LOWER SEMICONTINUITY AND THE THEOREM OF DATKO AND PAZY

J.M.A.M. VAN NEERVEN

Abstract. Let T be a C_{0}−semigroup on a real or complex Banach space X and let J : C^{+}[0, ∞) → [0, ∞] be a lower semicontinuous and nondecreasing functional on C^{+}[0, ∞), the positive cone of C[0, ∞), satisfying J(c1) = ∞ for all c > 0. We prove the following result: if T is not uniformly exponentially stable, then the set

{x ∈ X : J(∥T (·)x∥) = ∞}

is residual in X.

A C_{0}−semigroup T = {T (t)}_{t≥0} on a (real or complex) Banach space X is said to be uniformly exponentially stable if there exist constants M ≥ 1 and ω > 0 such that

∥T (t)∥ ≤ Me^{−ωt}, t ≥ 0.

A well-known result of Datko and Pazy [7] states that T is uniformly exponentially stable if there exists p ∈ [1, ∞) such that

∫_{0}^{∞} ∥T (t)x∥^{p} dt < ∞, x ∈ X.

This result was improved by Rolewicz [8], who proved that T is uniformly exponentially stable if there exists a nondecreasing function ϕ : [0, ∞) → [0, ∞) with ϕ(t) > 0 for all t > 0 such that

∫_{0}^{∞} ϕ(∥T (t)x∥) dt < ∞, x ∈ X.

The same result was obtained independently by Littman [3]. In [6] it is shown that T is uniformly exponentially stable if there exists a Banach function space E over [0, ∞) with the property that

lim_{t→∞} ∥1_{[0,t]}∥_{E} = ∞, (1.1)

such that

∥T (·)x∥ ∈ E, x ∈ X.

The Datko-Pazy theorem follows from this by taking E = L^{p}[0, ∞). As is shown in [6], Rolewicz’s result can be derived as well by taking for E a suitable Orlicz space over [0, ∞). This is a somewhat artificial construction, however, and in this note we propose a more natural generalization of the theorems of Datko-Pazy and Rolewicz. It is based on the following lemma.

1991 Mathematics Subject Classification. Primary: 47D06, secondary: 47D03.

Key words and phrases. C_{0}−semigroup, uniformly exponentially stable, Datko-Pazy theorem, lower semicontinuous functional.

Lemma 1. Let T be a bounded linear operator on a Banach space X, and assume that its spectral radius satisfies r(T) ≥ 1. Then for all x ∈ X and δ > 0 there exists a constant C > 0 with the following property: for all n ∈ ℕ there exists y ∈ X such that ∥x − y∥ < δ and ∥T^{j}y∥ ≥ C for all j = 0, …, n.

Proof. If the essential spectral radius of T satisfies r_{ess}(T) < 1, then T has an eigenvalue λ with |λ| = 1, and we may proceed as in part A of the proof of [5, Theorem 2.3], which works for both real and complex scalars.

Suppose next that r_{ess}(T) = 1. Let c be the constant from [5, Lemma 2.2]. Fix n ∈ ℕ and let E denote the finite–dimensional linear subspace of X spanned by the set {T^{j}x : j = 0, …, n}. By [4, Lemma 1], there exists a closed subspace F of X of finite codimension such that

∥e + f∥ ≥ max\{\frac{1}{2}∥e∥, \frac{1}{4}∥f∥\}, e ∈ E, f ∈ F.

Let F′ = {f ∈ F : T^{j}f ∈ F, j = 0, … n}. The assumption r_{ess}(T) = 1 implies that X is infinite-dimensional, and therefore F′ is a nontrivial closed subspace of X of finite codimension. By [5, Lemma 2.2] there exists a vector f ∈ F′ with ∥f∥ = 1 and ∥T^{j}f∥ ≥ c, j = 0, …, n; note that the proof of this lemma also works for both real and complex scalars. Let y := x + \frac{1}{2}δf. Then ∥x − y∥ < δ and

∥T^{j}y∥ = ∥T^{j}x+\frac{1}{2}δT^{j}f∥ ≥ \frac{1}{8}δ∥T^{j}f∥ ≥ \frac{1}{8}c δ, j = 0, …, n.

For Hilbert spaces operators, one may take any constant 0 < C < δ; in this form the result is due to Beauzamy [1].

We denote by C[0, ∞) the space of all continuous functions on [0, ∞). With the topology of uniform convergence on compact sets, this is a separable Fréchet space. By C^{+}[0, ∞) we denote the positive cone of C[0, ∞).

Recall that a subset of a topological space is called residual if its complement is of the first category.

Theorem 2. Let J : C^{+}[0, ∞) → [0, ∞] be a map with the following properties: 1. J is lower semicontinuous;

2. J is nondecreasing, i.e. 0 ≤ f ≤ g implies J(f) ≤ J(g);

3. J(c1) = ∞ for all c > 0. Let T be a C_{0}−semigroup on a Banach space X which is not uniformly exponentially stable. Then the set

{x ∈ X : J(∥T (·)x∥) = ∞}

is residual.

Proof. For k = 1, 2, … let

X_{k} = {x ∈ X : J(∥T (·)x∥) > k}.

The lower semicontinuity of J implies that each X_{k} is open. It suffices to prove that each X_{k} is dense.

Fix k ≥ 1 and let B(x, δ) an open ball with centre x ∈ X and radius δ > 0. We will show that X_{k} ∩ B(x, δ) ≠ ∅.

Since T is not uniformly exponentially stable we have r(T (1)) ≥ 1. By Lemma 1 there exists a constant C > 0 with the following property: for each n = 0, 1, … there exists an y_{n} ∈ X with ∥x − y_{n}∥ < δ and ∥T (j)y_{n}∥ ≥ C for all j = 0, …, n. Then,

∥T (t)y_{n}∥ ≥ \frac{C}{M} , t ∈ [0, n],

where M := sup_{0≤s≤1} ∥T (s)∥.

Let (f_{n})_{n≥0} ⊆ C^{+}[0, ∞) be a sequence with

0 ≤ f_{n} ≤ \frac{C}{M} 1_{[0,n]}, n = 0, 1, …

and

lim_{n→∞} f_{n} = \frac{C}{M} 1 uniformly on compact sets.

Then,

∥T (t)y_{n}∥ ≥ \frac{C}{M} 1_{[0,n]}(t) ≥ f_{n}(t), t ∈ [0, ∞), n = 0, 1, ….

By the monotonicity and lower semicontinuity of J we obtain

lim inf_{n→∞} J(∥T (·)y_{n}∥) ≥ lim inf_{n→∞} J(f_{n}) ≥ J(\frac{C}{M} 1) = ∞.

In particular, there exists an index n_{0} such that J(∥T (·)y_{n_{0}}∥) > k. Therefore, y_{n_{0}} ∈ X_{k} ∩ B(x, δ), showing that the intersection is nonempty.

Rolewicz’s theorem follows from Theorem 2 by taking

J(f) = ∫_{0}^{∞} ϕ(f(t)) dt, f ∈ C^{+}[0, ∞).

This functional satisfies the three assumptions of Theorem 2; lower semicontinuity follows from Fatou’s lemma. In fact, if T is not uniformly exponentially stable, we obtain the somewhat stronger result that the set

{x ∈ X : ∫_{0}^{∞} ϕ(∥T (t)x∥) dt = ∞}

is residual.

The result from [6] mentioned above involving Banach function spaces satisfying (1.1) also follows from Theorem 2: take

J(f) := lim_{t→∞} ∥1_{[0,t]}f∥_{E} = sup_{t≥0} ∥1_{[0,t]}f∥_{E}.

To see that J is lower semicontinuous we argue as follows. For each t ≥ 0, the map J_{t}(f) := ∥1_{[0,t]}f∥_{E} is continuous. Indeed, if f_{n} → f uniformly on compact sets, then 1_{[0,t]}f_{n} → 1_{[0,t]}f uniformly. Therefore, given ε > 0, for n large enough we have

|1_{[0,t]}f_{n} − 1_{[0,t]}f| ≤ ε1_{[0,t]}

in E, and therefore by the triangle inequality,

|J(f_{n}) − J(f)| ≤ ∥1_{[0,t]}f_{n} − 1_{[0,t]}f∥_{E} ≤ ε∥1_{[0,t]}∥_{E}.

Being the supremum of a family of continuous maps, J is lower semicontinuous. Thus, if T is not uniformly exponentially stable, then

{x ∈ X : ∥T (·)x∥ ∉ E} is residual.

(1.2)

The norm of a Banach function space E is said to have the Fatou property if the following holds: if f is a measurable function and (f_{n})_{n≥0} is a sequence in E such that sup_{n≥0} ∥f_{n}∥_{E} < ∞ and 0 ≤ f_{n} ↑ f, then f ∈ E and lim_{n→∞} ∥f_{n}∥_{E} = ∥f∥_{E}.

For Banach function spaces E whose norm has the Fatou property, in particular for E = L^{p}[0, ∞) with 1 ≤ p < ∞, the result contained in (1.2) can be proved in a more elementary way as follows.

Suppose E is a Banach function space satisfying (1.1) whose norm has the Fatou property, and assume that the set of all x ∈ X with ∥T (·)x∥ ∈ E is of the second category. We will show that T is uniformly exponentially stable.

For k = 1, 2, … define

X_{k} = {x ∈ X : ∥ ∥T (·)x∥ ∥_{E} ≤ k}.

In order to prove that X_{k} is closed, suppose that x_{n} → x in X with x_{n} ∈ X_{k} for all n ≥ 0. Defining f_{n} := ∥T (·)x_{n}∥ ∈ E and f := ∥T (·)x∥, we have 1_{[0,j]}f ∈ E and 1_{[0,j]}f_{n} → 1_{[0,j]}f uniformly, and hence in E, as n → ∞. It follows that

∥1_{[0,j]}f∥_{E} = lim_{n→∞} ∥1_{[0,j]}f_{n}∥_{E} ≤ lim sup_{n→∞} ∥f_{n}∥_{E} ≤ k.

By the Fatou property, it follows that f ∈ E and

∥f∥_{E} = lim_{j→∞} ∥1_{[0,j]}f∥_{E} ≤ k.

Therefore x ∈ X_{k} and X_{k} is closed.

Since by assumption ⋃_{k≥1} X_{k} is of the second category, at least one X_{k_{0}} has nonempty interior. Let B(x_{0}, δ_{0}) be an open ball with centre x_{0} and radius δ_{0} contained in X_{k_{0}}. Then by the triangle inequality in E, the open ball B(0, δ_{0}) is contained in X_{2k_{0}}. But then for all nonzero x ∈ X and 0 < δ < δ_{0},

∥ ∥T (·)x∥ ∥_{E} = \frac{∥x∥}{δ} · ∥ ∥T (·)(δx/∥x∥)∥ ∥_{E} ≤ \frac{∥x∥}{δ} · 2k_{0} < ∞.

This shows that ∥T (·)x∥ ∈ E for all x ∈ E, and we may apply the result from [6] (or the Datko-Pazy theorem if E = L^{p}[0, ∞)) to conclude that E is uniformly exponentially stable.

Remark 3. L. Weis has kindly pointed out that for E = L^{p}[0, ∞) this, and related residuality results, have been obtained by V. Wrobel (preprint).

A function J satisfying the three assumptions of Theorem 2 cannot be finitely valued on the subset C_{0}^{+}[0, ∞) of all positive functions vanishing at infinity. Indeed, the existence of such J would imply that every strongly stable C_{0}−semigroup is exponentially stable. However, the C_{0}−semigroup of left translations on C_{0}[0, ∞) is strongly stable but not uniformly exponentially stable. Recall that T is said to be strongly stable if

lim_{t→∞} ∥T (t)x∥ = 0, x ∈ X.

In fact we have the following simple observation. First note that conditions 2 and 3 of Theorem 2 imply that J(f) = ∞ whenever f ≥ c1 for some c > 0.

Proposition 4. If J : C^{+}[0, ∞) → [0, ∞] is lower semicontinuous and satisfies J(f) = ∞ for all f ∈ C^{+}[0, ∞) with f ≥ c1 for some c > 0, then the set

{f ∈ C_{0}^{+}[0, ∞) : J(f) = ∞}

is residual in C_{0}^{+}[0, ∞), endowed with the topology of uniform convergence.

Proof. Suppose, for a contradiction, that the set F := {f ∈ C_{0}^{+}[0, ∞) : J(f) < ∞} is of the second category in C_{0}^{+}[0, ∞). For k = 1, 2, … let

F_{k} := {f ∈ C_{0}^{+}[0, ∞) : J(f) ≤ k}.

As a subset of C_{0}^{+}[0, ∞), each F_{k} is closed. Indeed, if f_{n} → f uniformly with f_{n} ∈ F_{k} for all n, then f_{n} → f uniformly on compact sets, and the lower semicontinuity of J gives J(f) ≤ lim inf_{n→∞} J(f_{n}) ≤ k. Since by assumption ⋃_{k≥1} F_{k} is of the second category in C_{0}^{+}[0, ∞), there is an F_{k_{0}} with nonempty interior relative to C_{0}^{+}[0, ∞).

Let B(f_{0}, δ_{0}) be an open ball in C_{0}^{+}[0, ∞) contained in F_{k_{0}}, and fix 0 < δ < δ_{0} arbitrary. Choose a sequence (g_{n})_{n≥0} in C_{0}^{+}[0, ∞) such that 0 ≤ g_{n} ≤ δ1 and g_{n} → δ1 uniformly on compact sets. We have f_{0} + g_{n} ∈ B(f_{0}, δ_{0}) for each n, and lim_{n→∞}(f_{0} +g_{n}) = f_{0} +δ1 uniformly on compact sets. By the lower semicontinuity of J,

J(f_{0} + δ1) ≤ lim inf_{n→∞} J(f_{0} + g_{n}) ≤ k_{0},

a contradiction.

We do not know whether Theorem 2 remains true if the conditions 2 and 3 are replaced by the condition

2′. J(f) = ∞ for all f ∈ C^{+}[0, ∞) with f ≥ c1 for some c > 0.

We are going to check next that neither one of the three conditions in Theorem 2 can be omitted.

Example 5. Define

J(f) := \{ \begin{matrix}0, & f ∈ C_{0}^{+}[0, ∞), \\ ∞, & otherwise, \end{matrix} f ∈ C^{+}[0, ∞).

Then J is nondecreasing, J(c1) = ∞ for all c > 0, but J is not lower semicontinuous. If T is a C_{0}−semigroup which is strongly stable, then J(∥T (·)x∥) = 0 for all x ∈ X, but T need not be uniformly exponentially stable.

In order to give an example showing that the second condition of Theorem 2 cannot be omitted we need some preparations.

Let us call a subset K of C^{+}[0, ∞) solid if from 0 ≤ f ≤ g and g ∈ K it follows that f ∈ K.

Proposition 6. Let K be a closed, convex, solid subset of C^{+}[0, ∞) not containing any nonzero constant function. If T is not uniformly exponentially stable, then the set of all x ∈ X with the property c ∥T (·)x∥ ∉ K for all c > 0 is residual.

Proof. Since K is closed and convex, its Minkowski functional

J_{K}(f) := inf{λ > 0 : f ∈ λK}, f ∈ C^{+}[0, ∞),

is lower semicontinuous. Since K is solid, 0 ≤ f ≤ g implies J_{K}(f) ≤ J_{K}(g). Since K does not contain any nonzero constant function we have J_{K}(c1) = ∞ for all c > 0. By Theorem 2, the set of all x ∈ X with J_{K}(∥T (·)x∥) = ∞ is residual. Noting that J_{K}(∥T (·)x∥) < ∞ if and only if c ∥T (·)x∥ ∈ K for some c > 0, this gives the result.

The solidity of K was needed only to verify condition 2 of Theorem 2. Thus if Theorem 2 were true for any functional J satisfying only conditions 1 and 3, then Proposition 6 would be true for any closed convex subset K of C^{+}[0, ∞). The following example shows that this is not true, however.

Example 7. Let T be a C_{0}−semigroup on a Banach space X which is strongly stable but not uniformly exponentially stable.

Let 0 < ε < 1 and n ≥ 0 be fixed and define

K_{ε,n} = {f ∈ C^{+}[0, ∞) : εf(n) ≥ f(n + 1)}.

This set is closed and convex, it contains no nonzero constant function, but it is not solid. In order to obtain a contradiction let us assume that Proposition 6 may be applied to the set K_{ε,n}. We then find that the set

X_{ε,n} = {x ∈ X : ε∥T (n)x∥ < ∥T (n + 1)x∥}

is residual. Let (ε_{k})_{k≥0} be a sequence with 0 < ε_{k} < 1 for all k ≥ 0 and ε_{k} ↑ 1 as k → ∞. Then

∥T (n)x∥ ≤ ∥T (n + 1)x∥

if and only if x ∈ ⋂_{k≥0} X_{ε_{k},n} =: X_{n}, and this set is residual. Next,

∥T (n)x∥ ≤ ∥T (n + 1)x∥, n = 0, 1, …

if and only if x ∈ ⋂_{n≥0} X_{n}, and this set is again residual. But since we assumed that T is strongly stable, ⋂_{n≥0} X_{n} = {0}, a contradiction.

The next example shows that condition 3 in Theorem 2 cannot be relaxed too much.

Example 8. Let J(f) := |{t ∈ [0, ∞) : f(t) > ε}|, where | · | denotes the Lebesgue measure and ε > 0 is fixed. Then J is lower semicontinuous and nondecreasing, and J(c1) = ∞ if and only if c > ε. If T is a strongly stable semigroup on X, then J(∥T (·)x∥) < ∞ for all x ∈ X, but T need not be uniformly exponentially stable.

For p ∈ [1, ∞) the functional

J_{p}(f) = ∫_{0}^{∞}(f(t))^{p} dt, f ∈ C^{+}[0, ∞), (1.3)

ocuuring in the Datko-Pazy theorem is not only nondecreasing but also convex. It is not possible, however, to replace ‘nondecreasing’ by ‘convex’ in Theorem 2, as is shown by the following example.

Example 9. Let K_{ε,n} be the closed convex set of Example 7. Clearly, λK_{ε,n} = K_{ε,n} for all λ > 0, and therefore its Minkowski functional J_{ε,n} is given by

J_{ε,n}(f) = \{ \begin{matrix}0, & f ∈ K_{ε,n}; \\ ∞, & else. \end{matrix}

In particular, J_{ε,n} is convex. As we have seen, J_{ε,n} is also lower semicontinuous and J_{ε,n}(c1) = ∞ for all c > 0. Now let us assume that the conclusion of Theorem 2 holds for the functionals J_{ε,n}. Then the conclusion of Proposition 6 holds for the sets K_{ε,n}, and it was shown in Example 7 that this leads to a contradiction.

For p ∈ (0, 1), the functional J_{p} defined by (1.3) is concave. Theorem 2 does remain true if we replace ‘nondecreasing’ by ‘concave’. To see this, for k = 1, 2, … let X_{k} be the open set

X_{k} := {x ∈ X : J(∥T (·)y∥) > k}.

Let B(x, δ) be an arbitrary open ball in X; we will show that X_{k} ∩ B(x, 2δ) ≠ ∅. Proceeding as in the proof of Theorem 2, we construct a sequence (y_{n})_{n≥0} contained in B(x, δ) and a sequence (f_{n})_{n≥0} in C^{+}[0, ∞) with

∥T (·)y_{n}∥ ≥ \frac{C}{M} 1_{[0,n]} ≥ f_{n}, n = 0, 1, …

and

lim_{n→∞} J(f_{n}) = ∞.

Let (α_{n})_{n≥0} be a sequence of real numbers satisfying 0 < α_{n} ≤ \frac{1}{2} for all n ≥ 0 and

lim_{n→∞} α_{n} = 0,  lim_{n→∞} α_{n}J(f_{n}) = ∞.

Noting that α_{n}/(1 − α_{n}) ≤ 1 for all n ≥ 0 and using the concavity of J, it follows that

lim inf_{n→∞} J(∥T (·)((1 − α_{n})y_{n})∥)

≥ lim inf_{n→∞} (α_{n}J(f_{n}) + (1 − α_{n})J (∥T (·)y_{n}∥ − \frac{α_{n}}{1 − α_{n}} f_{n}))

≥ lim_{n→∞} α_{n}J(f_{n}) = ∞.

Given a fixed integer k ≥ 1, it follows that there exists an index n_{0} such that

J(∥T (·)((1 − α_{n})y_{n})∥) > k, n ≥ n_{0}.

On the other hand, since lim_{n→∞} α_{n} = 0, there exists an index n_{1} ≥ n_{0} such that (1−α_{n})y_{n} ∈ B(x, 2δ) for all n ≥ n_{1}. For such n we have (1−α_{n})y_{n} ∈ X_{k} ∩B(x, 2δ), showing that the intersection is nonempty. This concludes the proof.

By a well-known result of Datko [2], a C_{0}−semigroup T on X is uniformly exponentially stable if and only if there exists p ∈ [1, ∞) sich that T ∗ f ∈ L^{p}([0, ∞); X) for all f ∈ L^{p}([0, ∞); X). Here, the convolution T ∗ f is defined by

(T ∗ f)(t) = ∫_{0}^{t} T (t − s)f(s) ds, t ≥ 0.

In fact, let x ∈ X be arbitrary and define f_{x} ∈ L^{p}([0, ∞); X) by

f_{x}(t) = \{ \begin{matrix}T (t)x, & t ∈ [0, 1), \\ 0, & t ≥ 1. \end{matrix}

Then,

(T ∗ f_{x})(t) = \{ \begin{matrix}t T (t)x, & t ∈ [0, 1), \\ T (t)x, & t ≥ 1. \end{matrix}

Since by assumption T ∗ f_{x} ∈ L^{p}([0, ∞); X) for all x ∈ X, it follows that T (·)x ∈ L^{p}([0, ∞); X) for all x ∈ X. Therefore T is uniformly exponentially stable by the Datko-Pazy theorem. An easy modification of this argument shows that it is enough to have T ∗ f ∈ L^{p}([0, ∞); X) for all f ∈ C_{c}((0, ∞); X), the space of continuous X−valued functions with compact support in (0, ∞); cf. the proof below.

The following result extends Datko’s theorem to the setting of lower semicontinuous functionals:

Theorem 10. Let T be a C_{0}−semigroup on X and let J : C^{+}[0, ∞) → [0, ∞] be a map with the following properties:

1. J is lower semicontinuous; 2. J is nondecreasing;

3. J(f) = ∞ for all f ∈ C^{+}[0, ∞) satisfying lim inf_{t→∞} f(t) > 0. If J(∥T ∗ f∥) < ∞ for all f ∈ C_{c}((0, ∞); X), then T is uniformly exponentially stable.

Proof. Fix an arbitrary nonzero 0 ≤ ϕ ∈ C_{c}((0, ∞); X), with support in [a, b] say, and define

ψ(t) = ∫_{0}^{t} ϕ(s) ds, t ≥ 0.

Then for all t ≥ b we have ψ(t) = ψ(b) > 0. For x ∈ X let f_{x} ∈ C_{c}((0, ∞); X) be defined by

f_{x}(t) := ϕ(t)T (t)x, t ≥ 0.

Then J(∥T ∗ f_{x}∥) < ∞ by assumption. By the Baire category theorem, there is a ball B(x_{0}, r) in X and an N ∈ ℕ such that J(∥T ∗ f_{x}∥) ≤ N for all x ∈ B(x_{0}, r). Now suppose T were not uniformly exponentially stable. Then r(T) ≥ 1, and by Lemma 1 there exists a constant c > 0 such that for all n ∈ ℕ we can find y_{n} ∈ B(x_{0}, r) with

∥T (t)y_{n}∥ ≥ c, t ∈ [0, n].

Noting that

(T ∗ f_{y_{n}})(t) = ∫_{0}^{t} ϕ(s)T (t − s)T (s)y_{n} ds = ψ(t)T (t)y_{n}, t ≥ 0,

we see that

J(cψ) ≤ lim inf_{n→∞} J(ψ · c1_{[0,n]}) ≤ J(T ∗ f_{y_{n}}) ≤ N.

On the other hand, from lim inf_{t→∞} cψ(t) = cψ(b) > 0 it follows that J(cψ) = ∞, and we have arrived at a contradiction.

References

[1] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North Holland, 1988.

[2] R. Datko, Uniform asymptotic stability of evolutionary processes in in a Banach space, SIAM J. Math. Anal. 3 (1972), 428–445.

[3] W. Littman, A generalization of a theorem of Datko and Pazy, in: Lect. Notes in Control and Inform. Sci. 130, Springer-Verlag, 1989, 318–323.

[4] V. Müller, Local behaviour of the polynomial calculus of operators, J. Reine Angew. Math. 430 (1992), 61–68.

[5] V. Müller, Orbits, weak orbits and local capacity of operators, preprint, 1999.

[6] J.M.A.M. van Neerven, Exponential stability of operators and operator semigroups, J. Func. Anal. 130 (1995), 293–309.

[7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.

[8] S. Rolewicz, On uniform N-equistability, J. Math. Anal. Appl. 115 (1986), 434–441.

Department of Applied Mathematical Analysis, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands

   E-mail address: J.vanNeerven@its.tudelft.nl