in [4]). Let (T (t)) t≥0 be a bounded C 0 -semigroup on a Banach space X, with generator A such that iR ⊂ ̺(A). Then, there exists C > 0 such that

Category theorems for Schrödinger semigroups

“…It is natural to investigate the asymptotic behaviour of the orbits of the semigroups generated by the operators in the typical set in Theorem 1.2. The next result, which is a particular case of Theorem 1.2 in [1], says something in this direction. Suppose that (e tA ) t≥0 is stable but is not exponential stable.…”

“…Theorem 1.2 in[1]). Let A be a negative self-adjoint operator in H and α, β : R + −→ (0, ∞) functions so that, for each ǫ > 0, lim t→∞ α(t) = ∞ and lim t→∞ β(t)e −tǫ = 0.…”

Slow dynamics for self-adjoint semigroups and unitary evolution groups

“…Indeed, these identities were proven in [3] (note that since it is not possible to compare directly the two terms on the right-hand side of ( 5), some caution should be exercised when checking (6) and ( 7)). We use such identities in the proof of Theorem 2.1.…”

“…Lemma 2.1 (Lemma 2.1 in [3]). Let T be a negative self-adjoint operator with 0 ∈ σ(T ), and let α : R + −→ (0, ∞) be such that…”