A SUFFICIENT CONDITION FOR EXPONENTIAL DICHOTOMY OF PARABOLIC EVOLUTION EQUATIONS

ROLAND SCHNAUBELT

1. Introduction

In this note we establish the exponential dichotomy of the abstract Cauchy problem

(CP) \{ \begin{matrix}\frac{d}{dt}u(t) & = & A(t)u(t), t ≥ s, \\ u(s) & = & x, \end{matrix}

assuming that A(t), t ∈ ℝ, generates an analytic semigroup (e^{τA(t)})_{τ≥0} and A(·)^{−1} satisfies a certain Hölder condition (see (P) below). Moreover, each semigroup (e^{τA(t)})_{τ≥0} is supposed to have exponential dichotomy and the Hölder constant of A(·)^{−1} must be sufficiently small. Already simple matrix examples, [6, p.3], [16, Ex.3.4], show that one cannot omit this smallness condition.

Similar sufficient conditions for exponential dichotomy are known for matrices A(t), [6], bounded operators A(t), [4], and delay equations, [13]. In [16] we have shown the main Theorem 3.2 of the present paper for the special case of time independent domains D(A(t)) ≡ D(A(0)). This seems to be the first result of this kind for parabolic evolution equations; but the hypothesis of constant domains is rather restrictive because it excludes, for instance, Neumann type boundary conditions in the context of parabolic partial differential equations, see [17, §4]. The results in [4] and [16] are further improved by Proposition 3.3 which determines the rank of the splitting projections for (CP).

Our approach is based on a characterization of the exponential dichotomy of (CP) by means of the associated evolution semigroup defined in (2.10), cf. [5] or [8, §VI.9]. In order to prove our main result in the third section we use ideas from [4] and [16], but we supplement them by the theory of inter- and extrapolation spaces as studied in, e.g., [3], [8], [14]. The necessary tools are introduced in the next section.

2. Preparations

An operator A on a Banach space X is called sectorial of type (ϕ, K, w) if its domain D(A) is dense, Σ ⊆ ρ(A), and ∥R(λ, A(t))∥ ≤ \frac{K}{1+|λ−w|} for λ ∈ Σ := \{w\} ∪ \{λ : | arg(λ − w)| < ϕ\} and constants ϕ ∈ (\frac{π}{2}, π] and K, w ≥ 0. Recall that a sectorial operator generates an analytic semigroup. Given an invertible sectorial operator A and 0 < α < 1, we define the interpolation space

X_{α}^{A} := \overline{X}^{∥·∥_{α}^{X}} with norm ∥x∥_{α}^{A} := sup_{r>w} ∥r^{α}AR(r, A)x∥.

By X_{1}^{A} we denote D(A) endowed with the norm ∥Ax∥, and X_{0}^{A} := X. We further introduce the completion X_{−1}^{A} of X with respect to the norm ∥x∥_{−1}^{A} := ∥A^{−1}x∥. Then A has a unique continuous extension A_{−1} : X → X_{−1}^{A} which is sectorial of type (ϕ, K, w). So we can define X_{α−1}^{A} := (X_{−1}^{A})_{α}^{A_{−1}}, 0 < α < 1, equipped with the norm

1991 Mathematics Subject Classification. 34G10, 47D06.

∥x∥_{α−1}^{A} := sup_{r>w} ∥r^{α}R(r, A_{−1})x∥.

The continuous and dense embeddings

X_{1} ↪ X_{β}^{A} ↪ D((w − A)^{α}) ↪ X_{α}^{A} ↪ X ↪ X_{α−1}^{A} ↪ X_{−1}^{A}

hold for 0 ≤ α < β ≤ 1, where the domain of the fractional power is endowed with the graph norm and the norms of the embeddings depend only on the type of A and ∥A^{−1}∥, α, β. Moreover, the operator A can be extended to an isomorphism A_{α−1} : X_{α}^{A} → X_{α−1}^{A} for 0 ≤ α ≤ 1 which coincides with the part of A_{−1} in X_{α−1}^{A}. The semigroup e^{tA} restricts to an analytic semigroup on X_{α}^{A} and extends to an analytic semigroup on X_{α−1}^{A} generated by A_{α−1}, 0 ≤ α ≤ 1, and it satisfies

∥A^{k} e^{tA_{α−1}}x∥_{X} ≤ C t^{α−k−1} ∥x∥_{α−1}^{A}  (2.1)

for k = 0, 1, 0 ≤ α ≤ 1, x ∈ X_{α−1}^{A}, 0 < t ≤ 1, and a constant C depending only on the type of A and ∥A^{−1}∥. These facts follow from, e.g., [3, §V.1, V.2.1], [8, §II.5], [14, §1, 2.2].

The next assumption was introduced by P. Acquistapace and B. Terreni, [2], in a slightly different form. (P) A(t), t ∈ ℝ, is a sectorial operator of type (ϕ, K, w), with a uniformly bounded

inverse, and ∥A(t)^{−1} − A(s)^{−1}∥_{L(X,X_{ν}^{t})} ≤ L |t − s|^{μ} holds for t, s ∈ ℝ and constants L ≥ 0 and μ, ν ∈ (0, 1] with μ + ν > 1.

Here we set X_{α}^{t} := X_{α}^{A(t)} and ∥ · ∥_{α}^{t} := ∥ · ∥_{α}^{A(t)}. Assumption (P) implies that there is a unique evolution family (U(t, s))_{t≥s} in the space L(X) of bounded linear operators, i.e.,

U(t, s) = U(t, r)U(r, s), U(s, s) = I, (t, s) ↦ U(t, s) is strongly continuous

for t ≥ r ≥ s, t, r, s ∈ ℝ, such that U(·, s)x ∈ C^{1}([s, ∞), X) solves (CP) if x ∈ D(A(s)), see [1, Thm.2.3, Prop.3.2] and also [2], [3], [17], [18]. Moreover, U(t, s)X ⊂ D(A(t)) and

∥(w − A(t))^{α} U(t, s)∥ ≤ C′ (t − s)^{−α},  (2.2)
∥A(t)U(t, s)(w − A(s))^{−α}∥ ≤ C′ (t − s)^{α−1},  (2.3)
∥(w − A(t))^{α} U(t, s)(w − A(s))^{−α}∥ ≤ C′,  (2.4)
∥U(t, s)(w − A(s))^{α}x∥ ≤ \frac{C′}{μ − α} (t − s)^{−α} ∥x∥,  (2.5)

for 0 ≤ α ≤ 1 in (2.2)–(2.4), 0 ≤ α < μ and x ∈ D((w − A(s))^{α}) in (2.5), 0 < t − s ≤ 1, and a constant C′ ≥ 0 only depending on the constants in (P), see [18, Thm.2.1, (2.13)]. Thus U(·, ·) is exponentially bounded, i.e., there are constants M ≥ 1 and γ ∈ ℝ such that

∥U(t, s)∥ ≤ Me^{γ(t−s)} for t ≥ s.  (2.6)

By [1, Thm.2.3] and [18, Thm.2.3] the mappings

{(τ, σ) ∈ ℝ^{2} : τ > σ} ∋ (t, s) ↦ U(t, s) ∈ L(X)  (2.7)
{(τ, σ) ∈ ℝ^{2} : τ ≥ σ} ∋ (t, s) ↦ (w − A(t))^{α} U(t, s)(w − A(s))^{−α}x ∈ X, x ∈ X, (2.8)

are continuous for 0 ≤ α ≤ 1. Using the estimates (2.5) and ∥A_{w}(s)^{θ−1}x∥ ≤ c_{θ,β} ∥x∥_{β−1}^{s} for 0 ≤ θ < β ≤ 1, we can uniquely extend U(t, s), t > s, to a bounded operator Ũ(t, s) : X_{β−1}^{s} → X for 1 > β > 1 − μ. Finally we note that (P) implies

∥(w − A(t))^{α−1} − (w − A(s))^{α−1}∥ ≤ L_{α} |t − s|^{μ}  (2.9)

for t, s ∈ ℝ, 0 ≤ α < ν, and a constant L_{α} depending only on α and the constants in (P).

Let E := C_{0}(ℝ, X) be endowed with the sup–norm. For a given evolution family U(·, ·) on X satisfying (2.6), we define the evolution semigroup T (·) on E by

(T (t)f) (s) := U(s, s − t)f(s − t),  s ∈ ℝ, t ≥ 0, f ∈ E,  (2.10)

compare [5], [8, §VI.9], [11], [12], [15], [16], and the references therein. Clearly, T (·) is a strongly continuous semigroup on E. Its generator is denoted by G. It is easy to see that the resolvent of G is given by

(R(λ, G)f) (t) = ∫_{−∞}^{t} e^{−λ(t−s)}U(t, s)f(s) ds, t ∈ ℝ,  (2.11)

for Re λ > γ. To obtain a representation of G, we introduce the space C_{0}^{1}(ℝ, X) := {f ∈ C^{1}(ℝ, X) : f, f′ ∈ E} and the multiplication operator A(·) on E with domain D(A(·)) := {f ∈ E : f(s) ∈ D(A(s)) : A(·)f(·) ∈ E}. Notice that A(·) is sectorial of type (ϕ, K, w) with resolvent R(λ, A(·)) for λ ∈ Σ ∪ {0}.

Lemma 2.1. Let (P) hold. Then G extends the operator −\frac{d}{dt} +A(·) defined on C_{0}^{1}(ℝ, X)∩ D(A(·)). Moreover, D(A(·)) ∩ D(G) ⊂ C_{0}^{1}(ℝ, X).

Proof. For f ∈ C_{0}^{1}(ℝ, X) ∩ D(A(·)), t > 0, and s ∈ ℝ, we compute

\frac{1}{t} (T (t)f(s) − f(s)) + f′(s) − A(s)f(s)

= \frac{1}{t} (U(s, s − t)f(s − t) − f(s)) + \frac{1}{t} ∫_{s−t}^{s} U(s, τ)f′(τ) dτ − A(s)f(s)

− \frac{1}{t} ∫_{s−t}^{s}(U(s, τ)f′(τ) − f′(s)) dτ

= \frac{1}{t} ∫_{s−t}^{s} (U(s, τ)A(τ)f(τ) − A(s)f(s) − U(s, τ)f′(τ) + f′(s))dτ,

where we have integrated by parts and used \frac{∂^{+}}{∂s} U(t, s)x = −U(t, s)A(s)x for t > s and x ∈ A(s), [1, Thm.2.3]. This shows the first assertion. The second one can be established exactly as [16, Lem.3.6] using (2.4) and (2.8).

We further need the inter- and extrapolation spaces E_{α} := E_{α}^{A(·)} and E_{α−1} := E_{α−1}^{A(·)} , 0 ≤ α ≤ 1, of the operator A(·) on E. By [9, Thm.4.7], the extrapolated operator A(·)_{−1} : E → E_{−1} is given by the multiplication operator A_{−1}(·) and f ∈ E_{−1} can be identified with an element of ∏_{t∈ℝ} X_{−1}^{t}. Note that E_{α−1} is the closure of E_{1} in E_{−1} with respect to the norm

∥f∥_{α−1} := sup_{r>w} sup_{s∈ℝ}∥r^{α}R(r, A(s))f(s)∥,

A_{α−1}(·) : E_{α} → E_{α−1} is an isomorphism for 0 ≤ α ≤ 1, and E_{β} ↪ D((w − A(·))^{α}) ↪ E_{α} for 0 ≤ α < β ≤ 1. Due to (2.5) and (2.11) we can extend R(λ, G) to a bounded operator R̃_{λ} : E_{α−1} → E, 1 − μ < α < 1, λ > γ, given by

(R̃_{λ}f) (t) = ∫_{−∞}^{t} e^{−λ(t−s)}Ũ(t, s)f(s) ds, t ∈ ℝ, f ∈ E_{α−1}.

The following lemma will be crucial for the proof of our main result.

Lemma 2.2. Let (P) hold, λ > γ, and f ∈ C_{0}^{1}(ℝ, X) ∩ D((w − A(·))^{α}) for α ∈ (1 − μ, ν). Then we have R(λ, G)f′ = f − λR(λ, G)f + R̃_{λ}A_{α−1}(·)f.

Proof. By rescaling we may assume that λ = 0 > γ and w = 0. First, for s ∈ ℝ and 0 < h ≤ 1, we write

D_{h,f}(s) := (−A(s + h))^{α−1} \frac{1}{h} [U(s + h, s) − I]f(s) + (−A(s))^{α}f(s)

= \frac{1}{h} ∫_{0}^{h} [(−A(s + h))^{α−1} − (−A(s + τ))^{α−1}] A(s + τ)U(s + τ, s)f(s) dτ

− \frac{1}{h} ∫_{0}^{h} [(−A(s + τ))^{α} U(s + τ, s)(−A(s))^{−α} − I] (−A(s))^{α}f(s) dτ.

Using (2.9), (2.3) for the first integral and (2.4), (2.8) for the second one, we derive

lim_{h→0} ∥D_{h,f}∥_{E} = 0.  (2.12)

Next, we compute

∫_{−∞}^{t} U(t, s)f′(s) ds = lim_{h→0} ∫_{−∞}^{t−h} U(t, s + h) \frac{1}{h}(f(s + h) − f(s)) ds

= lim_{h→0} \frac{1}{h} ∫_{−∞}^{t−h}(U(t, s + h)f(s + h) − U(t, s)f(s)) ds

+ lim_{h→0} \frac{1}{h} ∫_{−∞}^{t−h} U(t, s + h)[U(s + h, s) − I]f(s) ds

= f(t) + lim_{h→0} \frac{1}{h} ∫_{−∞}^{t−h} U(t, s + h)[U(s + h, s) − I]f(s) ds.

To determine the remaining limit, we note that U(t, s)(−A(s))^{1−α}, t > s, has a unique bounded extension V (t, s) : X → X satisfying

∥V (t, s)∥_{L(X)} ≤ C′ (μ + α − 1)^{−1} (t − s)^{α−1}  (2.13)

for 0 < t − s ≤ 1 by (2.5). Hence,

\frac{1}{h} ∫_{−∞}^{t−h} U(t, s + h)[U(s + h, s) − I]f(s) ds − ∫_{−∞}^{t} Ũ(t, s)A_{α−1}(s)f(s) ds
= ∫_{−∞}^{t−h} V (t, s + h) \{(−A(s + h))^{α−1} \frac{1}{h} [U(s + h, s) − I]f(s) + (−A(s))^{α}f(s)\} ds
− ∫_{−∞}^{t−h} V (t, s + h) [(−A(s))^{α}f(s) − (−A(s + h))^{α}f(s + h)] ds.

As consequence of (2.13), (2.12), and (−A(·))^{α}f(·) ∈ E, the right hand side tends to 0 in X, and the assertion follows.

We say that an evolution family U(·, ·) has exponential dichotomy if there are projections P (t) ∈ L(X), t ∈ ℝ, and constants N, δ > 0 such that P (·) is strongly continuous and (a) P (t)U(t, s) = U(t, s)P (s) =: U_{P} (t, s) for t ≥ s,

(b) the restriction U_{Q}(t, s) : Q(s)X → Q(t)X is invertible for t ≥ s (and we set U_{Q}(s, t) := U_{Q}(t, s)^{−1}),

(c) ∥U_{P} (t, s)P (s)∥ ≤ Ne^{−δ(t−s)} and ∥U_{Q}(s, t)Q(t)∥ ≤ Ne^{−δ(t−s)} for t ≥ s. Here and below, we let Q = I − P for a projection P. We remark that the dichotomy projections P (t) are uniquely determined, see [16, Cor.3.3]. The Green’s function of an exponentially dichotomic evolution family is given by

Γ(t, s) := \{ \begin{matrix}U_{P} (t, s)P (s), & t ≥ s, \\ −U_{Q}(t, s)Q(s), & t < s. \end{matrix}

Now we can formulate our second main assumption supposing that (P) is valid. (ED) Each semigroup (e^{τA(t)})_{τ≥0}, t ∈ ℝ, has exponential dichotomy with constants N, δ > 0

and projection P_{t}. Observe that (ED) implies ∥A(t)^{−1}∥ ≤ \frac{2N}{δ}. It is not difficult to show that (ED) holds if [−δ, δ]+iℝ ⊂ ρ(A(t)) and R(λ, A(t)) is uniformly bounded for λ ∈ ±δ +iℝ, cf. [8, §V.1.c], [14, §2.3]. Assuming (P) and (ED), we set

Γ_{t}(s) := \{ \begin{matrix}e^{sA_{P} (t)}P_{t}, & s ≥ 0, t ∈ ℝ, \\ −e^{sA_{Q}(t)}Q_{t}, & s < 0, t ∈ ℝ, \end{matrix}

where A_{P} (t) and A_{Q}(t) denote the restrictions of A(t) to P_{t}X and Q_{t}X, respectively. One sees as in [16, Lem.3.5] that the mappings

t ↦ P_{t} ∈ L(X) and (t, s) ↦ Γ_{t}(s) ∈ L(X) are continuous  (2.14)

for t ∈ ℝ and s ∈ ℝ \ {0}. Clearly, P_{t} and Γ_{t}(s) leave X_{α}^{t} invariant and have unique extensions to X_{α−1}^{t}, 0 ≤ α ≤ 1, t ∈ ℝ. The following estimate easily follows from (2.1) (where the second integral in the first line is understood as an upper integral).

Lemma 2.3. Assume that (P) and (ED) hold. Then

q := sup_{t∈ℝ}max {∫_{ℝ} ∥A(t)^{2}Γ_{t}(τ)∥_{L(X_{ν}^{t},X)} |τ|^{μ} dτ, ∫_{ℝ} ∥A(s)Γ_{s}(t − s)∥_{L(X_{ν−1}^{s},X)} |t − s|^{μ} ds}

≤ NC (∫_{0}^{1} τ^{ν+μ−2} dτ + ∫_{0}^{∞}(|τ + 1|^{μ} + e^{−δ} |τ|^{μ}) e^{−δτ} dτ) < ∞.

3. Main results

We start with an auxiliary result which is an interesting refinement of the following fact: An exponentially bounded evolution family has exponential dichotomy if and only if the generator G of the associated evolution semigroup T (·) on E = C_{0}(ℝ, X) is bijective. In this case the unit circle 𝕋 belongs to ρ(T (1)),

P (·) = \frac{1}{2πi} ∫_{𝕋} R(λ, T (1)) dλ, and G^{−1}f = − ∫_{ℝ} Γ(·, s)f(s) ds,  (3.1)

see e.g. [11], [12], [15], and also [5], [8, §VI.9] for further references.

Lemma 3.1. Let U(·, ·) be an evolution family on a Banach space X with ∥U(t, s)∥ ≤ Me^{γ(t−s)} and let G be the generator of the associated evolution semigroup on E = C_{0}(ℝ, X). If G is bijective and ∥G^{−1}∥ ≤ 1/η, then U(·, ·) has exponential dichotomy and the dichotomy constants depend only on η > 0.

Proof. From σ(G) = σ(G)+iℝ, see [11, Thm.3.1], follows that (−η, η)+iℝ ⊂ ρ(G). Using rescaling and the above cited results, we see that U(·, ·) has exponential dichotomy with exponent δ′ ∈ (0, η).

Fix δ′ ∈ (0, η). Suppose the dichotomy constant N′ were not uniform in η. Then there would exist evolution families U_{n}(·, ·) on Banach spaces X_{n} such that ∥U_{n}(t, s)∥ ≤ Me^{γ(t−s)} and ∥G_{n}^{−1}∥ ≤ 1/η for the generator G_{n} of the associated evolution semigroup on C_{0}(ℝ, X_{n}), but there are x_{n} ∈ X_{n} and t_{n}, s_{n} ∈ ℝ with ∥x_{n}∥ = 1 and

lim_{n→∞} e^{δ′|t_{n}−s_{n}|} ∥Γ_{n}(t_{n}, s_{n})x_{n}∥ = ∞,

where Γ_{n}(·, ·) is the Green’s function of U_{n}(·, ·). Set f_{n}(s) := φ_{n}(s)e^{±δ′(s−s_{n})} U_{n}(s, s_{n})x_{n} for s ≥ s_{n} and f_{n}(s) := 0 for s < s_{n}, where φ_{n} ∈ C(ℝ) has compact support in (s_{n}, s_{n} +2), 0 ≤ φ_{n} ≤ 1, and ∫ φ_{n}(s) ds = 1, and we take +δ′ if t_{n} ≥ s_{n} and −δ′ if t_{n} < s_{n}. As a result,

R(∓δ′, G_{n})f_{n}(t_{n}) = ∫_{ℝ} e^{±δ′(t_{n}−s)}Γ_{n}(t_{n}, s)f_{n}(s) ds = e^{δ′|t_{n}−s_{n}|} Γ_{n}(t_{n}, s_{n})x_{n}.

This contradicts the uniform estimate of G_{n}^{−1}.

Theorem 3.2. Assume that (P) and (ED) hold and let q be given by Lemma 2.3. If qL < 1, then the evolution family (U(t, s))_{t≥s} solving (CP) has exponential dichotomy with constants N′, δ′ > 0 depending only on q and the constants in (P) and (ED).

Proof. (a) If Gf = 0 for some f ∈ D(G), then f(t) = (T (1)f)(t) = U(t, t − 1)f(t − 1) for t ∈ ℝ. Thus (2.2) and [1, Thm.2.3] yield f ∈ D(A(·)) so that f ∈ C_{0}^{1}(ℝ, X) by Lemma 2.1. We define on E a bounded operator L by

(Lg)(t) := ∫_{−∞}^{∞} Γ_{t}(t − s)g(s) ds, t ∈ ℝ.

Using Lemma 2.1 and integration by parts, we obtain

0 = LGf = −Lf′ + LA(·)f = LA(·)f − A_{−1}(·)Lf − f =: V f − f,

where we have set

(V f)(t) := ∫_{ℝ} Γ_{t}(t − s)(A(s) − A_{−1}(t))f(s) ds
= ∫_{ℝ} A(t)Γ_{t}(t − s)(A(t)^{−1} − A(s)^{−1})A(s)f(s) ds.

Here the integral in the first line is understood in the topology of X_{−1}^{t}. Then (P) and Lemma 2.3 imply ∥A(t) V f(t)∥ ≤ qL ∥A(·)f∥_{E} for each t ∈ ℝ. Since V f = f and A(·) is invertible, this estimate gives f = 0. Hence G is injective.

(b) Define on E a bounded operator R by

(Rf)(t) := ∫_{−∞}^{∞} Γ_{s}(t − s)f(s) ds.

Notice that due to (2.1) R extends to an operator R̃ : E_{α−1} → E for α ∈ (0, 1) with norm

∥R̃∥_{L(E_{α−1},E)} ≤ NC (\frac{1}{α} + \frac{1 + e^{−δ}}{δ} ) =: ρ_{α}.  (3.2)

For f ∈ D(A(·)), one easily sees that Rf ∈ C_{0}^{1}(ℝ, X) and \frac{d}{dt}Rf = f + RA(·)f. We let

(Sf)(t) := ∫_{ℝ}(A_{−1}(t) − A(s))Γ_{s}(t − s)f(s) ds
= A_{−1}(t) ∫_{ℝ}(A(s)^{−1} − A(t)^{−1})Γ_{s}(t − s)A(s)f(s) ds,

where first integral is understood in the topology of X_{−1}^{t}. This leads to

G̃Rf := −\frac{d}{dt} Rf + A_{−1}(·)Rf = (S − I)f.  (3.3)

We further need the Banach space F_{α−1} := {f ∈ E_{−1} : ∥f∥_{α−1} < ∞} endowed with the norm ∥ · ∥_{α−1} for 0 < α < 1. We note that E is in general not dense in F_{α−1} and that F_{ν−1} ↪ E_{α−1} for 0 ≤ α < ν by [8, Prop.II.5.14]. Using (P) and Lemma 2.3, we can extend S to a bounded operator S̃ : F_{ν−1} → F_{ν−1} with norm less than qL. Fix α ∈ (1 − μ, ν). Since (P) holds with ν replaced by α, S is also continuous with respect to ∥ · ∥_{α−1}.

For a given g ∈ E, we have f := (S̃ − I)^{−1}g ∈ F_{ν−1} ⊂ E_{α−1}. There are f_{n} ∈ D(A(·)) converging to f in ∥ · ∥_{α−1}. From (3.3) we derive

G̃Rf_{n} = (S − I)f_{n} ⟶ g in ∥ · ∥_{α−1} as n → ∞.

Moreover, A_{−1}(·)Rf_{n} ∈ E_{α−1} so that Rf_{n} ∈ E_{α} ⊂ D((w − A(·))^{β}) for 1 − μ < β < α < ν. Then Lemma 2.2 yields

R̃_{λ}G̃Rf_{n} = λR(λ, G)Rf_{n} − Rf_{n}

from which we deduce

R̃_{λ}g = R(λ, G)g = λR(λ, G)R̃f − R̃f.

Therefore R̃f ∈ D(G) and GR̃f = g; i.e., G is also surjective.

Considering (S̃ − I)^{−1} as an operator from E to F_{ν−1}, we obtain G^{−1} = R̃(S̃ − I)^{−1} and ∥G^{−1}∥ ≤ \frac{cρ}{1−qL} where ρ := ρ_{ν} is given by (3.2) and c is the norm of the embedding of E in F_{ν−1}. So Lemma 3.1 yields the assertion.

We further prove that the dimensions of the stable and unstable subspaces of U(t, s) and e^{τA(s)} coincide, cf. [6], [13].

Proposition 3.3. Under the assumptions of Theorem 3.2 we have dim P_{s}X = dim P (t)X and dim ker P_{s} = dim ker P (t) for t, s ∈ ℝ.

Proof. (a) By [10, p.298] the dimensions of the kernel and the range of two projections P and P′ coincide whenever ∥P − P′∥ < 1. So (2.14) shows that the dimensions of ker P_{t} and P_{t}X do not depend on t.

(b) We set A_{ε}(t) := A(εt) for 0 < ε ≤ 1 and t ∈ ℝ, and A_{0}(t) = A(0) for t ∈ ℝ. Observe that A_{ε}(·) satisfies (P) and (ED) with the same constants and the same q. So the corresponding evolution families U_{ε}(·, ·) have a uniform exponential bound and exponential dichotomy with uniform constants δ′, N′ > 0 and projections P_{ε}(t), where P_{1}(t) = P (t) and P_{0}(t) = P_{0} for t ∈ ℝ. Due to (a) it remains to show that ε ↦ P_{ε}(t) ∈ L(X) is continuous for each t ∈ ℝ. Clearly, U_{ε}(t, s) = Û_{ε}(εt, εs), where Û_{ε}(t, s) solves the Cauchy problem corresponding to the operators \frac{1}{ε}A(t), 0 < ε ≤ 1. We note that \frac{1}{ε}A(t) fulfills (P) with the same constants except for w replaced by w/ε. For 0 < ε, η ≤ 1 we compute

U_{ε}(s, s − 1) − U_{η}(s, s − 1) = Û_{ε}(εs, ε(s − 1)) − Û_{ε}(ηs, η(s − 1))
+ ∫_{η(s−1)}^{ηs} Û_{ε}(ηs, r)(\frac{1}{ε} − \frac{1}{η})A(r)Û_{η}(r, η(s − 1)) dr

(where the existence of the integral is verified in the next step). The first summand tends to 0 in L(X) uniformly for s in compact sets as η → ε by (2.7). The second summand is equal to

\frac{ε − η}{ε^{1−α}η^{α}} ∫_{η(s−1)}^{ηs} Û_{ε}(ηs, r) ε^{−α}(w − A(r))^{α} η^{α−1}(w − A(r))^{1−α} Û_{η}(r, η(s − 1)) dr

+ w (\frac{1}{ε} − \frac{1}{η}) ∫_{η(s−1)}^{ηs} Û_{ε}(ηs, r)Û_{η}(r, η(s − 1)) dr,

where 0 < α < μ. Using (2.5) and (2.2), we see that this expression tends to 0 in operator norm uniformly in s ∈ ℝ as η → ε. In the case η → 0, we write

U_{η}(s, s − 1) − e^{A(0)} = Û_{η}(ηs, η(s − 1)) − e^{η \frac{1}{η} A(η(s−1))} + e^{A(η(s−1))} − e^{A(0)}.

The right hand side converges to 0 in L(X) uniformly for s in compact subsets due to [1, (2.6), Lem.2.2] and (P).

(c) Let T_{ε}(·) be the evolution semigroup on E = C_{0}(ℝ, X) associated with U_{ε}(·, ·). Notice that

R(λ, T_{ε}(1))f = ∑_{n=0}^{∞} λ^{−(n+1)}T_{ε}(n)P_{ε}(·)f − ∑_{n=1}^{∞} λ^{n−1}(T_{ε}(n)Q_{ε}(·))^{−1}f

= ∑_{n=0}^{∞} λ^{−(n+1)}U_{ε,P} (·, · − n)f(· − n) − ∑_{n=1}^{∞} λ^{n−1}U_{ε,Q}(·, · + n)f(· + n)

for f ∈ E. In particular, ∥R(λ, T_{ε}(1))∥ ≤ \frac{2N′}{δ′} =: c for each |λ| = 1, 0 ≤ ε ≤ 1. Given t ∈ ℝ and x ∈ X, we set f = φ(·)x for a function φ ∈ C(ℝ) with support in (t − 1, t + 1), φ(t) = 1, and 0 ≤ φ ≤ 1. Then (3.1) and (2.6) yield

∥P_{η}(t)x − P_{ε}(t)x∥ ≤ ∥P_{η}(·)f − P_{ε}(·)f∥_{∞} ≤ c sup_{|λ|=1} ∥(T_{η}(1) − T_{ε}(1))R(λ, T_{ε}(1))f∥_{∞}

≤ c max{ sup_{|s|≤r,|λ|=1} ∥U_{ε}(s, s − 1) − U_{η}(s, s − 1)∥ ∥R(λ, T_{ε}(1))f(s − 1)∥,

sup_{|s|≥r,|λ|=1} 2Me^{γ}∥R(λ, T_{ε}(1))f(s − 1)∥}

≤ max{c^{2} sup_{|s|≤r} ∥U_{ε}(s, s − 1) − U_{η}(s, s − 1)∥, 4cMN′e^{γ} ∑_{n>N(r)} e^{−δ′n}}∥x∥

for each r > max{0, 1 − t, t + 1}, where N(r) is the maximum of the integer parts of r − t − 1 and r + t − 1. Now part (b) easily implies the assertion.

Remark 3.4. Theorem 3.2 and Proposition 3.3 can be generalized to operators A(t) satisfying (P) and (ED) for t ≥ t_{0} by setting A(t) := A(t_{0}) for t < t_{0}, cf. [16, §3].

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Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

E-mail address: rosc@michelangelo.mathematik.uni-tuebingen.de