ON THE SPECTRAL MAPPING THEOREM FOR PERTURBED STRONGLY CONTINUOUS SEMIGROUPS

SIMON BRENDLE, RAINER NAGEL AND JAN POLAND

Abstract. We consider a strongly continuous semigroup (T (t))_{t≥0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))_{t≥0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))_{t≥0}. The results are applied to a cell equation with age structure and spatial distribution.

1. Introduction

R.S. Phillips, in his fundamental papers [13] and [14], started the investigation of qualitative and spectral properties of perturbed strongly continuous semigroups on Banach spaces. In particular, he showed that, even by bounded perturbation, important properties as eventual norm continuity or the spectral mapping theorem may be destroyed. Also, in many applications of semigroup theory to concrete evolution equations like transport equations (e.g.

[17] or [3]) or population equations (e.g. [4] or [10],C-IV, Ex. 2.15), perturbation methods are essential. Therefore, the following problem has been studied by many authors.

Let 𝒯 = (T (t))_{t≥0} be a strongly continuous semigroup with generator A on a Banach space X and assume that B is a perturbing operator such that A + B generates a strongly continuous semigroup 𝒮 = (S(t))_{t≥0}. Under which conditions can one determine the spectrum σ(S(t)) by the spectrum σ(A + B) of the perturbed generator and the spectrum σ(T (t)) of the unperturbed semigroup operators? In general, this remains a difficult problem, and for most purposes, e.g. for applications to stability problems, an estimate for the growth bound ω_{0}(𝒮) by the spectral bound s(A + B) := sup{Re λ : λ ∈ σ(A + B)} and the growth bound ω_{0}(𝒯) is already a desirable result.

The techniques used by Andreu, Martínez, Mazón [1], Voigt [16],[19] and many others to treat this problem are based on the concept of essential spectrum and need some kind of compactness assumption on the perturbation. In this paper, we use the notion of critical spectrum introduced in [12] in order to replace compactness by norm continuity assumptions. In this way, we weaken these assumptions and improve the quoted results. In particular, we obtain, under appropriate assumptions, a partial spectral mapping theorem for the perturbed semigroup (see Corollary 4.2 below), thereby answering a conjecture proposed by Mokhtar-Kharroubi in [8] and [9].

In the final section, we discuss an age structured cell equation with spatial distribution. In this case, the previous results using compactness and essential spectrum do not apply. However, we can show that the first term in the Dyson-Phillips series of the semigroup (S(t))_{t≥0} is norm continuous. This yields a precise description of the spectra σ(S(t)).

We first develop the necessary tools and recall the concept of critical spectrum of a strongly continuous semigroup.

2. The Critical Spectrum

For a Banach space X and a strongly continuous semigroup (T (t))_{t≥0} on X, we consider the Banach space X̃ := ℓ^{∞}(X) of all bounded sequences in X. We extend the semigroup (T (t))_{t≥0} to this space and obtain a new semigroup 𝒯̃ = (T̃(t))_{t≥0} by

T̃(t)x̃ := (T (t)x_{n})_{n∈ℕ} for x̃ = (x_{n})_{n∈ℕ}.

For this extended semigroup we consider its space of strong continuity

X̃_{𝒯} := {x̃ ∈ X̃ : lim_{h↓0} ∥T̃(h)x̃ − x̃∥ = 0}.

This subspace is closed and (T̃(t))_{t≥0}-invariant. Therefore, on the quotient space X̂ := X̃/X̃_{𝒯}, the semigroup (T̃(t))_{t≥0} induces a quotient semigroup 𝒯̂ = (T̂(t))_{t≥0} which is given by

T̂(t)x̂ := T̃(t)x̃ + X̃_{𝒯} for x̂ = x̃ + X̃_{𝒯}.

The critical spectrum of T (t) is then defined as

σ_{crit}(T (t)) := σ(T̂(t)),

while its critical spectral radius is defined as

r_{crit}(T (t)) := r(T̂(t)).

Moreover, the critical growth bound is defined as

ω_{crit}(𝒯) := ω_{0}(𝒯̂).

With these definitions, the following theorem holds.

2.1. Theorem. For a strongly continuous semigroup (T (t))_{t≥0} the following statements hold.

(i) σ_{crit}(T (t)) ⊂ σ(T (t)). (ii) r_{crit}(T (t)) = e^{ω_{crit}(𝒯) t}. (iii) σ(T (t)) \ {0} = e^{tσ(A)} ∪ σ_{crit}(T (t)) \ {0}. (iv) ω_{0}(𝒯) = max{s(A), ω_{crit}(𝒯)}.

For the proofs we refer to [12].

The next section is devoted to the Dyson-Phillips expansion of a perturbed semigroup. In particular, we study how this expansion behaves under the above construction leading to the critical spectrum.

3. The Dyson-Phillips Expansion

Let B be an A-bounded operator on X. In order to guarantee that A + B generates a strongly continuous semigroup, we make the following assumption.

3.1. Assumption. There is a function q : ℝ_{+} → ℝ_{+} satisfying lim_{t↓0} q(t) = 0 such that

∫_{0}^{h} ∥BT (s)x∥ ds ≤ q(h)∥x∥

for each x ∈ D(A) and h ≥ 0.

The Miyadera-Voigt perturbation theorem (see [15],[18], or [2], Theorem III.3.15) implies that A+B generates a strongly continuous semigroup (S(t))_{t≥0} which is given by the Dyson-Phillips series

S(t) = ∑_{j=0}^{∞} S_{j}(t). (DPS)

The operators S_{j}(t) are defined inductively as

S_{0}(t)x := T (t)x

and

S_{j}(t)x := ∫_{0}^{t} S_{j−1}(t − s)BT (s)x ds

for each x ∈ D(A) and t ≥ 0.

We first prove a useful algebraic identity for the terms S_{j}(t) in the Dyson-Phillips series above.

3.2. Lemma. We have

S_{j}(t + s) = ∑_{i=0}^{j} S_{i}(t)S_{j−i}(s)

for all t, s ≥ 0.

Proof. Suppose that the assertion holds for j − 1. Then we have

S_{j}(t + s)x = ∫_{0}^{s} S_{j−1}(t + s − r)BT (r)x dr + ∫_{s}^{t+s} S_{j−1}(t + s − r)BT (r)x dr

= ∑_{i=0}^{j−1} S_{i}(t) ∫_{0}^{s} S_{j−i−1}(s − r)BT (r)x dr + ∫_{0}^{t} S_{j−1}(t − r)BT (r)T (s)x dr

= ∑_{i=0}^{j−1} S_{i}(t)S_{j−i}(s)x + S_{j}(t)S_{0}(s)x

= ∑_{i=0}^{j} S_{i}(t)S_{j−i}(s)x

for all x ∈ D(A). Therefore, the assertion holds for j.

The next two lemmata contain norm estimates relating the semigroups (T (t))_{t≥0} and (S(t))_{t≥0} and the Dyson-Phillips terms S_{j}(t). These estimates will be used in Lemma 3.5 and Lemma 3.6 below.

3.3. Lemma. We have

∥S_{j}(h)∥ ≤ q(h)^{j} sup_{0≤s≤h} ∥T (s)∥

for each j ∈ ℕ_{0} and h ≥ 0.

Proof. Suppose that the assertion holds for j − 1. Then we have

∥S_{j}(h)x∥ ≤ ∫_{0}^{h} ∥S_{j−1}(h − s)∥ ∥BT (s)x∥ ds

≤ q(h) sup_{0≤s≤h} ∥S_{j−1}(s)∥ ∥x∥

≤ q(h)^{j} sup_{0≤s≤h} ∥T (s)∥ ∥x∥

for each x ∈ D(A) and h ≥ 0. This implies

∥S_{j}(h)∥ ≤ q(h)^{j} sup_{0≤s≤h} ∥T (s)∥

for each h ≥ 0. Therefore, the assertion holds for j.

3.4. Lemma. For each j ∈ ℕ_{0} and t ≥ 0, the following statements hold.

(i) lim_{h↓0} ∥S(t + h) − T (h)S(t)∥ = lim_{h↓0} ∥S(t + h) − S(t)T (h)∥ = 0. (ii) lim_{h↓0} ∥T (h)S(t) − S(t)T (h)∥ = 0. (iii) lim_{h↓0} ∥S_{j}(t + h) − T (h)S_{j}(t)∥ = lim_{h↓0} ∥S_{j}(t + h) − S_{j}(t)T (h)∥ = 0. (iv) lim_{h↓0} ∥T (h)S_{j}(t) − S_{j}(t)T (h)∥ = 0.

Proof.

(i) Since (S(t))_{t≥0} is a semigroup, it suffices to prove the assertion for t = 0 only. By the variation of parameters formula we have

∥S(h)x − T (h)x∥ ≤ ∫_{0}^{h} ∥S(h − s)∥ ∥BT (s)x∥ ds

≤ q(h) sup_{0≤s≤h} ∥S(s)∥ ∥x∥

for each x ∈ D(A) and h ≥ 0. This implies

∥S(h) − T (h)∥ ≤ q(h) sup_{0≤s≤h} ∥S(s)∥.

(ii) This follows immediately from the two statements in (i).

(iii) By Lemma 3.2 we have

∥S_{j}(t + h) − T (h)S_{j}(t)∥ ≤ ∑_{i=1}^{j} ∥S_{i}(h)∥ ∥S_{j−i}(t)∥

and

∥S_{j}(t + h) − S_{j}(t)T (h)∥ ≤ ∑_{i=0}^{j−1} ∥S_{i}(t)∥ ∥S_{j−i}(h)∥.

Therefore, the assertion follows from Lemma 3.3.

(iv) This follows immediately from the two statements in (iii).

In the next step, we extend both semigroups (T (t))_{t≥0} and (S(t))_{t≥0} to semigroups (T̃(t))_{t≥0} and (S̃(t))_{t≥0} on X̃. Lemma 3.4 (i) immediately implies that their spaces of strong continuity coincide.

3.5. Lemma. We have X̃_{𝒮} = X̃_{𝒯}.

Analogously, we extend the operators S_{j}(t) from the Dyson-Phillips series to operators S̃_{j}(t) on X̃. The next lemma then shows that each S̃_{j}(t) induces a quotient operator Ŝ_{j}(t) on the quotient space X̂ = X̃/X̃_{𝒯}.

3.6. Lemma. Let x̃ ∈ X̃_{𝒯}. Then S̃_{j}(t)x̃ ∈ X̃_{𝒯} for every j ∈ ℕ_{0} and t ≥ 0.

Proof. Since x̃ ∈ X̃_{𝒯}, we have

lim_{h↓0} ∥S̃_{j}(t)T̃(h)x̃ − S̃_{j}(t)x̃∥ = 0.

Lemma 3.4 (iv) implies that

lim_{h↓0} ∥T̃(h)S̃_{j}(t)x̃ − S̃_{j}(t)x̃∥ = 0,

hence S̃_{j}(t)x̃ ∈ X̃_{𝒯}.

As it turns out, the operators Ŝ_{j}(t) satisfy the same exponential estimate as the unperturbed semigroup operators T̂(t) on X̂.

3.7. Proposition. For every w > ω_{crit}(𝒯) and every j ∈ ℕ_{0}, we can find a real number M such that

∥Ŝ_{j}(t)∥ ≤ Me^{wt}

for all t ≥ 0.

Proof. For j = 0 the assertion follows from the definition of ω_{crit}(𝒯). Suppose that the assertion holds for all natural numbers less than j. Since w > ω_{crit}(𝒯), there exists t_{0} > 0 such that

∥T̂(t_{0})∥ ≤ \frac{1}{2}e^{wt_{0}}.

By Lemma 3.2, we have

∥Ŝ_{j}(t)∥ ≤ ∑_{i=0}^{j} ∥Ŝ_{i}(t_{0})∥ ∥Ŝ_{j−i}(t − t_{0})∥ ≤ \frac{1}{2}e^{wt_{0}}∥Ŝ_{j}(t − t_{0})∥ + M′e^{wt}

for each t ≥ t_{0} with a suitable real number M′. This recursion formula implies that

∥Ŝ_{j}(t)∥ ≤ Me^{wt}

for each t ≥ 0 with a suitable real number M. Therefore, the assertion holds for j, and the proof is complete.

Finally, we consider (S(t))_{t≥0} as the unperturbed semigroup and take −B as perturbation. The perturbed semigroup then becomes just the original semigroup (T (t))_{t≥0}. In addition, Assumption 3.1 implies the estimate

∫_{0}^{h} ∥BS(s)x∥ ds ≤ \frac{q(h)}{1 − q(h)}∥x∥

for each x ∈ D(A) and each h ≥ 0 with q(h) < 1 (cf. [2], Corollary III.3.17). Hence, an analogue of Assumption 3.1 holds, and the semigroup (T (t))_{t≥0} can be written in the form

T (t) = ∑_{i=0}^{∞} T_{i}(t),

where the operators T_{i}(t) are defined as

T_{0}(t)x := S(t)x

and

T_{i}(t)x := − ∫_{0}^{t} T_{i−1}(t − s)BS(s)x ds

for x ∈ D(A). In order to relate the operators T_{i}(t) to the operators S_{j}(t) in the original Dyson-Phillips series, we need two lemmata.

3.8. Lemma. Assume that B is closed. If f is in L^{1}([0, t], X), t ≥ 0, then the function

(r, s) ↦ BT (r − s)f(s)

is defined almost everywhere and belongs to L^{1}(Δ_{t}, X), where Δ_{t} := {(r, s) : 0 ≤ s ≤ r ≤ t}.

Proof. Since f ∈ L^{1}([0, t], X), we can find a sequence (f_{n})_{n∈ℕ} of simple functions with values in D(A) which converges to f in L^{1}([0, t], X). For each n ∈ ℕ, we define a function g_{n} as

g_{n}(r, s) := BT (r − s)f_{n}(s).

By Assumption 3.1, (g_{n})_{n∈ℕ} is a Cauchy sequence in L^{1}(Δ_{t}, X), hence converges to a function g ∈ L^{1}(Δ_{t}, X). By passing to a suitable subsequence, we may assume that both (f_{n})_{n∈ℕ} and (g_{n})_{n∈ℕ} converge pointwise almost everywhere. Since B is closed, we obtain that

T (r − s)f(s) ∈ D(B)

and

g(r, s) = BT (r − s)f(s)

for almost all (r, s) ∈ Δ_{t}.

3.9. Lemma. If B is closed, then we have

∫_{0}^{t} S_{k}(t − s)BS(s)x ds = ∑_{j=k+1}^{∞} S_{j}(t)x

for each x ∈ D(A), k ∈ ℕ_{0} and t ≥ 0.

Proof. Suppose that the assertion holds for j − 1. By Lemma 3.8, we can apply Fubini’s theorem and obtain

∫_{0}^{t} S_{k}(t − s)BS(s)x ds = ∫_{0}^{t} [ ∫_{s}^{t} S_{k−1}(t − r)BT (r − s)BS(s)x dr] ds

= ∫_{0}^{t} [ ∫_{0}^{r} S_{k−1}(t − r)BT (r − s)BS(s)x ds] dr

= ∫_{0}^{t} S_{k−1}(t − r)BS(r)x dr − ∫_{0}^{t} S_{k−1}(t − r)BT (r)x dr

= ∑_{j=k}^{∞} S_{j}(t)x − S_{k}(t)x

= ∑_{j=k+1}^{∞} S_{j}(t)x

for all x ∈ D(A) and t ≥ 0.

We can now express the operators T_{i}(t) in terms of the operators S_{j}(t). This will be used in Theorem 4.5 to deduce regularity properties of T_{i}(t) from the corresponding regularity properties of S_{j}(t).

3.10. Proposition. If B is closed, then we have

T_{i}(t) = (−1)^{i} ∑_{j=i}^{∞} \binom{j}{i}S_{j}(t)

for each i ∈ ℕ_{0} and t ≥ 0. Moreover, the series on the right hand side converges uniformly for t in compact intervals.

Proof. Suppose that the assertion holds for i − 1. Using Lemma 3.9, we obtain

T_{i}(t)x = − ∫_{0}^{t} T_{i−1}(t − s)BS(s)x ds

= (−1)^{i} ∑_{k=i−1}^{∞} \binom{k}{i − 1} ∫_{0}^{t} S_{k}(t − s)BS(s)x ds

= (−1)^{i} ∑_{k=i−1}^{∞} \binom{k}{i − 1} ∑_{j=k+1}^{∞} S_{j}(t)

= (−1)^{i} ∑_{j=i}^{∞} ∑_{k=i−1}^{j−1} \binom{k}{i − 1}S_{j}(t)x

= (−1)^{i} ∑_{j=i}^{∞} \binom{j}{i}S_{j}(t)x

for all x ∈ D(A) and t ≥ 0. Since D(A) is dense, the identity holds for all x ∈ X and t ≥ 0.

4. The Regularity Assumption

We keep the assumptions from the previous section and consider the remainder terms

R_{k}(t) := ∑_{j=k}^{∞} S_{j}(t)

of the Dyson-Phillips series of the semigroup (S(t))_{t≥0}. Then

S(t) = ∑_{j=0}^{k−1} S_{j}(t) + R_{k}(t),

and so we obtain for the induced quotient operators that

Ŝ(t) = ∑_{j=0}^{k−1} Ŝ_{j}(t) + R̂_{k}(t).

In order to obtain a relation beween the critical growth bounds of (S(t))_{t≥0} and (T (t))_{t≥0}, we need an additional assumption on these remainder terms.

4.1. Theorem. Suppose that the perturbing operator B satisfies Assumption 3.1. If the mapping t ↦ R_{k}(t) is norm right continuous for t ≥ t_{0} and some k ∈ ℕ, then one has

ω_{crit}(𝒮) ≤ ω_{crit}(𝒯).

Proof. Using the norm continuity of R_{k}(t) and Lemma 3.4 (i) and (iii), we obtain

lim_{h↓0} ∥T (h)R_{k}(t) − R_{k}(t)∥ ≤ lim_{h↓0} ∥T (h)R_{k}(t) − R_{k}(t + h)∥ + ∥R_{k}(t + h) − R_{k}(t)∥ = 0

for every t ≥ t_{0}. This implies that R̃_{k}(t) maps X̃ into X̃_{𝒯}, hence R̂_{k}(t) = 0 for t ≥ t_{0}. Therefore, we have

Ŝ(t) = ∑_{j=0}^{k−1} Ŝ_{j}(t)

for t ≥ t_{0}. By Proposition 3.7, we can find for every w > ω_{crit}(𝒯) a real number M such that

∥Ŝ(t)∥ ≤ Me^{wt}

for every t ≥ t_{0}. By definition of ω_{crit}(𝒮), it follows that

ω_{crit}(𝒮) ≤ ω_{crit}(𝒯).

In combination with Theorem 2.1 (iii) and (iv), Theorem 4.1 yields the following partial spectral mapping theorem.

4.2. Corollary. Suppose that the operator B satisfies Assumption 3.1. If the mapping t ↦ R_{k}(t) is norm right continuous for t ≥ t_{0} and some k ∈ ℕ, then one has

ω_{0}(𝒮) ≤ max{s(A + B), ω_{crit}(𝒯)}

and

σ(S(t)) \ {0} ⊂ e^{tσ(A+B)} ∪ {λ : |λ| ≤ r_{crit}(T (t))}

for each t ≥ 0.

If the norm continuity already holds for R_{1}(t), the above assertions can be improved considerably.

4.3. Theorem. Suppose that the perturbing operator B satisfies Assumption 3.1. If the mapping t ↦ R_{1}(t) is norm right continuous for t ≥ t_{0}, then one has

σ_{crit}(S(t)) = σ_{crit}(T (t))

for each t ≥ t_{0}.

Proof. Using the norm continuity of R_{1}(t) and Lemma 3.4 (i) and (iii), we obtain

lim_{h↓0} ∥T (h)R_{1}(t) − R_{1}(t)∥ ≤ lim_{h↓0} ∥T (h)R_{1}(t) − R_{1}(t + h)∥ + ∥R_{1}(t + h) − R_{1}(t)∥ = 0

for every t ≥ t_{0}. This implies that R̃_{1}(t) maps X̃ into X̃_{𝒯}, hence R̂_{1}(t) = 0 for t ≥ t_{0}. Therefore, we have

Ŝ(t) = T̂(t)

for t ≥ t_{0}. In particular, we have

σ_{crit}(S(t)) = σ_{crit}(T (t))

for t ≥ t_{0}.

4.4. Corollary. Suppose that the operator B satisfies Assumption 3.1. If the mapping t ↦ R_{1}(t) is norm right continuous for t ≥ t_{0}, then one has

ω_{0}(𝒮) = max{s(A + B), ω_{crit}(𝒯)}

and

σ(S(t)) \ {0} = e^{tσ(A+B)} ∪ σ_{crit}(T (t)) \ {0}

for each t ≥ t_{0}.

If, in addition, we require B to be closed, it is possible to give a condition implying equality of the critical growth bounds of (S(t))_{t≥0} and (T (t))_{t≥0}.

4.5. Theorem. Suppose that B is closed and satisfies Assumption 3.1. If the mapping t ↦ R_{k}(t) is norm continuous for t ≥ 0 and some k ∈ ℕ, then one has

ω_{crit}(𝒮) = ω_{crit}(𝒯).

Proof. By Theorem 4.1, we have ω_{crit}(𝒮) ≤ ω_{crit}(𝒯). On the other hand, since the mapping t ↦ S_{k}(t) is norm continuous, Proposition 3.10 implies that the mapping t ↦ T_{k}(t) is norm continuous as well. Consequently, we can interchange the roles of (S(t))_{t≥0} and (T (t))_{t≥0} and obtain ω_{crit}(𝒯) ≤ ω_{crit}(𝒮).

4.6. Corollary. Suppose that B is closed and satisfies Assumption 3.1. If the mapping t ↦ R_{k}(t) is norm continuous for t ≥ 0 and some k ∈ ℕ, then one has

ω_{0}(𝒮) = max{s(A + B), ω_{crit}(𝒯)}

and

σ(S(t)) \ {0} ⊂ e^{tσ(A+B)} ∪ {λ : |λ| ≤ r_{crit}(T (t))}

for each t ≥ 0.

The norm continuity of R_{k}(t) for t ≥ 0 is characterized in our final proposition (compare [11], Lemma 6.2 for the case of a bounded perturbation).

4.7. Proposition. Suppose that B satisfies Assumption 3.1. Then, for each k ∈ ℕ, the following statements are equivalent:

(i) The mapping t ↦ S_{k}(t) is norm continuous for t ≥ 0. (ii) The mapping t ↦ S_{j}(t) is norm continuous for t ≥ 0 and all j ≥ k. (iii) The mapping t ↦ R_{k}(t) is norm continuous for t ≥ 0.

Proof.

(i) ⟹ (ii): Suppose that the mapping t ↦ S_{j−1}(t) is norm continuous for t ≥ 0. Since

S_{j}(t)x = ∫_{0}^{t} S_{j−1}(t − s)BT (s)x ds,

we obtain

∥S_{j}(t + h)x − S_{j}(t)x∥ ≤ q(t) sup_{0≤s≤t} ∥S_{j−1}(s + h) − S_{j−1}(s)∥ ∥x∥

+ q(h) sup_{0≤s≤h} ∥S_{j−1}(s)∥ ∥T (t)∥ ∥x∥

for all x ∈ D(A), t ≥ 0 and h ≥ 0. Therefore, the mapping t ↦ S_{j}(t) is norm continuous for t ≥ 0 as well.

(ii) ⟹ (iii): This follows immediately from the uniform convergence of the Dyson-Phillips series.

(iii) ⟹ (i): Suppose that the mapping t ↦ R_{k}(t) is norm continuous for t ≥ 0. Since

S_{k}(t)x = R_{k}(t)x − ∫_{0}^{t} R_{k}(t − s)BT (s)x ds,

we obtain

∥S_{k}(t + h)x − S_{k}(t)x∥ ≤ ∥R_{k}(t + h)x − R_{k}(t)x∥

+ q(t) sup_{0≤s≤t} ∥R_{k}(s + h) − R_{k}(s)∥ ∥x∥

+ q(h) sup_{0≤s≤h} ∥R_{k}(s)∥ ∥T (t)∥ ∥x∥

for all x ∈ D(A), t ≥ 0 and h ≥ 0. Therefore, the mapping t ↦ S_{k}(t) is norm continuous for t ≥ 0.

4.8. Remarks. (i) Under considerably stronger assumptions, Andreu, Martínez and Mazón proved a partial spectral mapping theorem similar to that given in Theorem 4.1 (see [1]). Our result can also be viewed as an answer to Mokhtar-Kharroubi’s conjecture from [8] and [9]. (ii) A perturbation theorem analogous to Theorem 4.1 has been proved by Voigt (see [19]). Instead of assuming norm continuity of the remainder terms, he required compactness and showed that ω_{ess}(𝒮) ≤ ω_{ess}(𝒯).

5. Examples

In this final section we will give two examples. The first one shows that even if the unperturbed semigroup is nilpotent, the perturbed semigroup need not be norm continuous at infinity (see [6]), and therefore not eventually norm continuous. This considerably simplifies Phillips’ example from [14].

5.1. Example. Take X := L^{1}([0, 2]) and define the operator A as

Af(u) := −f′(u)

with domain

D(A) := {f ∈ W^{1,1}([0, 2]) : f(0) = 0}.

The perturbation B ∈ ℒ(X) is defined as

Bf(u) := \begin{cases}(u + 1) & for 0 ≤ u ≤ 1, \\ & otherwise. \end{cases}

The operator A generates the nilpotent translation semigroup (T (t))_{t≥0} given by

T (t)f(u) = \begin{cases}(u − t) & for u − t ≥ 0, \\ & otherwise, \end{cases}

and A + B generates a strongly continuous semigroup (S(t))_{t≥0}. If we define

f_{n}(u) := \begin{cases}^{−2πinu} & for 0 ≤ u ≤ 1, \\ e^{−2πinu} & otherwise, \end{cases}

for each n ∈ ℕ, then f_{n} ∈ D(A) and

(A + B)f_{n} = 2πinf_{n}.

Hence, we have

s(A + B) ≥ 0.

On the other hand, since

S(t)f_{n} = e^{2πint}f_{n}

holds, the sequence (f_{n})_{n∈ℕ} does not belong to X̃_{𝒯} and forms an eigenvector of S̃(1). Hence, we have

1 ∈ σ_{crit}(S(1))

and

ω_{crit}(𝒮) ≥ 0.

Computing the terms of the Dyson-Phillips series explicitely, we find

|S_{j}(t)f(u)| ≤ \begin{cases}⌊f(u − t + j)| & for 0 ≤ u − t + j ≤ 2, \\ & otherwise. \end{cases}

This implies that ∥S_{j}(t)∥ ≤ 1 for |t − j| < 2 and ∥S_{j}(t)∥ = 0 for |t − j| ≥ 2. Thus

∥S(t)∥ ≤ 4

for all t ≥ 0. Consequently, we have

ω_{0}(𝒮) = ω_{crit}(𝒮) = s(A + B) = 0.

Therefore, the semigroup (S(t))_{t≥0} is not norm continuous at infinity in the sense of [6].

In the second example we apply our results to an equation from population dynamics.

5.2. Example. Consider the equation

\frac{∂}{∂t}n(t, a, x) = −\frac{∂}{∂a}n(t, a, x) − μ(a, x)n(t, a, x) − b(a, x)n(t, a, x) + 4b(2a, x)n(t, 2a, x)

with boundary condition

n(t, \frac{α}{2}, x) = 0

for all t ∈ [0, ∞[, x ∈ ℝ^{N} and some fixed α ∈]0, 1[, and with the initial condition

n(0, a, x) = n_{0}(a, x)

for all a ∈ [\frac{α}{2}, ∞[ and x ∈ ℝ^{N}. This equation describes the change of the density n(t, a, x) of a cell population at time t, age a, and position x. The function μ, representing the death rate, and the function b, representing the division rate are positive and continuous on [\frac{α}{2}, ∞[×ℝ^{N} with compact support. Moreover, we assume

b(a, x) = 0 for \frac{α}{2} ≤ a ≤ α and a ≥ 1.

This means that only cells with age between α and 1 divide into two daughter cells of equal size. For more information on the biological interpretation of this equation, however without the spatial distribution, we refer to [7].

In the Banach space X := \{f ∈ L^{1}(ℝ × ℝ^{N}) : f(a, x) = 0 for all a ≤ \frac{α}{2} and x ∈ ℝ^{N}\} this equation can be rewritten as

\frac{∂}{∂t}n(t, a, x) = (A + B)n(t, a, x),

n(0, a, x) = n_{0}(a, x).

Here, the operator A is defined as

Af(a, x) := −\frac{∂}{∂a}f(a, x) − μ(a, x)f(a, x) − b(a, x)f(a, x)

with domain

D(A) := \{f ∈ W^{1,1}(ℝ × ℝ^{N}) : f(a, x) = 0 for all a ≤ \frac{α}{2} and x ∈ ℝ^{N}\}.

The perturbation B ∈ ℒ(X) is defined as

Bf(a, x) := 4b(2a, x)f(2a, x).

Then the closure \overline{A} generates a strongly continuous semigroup (T (t))_{t≥0} which is given by

T (t)f(a, x) = E(a − t, a, x)f(a − t, x)

where

E(a − t, a, x) := e^{− ∫_{a−t}^{a}(μ(τ,x)+b(τ,x)) dτ}

for all f ∈ X and t ≥ 0. From this representation we obtain

∥T (t)∥ = sup_{\begin{matrix}a−t≥ \frac{α}{2} , \\ x∈ℝ^{N} \end{matrix}} e^{− ∫_{a−t}^{a}(μ(τ,x)+b(τ,x)) dτ},

hence

log ∥T (t)∥ = − inf_{\begin{matrix}a−t≥ \frac{α}{2} , \\ x∈ℝ^{N} \end{matrix}} ∫_{a−t}^{a}(μ(τ, x) + b(τ, x)) dτ,

and therefore

ω_{0}(𝒯) = − sup_{t>0} inf_{\begin{matrix}a−t≥ \frac{α}{2} , \\ x∈ℝ^{N} \end{matrix}} \frac{1}{t} ∫_{a−t}^{a}(μ(τ, x) + b(τ, x)) dτ.

Moreover, since σ(\overline{A}) is invariant under translations along iℝ, we have

ω_{crit}(𝒯) = ω_{0}(𝒯).

Consider now the strongly continuous semigroup (S(t))_{t≥0} generated by \overline{A} + B. The first Dyson-Phillips term for the perturbed semigroup can be computed as

S_{1}(t)f(a, x) = ∫_{2a−2t}^{2a−t} T (2a − t − s)BT (2t + s − 2a)f(a, x) ds

= ∫_{2a−2t}^{2a−t} 4E(t + s − a, a, x)E(s, 2t + 2s − 2a, x)b(2t + 2s − 2a, x)f(s, x) ds

= ∫_{2a−2t}^{2a−t} G(t, s, a, x)f(s, x) ds,

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