Spectral Theory for Positive Semigroups
We have discovered in the finite-dimensional case that exponential functions enjoy some rather special spectral properties. Such properties are, for example, that the spectrum of a semigroup operator is determined by the spectrum of its generator, or, tha
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Spectral Theory for Positive Semigroups We have discovered in the finite-dimensional case that exponential functions enjoy some rather special spectral properties. Such properties are, for example, that the spectrum of a semigroup operator is determined by the spectrum of its generator, or, that the stability of a semigroup is guaranteed whenever the spectrum of its generator lies in the left half-plane. Unfortunately, since strongly continuous semigroups are not exactly exponential functions, these properties fail to hold in general. However, we will see that positivity has significant impact on the spectrum of the semigroup. We will show, for example, that the spectral bound is always an element of the spectrum of the generator of a positive semigroup and we will be able to make some more results analogous to the finite-dimensional case. Throughout this chapter we suppose that E is a complex Banach lattice and X is a Banach space.
12.1 Asymptotic Stability of Semigroups We are interested in the asymptotic behavior of the solution of the abstract Cauchy problem u(t) ˙ = Au(t), t ≥ 0, u(0) = f ∈ X, where A is the generator of a C0 -semigroup (T (t))t≥0 on X. Recall from Proposition 9.15 that the solution to this Cauchy problem is given by u(t) = T (t)f . In Chapter 9 we have also already defined the growth bound of the semigroup (T (t))t≥0 as ω0 (T ) := inf{ω ∈ R : there is M = Mω ≥ 1 with T (t) ≤ M eωt for all t ≥ 0}. © Springer International Publishing AG 2017 A. Bátkai et al., Positive Operator Semigroups, Operator Theory: Advances and Applications 257, DOI 10.1007/978-3-319-42813-0_12
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Chapter 12. Spectral Theory for Positive Semigroups
There is an important connection between the growth bound and the spectral radius of semigroup operators. Proposition 12.1. Let (T (t))t≥0 be a C0 -semigroup on X. a) We have that ω0 (T ) = lim
t→∞
log T (t) log T (t) = inf . t>0 t t
(12.1)
b) For every t ≥ 0, the spectral radius r(T (t)) of the operator T (t) satisfies r(T (t)) = etω0 (T ) . Proof. a) By Exercise 1, lim
t→∞
log T (t) log T (t) = inf . t>0 t t
Setting η := inf
t>0
log T (t) log T (t) = lim , t→∞ t t
we obtain eηt ≤ T (t) for all t ≥ 0. So, by the definition of ω0 (T ), we infer that η ≤ ω0 (T ). Take now ω > η. Then there is a τ > 0 such that log T (t) ≤ ω, t
for all t ≥ τ.
Hence, T (t) ≤ eωt for all t ≥ τ . Since the function t → T (t) is bounded on [0, τ ], we see that T (t) ≤ M eωt for all t ≥ 0 and some constant M ≥ 1. This implies that ω0 (T ) ≤ η and therefore ω0 (T ) = η. b) Since r(T (t)) = lim T (kt) 1/k , k→∞
we obtain for t > 0 that r(T (t)) = lim et(kt) k→∞
−1
log T (kt)
= etω0 (T ) .
As in finite dimensions (compare with Definition 4.6) we define the spectral bound of A by s(A) := sup{Re λ : λ ∈ σ(A)}. Motivated by the finite-dimensional case, see Corollary 4.8, one may ask whether for a generator A of a C0 -semigroup (T (t))t≥0 on X we have ω0 (T ) = s(A). We will, however, see later that this equality is in general not even true for positive C0 -s
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