Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

Nguyen Van Minh^{∗}, Frank Räbiger and Roland Schnaubelt^{∗∗} Department of Mathematics University of Tübingen D-72076-Tübingen, Germany

Abstract

For an evolution family on the half-line U = (U(t, s))_{t≥s≥0} of bounded linear operators on a Banach space X we introduce operators G_{0}, G_{X} and I_{X} on certain spaces of Xvalued continuous functions connected with the integral equation u(t) = U(t, s)u(s) + ∫_{s}^{t} U(t, ξ)f(ξ)dξ. We characterize exponential stability, exponential expansiveness and exponential dichotomy of U by properties of G_{0}, G_{X} and I_{X}, respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

1991 Mathematics Subject Classification: Primary: 34G10, 47D06, Secondary: 47H20 Keywords and Phrases: evolution equation, evolution family, evolution semigroup, exponential stability, exponential expansiveness, exponential dichotomy, integral equation.

Introduction

We consider the non-autonomous linear evolution equation

\frac{d}{dt}u(t) = A(t)u(t), t ∈ J = ℝ_{+} or ℝ,  (NCP)

∗ This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement. Permanent address: Dept. of Mathematics, University of Hanoi, 90 Nguyen Trai, Hanoi, Vietnam. ∗∗ Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged. on a Banach space X. If the operators A(t), t ∈ J, are bounded or if X is finite dimensional there is an extensive literature initiated by the work of Perron which connects asymptotic properties (of the solutions) of (NCP) with specific properties of the operator L defined by

Lu(t) = \frac{d}{dt}u(t) − A(t)u(t), t ∈ J,

on a space of X-valued functions. For example, the existence of an exponential dichotomy of (NCP) is related to the invertibility of L if J = ℝ and to Fredholm properties of L if J = ℝ_{+} (see [BeG], [BGK], [Cop], [DaK], [MaS], [Pal] and the references therein). If the operators A(t), t ∈ J, are unbounded the situation becomes more delicate and usually one has to impose rather restrictive conditions in order to obtain similar results referring to the operator L (e.g. see [Hen], [Zha]).

A less restrictive assumption is the well-posedness of (NCP) in the sense that the solutions of (NCP) yield an evolution family U = (U(t, s))_{t≥s, t,s∈J} of bounded linear operators on X (see [Nic], [Paz], [RRS], [RS2], [Sch] and [Tan] for sufficient conditions and examples). Note that solution does not necessarily mean ‘classical’ solution. Then instead of the operator L one investigates the integral equation

u(t) = U(t, s)u(s) + ∫_{s}^{t} U(t, ξ)f(ξ)dξ, t ≥ s, t, s ∈ J,  (IE)

and its connection with asymptotic properties of the evolution family U (see [Bus], [DaK], [Dat], [LRS], [LeZ], [Zhi]). For instance, if J = ℝ then it is shown in [LRS] that U has an exponential dichotomy if and only if for every f ∈ C^{b}(ℝ, X) there is a unique solution u ∈ C^{b}(ℝ, X) of (IE).

Another approach uses the so-called evolution semigroup T = (T (t))_{t≥0} on a space of X-valued functions induced by the evolution family U (see [AuM], [LaM], [LaR], [LMR1], [LMR2], [Mi1], [Mi2], [Nic], [RRS], [RS1], [RS2], [Rau], [Sch] and the references therein for this concept). Roughly speaking (if J = ℝ) the evolution semigroup T is given by

T (t)f(ξ) = U(ξ, ξ − t)f(ξ − t), t ≥ 0,

(see (1.1) for the exact definition in the present context) and in certain situations its generator G is an extension of −L = −\frac{d}{dt} + A(·) (see [LMR1, Prop. 2.9], [Sch, Prop. 1.13]). It turns out that in the spirit of Perron’s observations asymptotic properties of U like exponential stability or exponential dichotomy can be described by spectral properties of the operator G (see [AuM], [LaM], [LaR], [LMR1], [LMR2], [Mi1], [Mi2], [RS1], [Rau], [Sch]). However, we have to point out that most results in this direction are restricted to the line case J = ℝ.

In the present paper we characterize exponential stability, exponential expansiveness and exponential dichotomy of an evolution family U on the half-line J = ℝ_{+}. Our approach is based on the use of (generators of) evolution semigroups and their connection with the integral equation (IE) (see Section 1). Exponential stability and exponential expansiveness of U is characterized in Theorem 2.2 and Theorem 2.5, respectively. Section 3 deals with exponential stability of individual orbits. Our main interest, however, is directed to the exponential dichotomy of U which will be characterized in Theorem 4.3 and Theorem 4.5. Concerning the relevance of exponential dichotomy and its far-reaching applications we refer to [SaS] and the references therein.

1. Preliminaries

Recall that a family of operators U = (U(t, s))_{t≥s≥0} of bounded linear operators on a Banach space X is a (strongly continuous, exponentially bounded) evolution family on the half-line if

1. U(t, t) = Id and U(t, r)U(r, s) = U(t, s) for t ≥ r ≥ s ≥ 0,

2. (t, s) ↦ U(t, s)x is continuous for every x ∈ X,

3. there are constants K ≥ 0 and α ∈ ℝ such that ∥U(t, s)∥ ≤ Ke^{α(t−s)} for t ≥ s ≥ 0.

Then ω(U) := inf{α ∈ ℝ : there is K ≥ 0 such that ∥U(t, s)∥ ≤ Ke^{α(t−s)}, t ≥ s ≥ 0} is called the growth bound of U. Throughout the whole paper the following function spaces (endowed with the supnorm) play an important role:

C_{0} := {v : [0, ∞) → X : v is continuous, v(0) = 0 = lim_{t→∞} v(t)},
C_{X} := {v : [0, ∞) → X : v is continuous, lim_{t→∞} v(t) = 0},
C_{X}(t_{0}) := {v|_{[t_{0},∞)} : v ∈ C_{X}}, t_{0} ≥ 0,
C^{b} := {v : [0, ∞) → X : v is continuous and bounded}.

An evolution family U on X always defines an evolution semigroup T = (T (t))_{t≥0} of bounded linear operators on C_{0} and C_{X} by setting

[T (t)v](s) = \{ \begin{matrix}U(s, s − t)v(s − t), & s ≥ t, \\ U(s, 0)v(0), & 0 ≤ s ≤ t, \end{matrix}  (1.1)

for v in C_{0} and C_{X}, respectively. It can be easily seen that T is strongly continuous. We denote the infinitesimal generator of T on C_{0} and C_{X} by G_{0} and G_{X}, respectively. Note that D(G_{0}) = D(G_{X}) ∩ C_{0} and G_{0}v = G_{X}v for v ∈ D(G_{0}), i.e. G_{0} is the part of G_{X} in C_{0}.

The aim of this paper is to characterize asymptotic properties of a given evolution family U by (spectral) properties of the generators G_{0} and G_{X}. The following lemma is the key tool in our strategy. It connects the operators G_{0} and G_{X} with the following inhomogeneous integral equation

u(t) = U(t, s)u(s) + ∫_{s}^{t} U(t, ξ)f(ξ)dξ, t ≥ s ≥ 0.  (1.2)

1.1 Lemma. a) Let u, f ∈ C_{0}. Then u ∈ D(G_{0}) and G_{0}u = −f if and only if

u(t) = ∫_{0}^{t} U(t, ξ)f(ξ)dξ, t ≥ 0.  (1.3)

b) Let u ∈ C_{X} and f ∈ C_{0}. Then u ∈ D(G_{X}) and G_{X}u = −f if and only if (1.2) holds.

Proof. We only prove the first assertion. The proof of the second one follows the same lines. Let G_{0}u = −f. The general theory of linear semigroups (see e.g. [Paz, p.4-5]) yields

T (t)u − u = ∫_{0}^{t} T (ξ)G_{0}udξ = − ∫_{0}^{t} T (ξ)fdξ for t ≥ 0.

Thus,

u = T (t)u + ∫_{0}^{t} T (ξ)fdξ.

From the definition of T we easily obtain that u is a solution of the integral equation (1.2) and has the form (1.3). Conversely, if u, f ∈ C_{0} and u satisfies (1.3), then by reversing the above argument we obtain

T (t)u − u = − ∫_{0}^{t} T (ξ)fdξ for t ≥ 0.

In particular, this implies u ∈ D(G_{0}) and G_{0}u = −f. □

In the following remark we collect some additional properties of the operators G_{0} and G_{X}. For sake of convenience we set U(t, s) = 0 for 0 ≤ t < s.

1.2 Remarks. a) From Lemma 1.1 we immediately obtain that

• G_{0} is injective
• ker G_{X} = {u ∈ C_{X} : u(t) = U(t, 0)u(0), t ≥ 0}.

b) The range R(G_{X}) of G_{X} is always contained in C_{0} (in particular G_{X} is never invertible). In fact, if u ∈ D(G_{X}) then

[G_{X}u](0) = lim_{t↓0} \frac{T (t)u(0) − u(0)}{t} = lim_{t↓0} \frac{u(0) − u(0)}{t} = 0.

c) Let t_{0} ≥ 0, x ∈ X and φ : [0, ∞) → ℝ be continuously differentiable such that φ_{|[0,t_{0})} = 0. Set u(t) = φ(t)U(t, t_{0})x, t ≥ 0, and f(t) = φ′(t)U(t, t_{0})x, t ≥ 0. If u ∈ C_{X} and f ∈ C_{0} then an immediate application of Lemma 1.1 yields u ∈ D(G_{X}) and G_{X}u = −f.

Next we define an operator I_{X} on C_{X} connected with the integral equation (1.2). If u, f ∈ C_{X} satisfy (1.2) we set

I_{X}u := f,
D(I_{X}) := {u ∈ C_{X} : there is f ∈ C_{X} such that u and f satisfy (1.2)}.

1.3 Lemma. (I_{X}, D(I_{X})) is a well-defined closed linear operator on C_{X} and an extension of (−G_{X}, D(G_{X})).

Proof. Let I_{X}u = f and I_{X}u = g for u, f, g ∈ C_{X}. Then

∫_{s}^{t} U(t, ξ)(f(ξ) − g(ξ))dξ = 0 for t ≥ s ≥ 0.

By the continuity of the integrand we obtain 0 = U(t, t)(f(t)−g(t)) = f(t)−g(t), t ≥ 0, i.e. f = g. The closedness of I_{X} follows immediately from the definition of I_{X} by the integral equation (1.2). Finally, Lemma 1.1 shows that I_{X} is an extension of −G_{X}. □

1.4 Remarks. a) In general, −I_{X} is a proper extension of G_{X}, in particular, I_{X} can be surjective (cf. Theorem 4.3). Moreover, the spectra of −I_{X} and G_{X} are different. In fact, for λ ∈ ℂ with Reλ < −ω(U) the function u_{λ} : t ↦ e^{λt}U(t, 0)x, t ≥ 0, x ∈ X, belongs to C_{X} and I_{X}u_{λ} = λu_{λ}. Thus {λ ∈ ℂ : Reλ > ω(U)} is contained in the (point) spectrum of −I_{X} whereas the spectrum of G_{X} is contained in a left half-plane (see [Paz, Remark 1.5.4]). Note, however, that we always have ker I_{X} = ker G_{X} = {u ∈ C_{X} : u(t) = U(t, 0)u(0), t ≥ 0}.

b) From Lemma 1.1 it easily follows that G_{0} is the part of −I_{X} in C_{0}, i.e. D(G_{0}) = {u ∈ D(I_{X}) ∩ C_{0} : I_{X}u ∈ C_{0}} and G_{0}u = −I_{X}u for u ∈ D(G_{0}).

2. Exponential stability and exponential expansiveness

In this section we characterize exponentially stable and exponentially expansive evolution families U = (U(t, s))_{t≥s≥0} by means of the operators G_{0} and I_{X} and the integral equation (1.2), respectively. First we give the definition of exponential stability and exponential expansiveness of an evolution family.

2.1 Definition. The evolution family U on the Banach space X is said to be

a) exponentially stable if there are constants N, ν > 0 such that ∥U(t, s)∥ ≤ Ne^{−ν(t−s)} for all t ≥ s ≥ 0 (or, in other words, if ω(U) < 0),

b) exponentially expansive if U(t, s) is invertible and there are constants N, ν > 0 such that ∥U(t, s)x∥ ≥ Ne^{ν(t−s)}∥x∥ for x ∈ X and t ≥ s ≥ 0.

We have the following characterization of exponentially stable evolution families.

2.2 Theorem. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Banach space X. Then the following assertions are equivalent:

(i) U is exponentially stable. (ii) G_{0} is invertible. (iii) For every f ∈ C_{0} the function t ↦ u_{f}(t) = ∫_{0}^{t} U(t, ξ)f(ξ)dξ belongs to C_{0}. (iv) For every f ∈ C_{0} the function t ↦ u_{f}(t) = ∫_{0}^{t} U(t, ξ)f(ξ)dξ belongs to C^{b}.

From the proof of the theorem we separate the following lemma for later use.

2.3 Lemma. Let χ : [t_{0}, t_{1}) → (0, ∞) be a continuous function and let c > 0 and K, α ≥ 0 be constants such that χ(t) ≤ Ke^{α(t−t_{0})} and ∫_{t_{0}}^{t} χ(t)χ(ξ)^{−1}dξ ≤ c for t ∈ [t_{0}, t_{1}). Then χ(t) ≤ max(cK, K)e^{α+ \frac{1}{c}} e^{− \frac{1}{c} (t−t_{0})}.

Proof of Lemma 2.3. If t ∈ [t_{0}, t_{1}) and t ≤ t_{0}+1, then χ(t) ≤ Ke^{α} ≤ Ke^{α+ \frac{1}{c}} e^{− \frac{1}{c} (t−t_{0})}. Now let t ∈ [t_{0}, t_{1}) and t > t_{0} + 1. Set ϕ(s) := ∫_{t_{0}}^{s} χ(ξ)^{−1}dξ for s ∈ [t_{0}, t_{1}). By our assumption ϕ(s) ≤ cϕ′(s), and hence ϕ(s_{1}) ≥ ϕ(s_{0})e^{\frac{1}{c} (s_{1}−s_{0})} for t_{1} > s_{1} ≥ s_{0} > t_{0}. Thus

χ(t) = \frac{1}{ϕ′(t)} ≤ \frac{c}{ϕ(t_{0} + 1)}e^{− \frac{1}{c} (t−t_{0}−1)} ≤ cKe^{α+ \frac{1}{c}} e^{− \frac{1}{c} (t−t_{0})}. □

Proof of Theorem 2.2. The implications (i) ⇒ (iii) ⇒ (iv) are obvious. The equivalence of (ii) and (iii) follows from Lemma 1.1. (iv) ⇒ (i): By assumption B : C_{0} → C^{b} : f ↦ u_{f} is linear and everywhere defined. Moreover, B is closed, and hence B is bounded. Let c := ∥B∥. Now fix t_{0} ≥ 0 and 0 ≠ y ∈ X. Let t_{1} := sup{t ≥ t_{0} : U(t, t_{0})y ≠ 0}. Then U(t, t_{0})y ≠ 0 for t ∈ [t_{0}, t_{1}) and U(t, t_{0})y = 0 for t ≥ t_{1}. Set χ(t) := ∥U(t, t_{0})y∥, t ∈ [t_{0}, t_{1}). For n ∈ ℕ sufficiently large choose a real continuous function φ_{n} on [0, ∞) such that φ_{n} has compact support contained in (t_{0}, t_{1}), 0 ≤ φ_{n} ≤ 1 and

φ_{n}(t) = 1 for t ∈ [t_{0} + \frac{1}{n}, min\{n, t_{1} − \frac{1}{n}\}].

Let

f_{n}(t) = \{ \begin{matrix}φ_{n}(t)χ(t)^{−1}U(t, t_{0})y & for t ∈ [t_{0}, t_{1}), \\ 0 & for t ∈ [0, t_{0}] ∪ [t_{1}, ∞). \end{matrix}

Then f_{n} ∈ C_{0} and

∫_{t_{0}}^{t} \frac{φ_{n}(ξ)}{χ(ξ)}dξ χ(t) = ∫_{t_{0}}^{t} \frac{φ_{n}(ξ)}{χ(ξ)}dξ ∥U(t, t_{0})y∥ = ∥ ∫_{t_{0}}^{t} U(t, ξ)f_{n}(ξ)dξ∥ ≤ c∥f_{n}∥ = c

for t ∈ [t_{0}, t_{1}). By letting n → ∞ we obtain ∫_{t_{0}}^{t} χ(t)χ(ξ)^{−1}dξ ≤ c for t ∈ [t_{0}, t_{1}). Moreover, by the exponential boundedness of U we have χ(t) ≤ Ke^{α(t−t_{0})}∥y∥ on [t_{0}, t_{1}). An application of Lemma 2.3 yields

χ(t) ≤ max\{cK, K\}e^{α+ \frac{1}{c}} e^{− \frac{1}{c} (t−t_{0})}∥y∥

for t ∈ [t_{0}, t_{1}). Thus

∥U(t, t_{0})y∥ ≤ max\{cK, K\}e^{α+ \frac{1}{c}} e^{− \frac{1}{c} (t−t_{0})}∥y∥

for t ≥ t_{0}. Since the constants K, α and c are independent of t_{0} and y the exponential stability of U follows. □

Remark. The proof of (iv) ⇒ (i) follows ideas used in [DaK, Proof of Theorem IV.3.3]. By different methods the equivalence of (i) and (iv) has been shown by C. Buşe [Bus, Theorem 1] (see also [Dat, Theorem 8] for a related result) and the equivalence of (i), (ii) and (iii) is proved by Y. Latushkin, S. Montgomery-Smith and T. Randolph [LMR2, Theorem 2.2, Corollary 2.3]. Note that Theorem 2.2 also holds if we replace C_{0} and C^{b} by L^{p}(ℝ_{+}, X), 1 ≤ p < ∞, and consider the evolution semigroup T on L^{p}(ℝ_{+}, X) induced by U (see [Dat, Theorem 6] and [LMR2, Theorem 2.2, Corollary 2.3]).

An immediate consequence of Theorem 2.2 is the spectral mapping theorem for the evolution semigroup on C_{0}. Recall that σ(B) denotes the spectrum of a linear operator B and ρ(B) := ℂ\σ(B) is the resolvent set. Moreover, s(B) := sup{Reλ : λ ∈ σ(B)} is the spectral bound and r(B) := sup{|λ| : λ ∈ σ(B)} is the spectral radius of B.

2.4 Corollary. Let U be an evolution family on the Banach space X. Then the evolution semigroup T on C_{0} satisfies the spectral mapping theorem

e^{tσ(G_{0})} = σ(T (t)) \ {0}, t ≥ 0.

Furthermore, σ(G_{0}) = {λ ∈ ℂ : Reλ ≤ s(G_{0})} is a left half-plane and σ(T (t)) = {λ ∈ ℂ : |λ| ≤ r(T (t))}, t > 0, is a disc.

Proof. Let λ ∈ ρ(G_{0}) and μ ∈ ℂ such that Reμ ≥ Reλ. Note that G_{0} − λId is the generator of the evolution semigroup T_{λ} = (e^{−λt}T (t))_{t≥0} on C_{0} induced by the evolution family U_{λ} = (e^{−λ(t−s)}U(t, s))_{t≥s≥0}. Theorem 2.2 implies that U_{λ} and hence U_{μ} is exponentially stable and, as a consequence, that G_{0} − μId is invertible, i.e. μ ∈ ρ(G_{0}). Thus σ(G_{0}) = {λ ∈ ℂ : Reλ ≤ s(G_{0})} is a left half-plane. If Reλ > s(G_{0}), then again by Theorem 2.2 the evolution family U_{λ} is exponentially stable, and hence the induced evolution semigroup T_{λ} is exponentially stable as well. In particular r(e^{−λt}T (t)) < 1, i.e. r(T (t)) < e^{Reλ·t} for t > 0. Thus r(T (t)) ≤ e^{s(G_{0})·t}, t ≥ 0. Together with the spectral inclusion theorem e^{tσ(G_{0})} ⊆ σ(T (t)), t ≥ 0 (see [Paz, 2.2.3]), it follows that σ(T (t)), t > 0, is a disc and that the spectral mapping theorem holds. □

Remark. The same result (with a different proof) can be found in [LMR2, Theorem 1.2, Theorem 2.2, Corollary 2.3] and [Sch, Theorem 5.3]. The observation that on C_{0} the spectrum of T (t), t ≥ 0, is always a disc goes back to [Rau, Proposition 2].

We now come to the characterization of exponentially expansive evolution families.

2.5 Theorem. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Banach space X. Then the following assertions are equivalent:

(i) U is exponentially expansive. (ii) For every t_{0} ≥ 0 and every f ∈ C_{X}(t_{0}) there exists a unique u_{f} ∈ C_{X}(t_{0}) such

that

u_{f}(t) = U(t, s)u_{f}(s) + ∫_{s}^{t} U(t, ξ)f(ξ)dξ, t ≥ s ≥ t_{0}.  (2.1)

Remark. Condition (ii) implies that the operator I_{X} is invertible. However, Example 4.6 shows that the exponential expansiveness of U (and hence (ii)) is not equivalent to the invertibility of I_{X}. On the contrary, if U(t, s) is surjective for all t ≥ s ≥ 0, then from the proof of Theorem 2.5 it follows that U is exponentially expansive if and only if I_{X} is invertible.

As above we separate from the proof of the theorem the following lemma which is essentially contained in [DaK, Proof of Theorem IV.3.3].

2.6 Lemma. Let χ : [0, ∞) → (0, ∞) be a continuous function and let c > 0 and K, α > 0 be constants such that χ(τ) ≤ Ke^{α(τ−t)}χ(t), τ ≥ t ≥ 0, and ∫_{t}^{∞} χ(t)χ(τ)^{−1}dτ ≤ c, t ≥ 0. Then there exists N ≥ 0 only dependent on K, α and c such that χ(t) ≥ Ne^{\frac{1}{c} (t−s)}χ(s) for t ≥ s ≥ 0.

Proof. Let ϕ(t) := ∫_{t}^{∞} χ(τ)^{−1}dτ, t ≥ 0. By our assumption ϕ(t) ≤ −cϕ′(t). Thus ϕ(t) ≤ ϕ(s)e^{− \frac{1}{c} (t−s)} for t ≥ s ≥ 0. On the other hand the exponential estimate of χ yields

\begin{aligned}χ(t)ϕ(t) & = χ(t) ∫_{t}^{∞} \frac{1}{χ(τ)}dτ ≥ K^{−1} ∫_{t}^{∞} e^{−α(τ−t)}dτ \\ & ≥ K^{−1} ∑_{k=1}^{∞} ∫_{t+(k−1)}^{t+k} e^{−α(t+k−t)}dτ = K^{−1} ∑_{k=1}^{∞} e^{−αk} =: Ñ. \end{aligned}

Moreover, by our assumption, χ(s)ϕ(s) ≤ c, t ≥ s ≥ 0. Thus

χ(t) ≥ \frac{Ñ}{ϕ(t)} ≥ \frac{Ñ}{ϕ(s)}e^{\frac{1}{c} (t−s)} ≥ \frac{Ñ}{c} e^{\frac{1}{c} (t−s)}χ(s) for t ≥ s ≥ 0. □

Now we are in a position to give the proof of Theorem 2.5.

Proof of Theorem 2.5. (i) ⇒ (ii): Let U be exponentially expansive and t_{0} ≥ 0. If f ∈ C_{X}(t_{0}) then the function t ↦ u_{f}(t) := − ∫_{t}^{∞} U(ξ, t)^{−1}f(ξ)dξ belongs to C_{X}(t_{0}). It is a straightforward computation to show that u_{f} satisfies (2.1). If u ∈ C_{X}(t_{0}) satisfies (2.1) with f = 0, then u(t) = U(t, t_{0})u(t_{0}), t ≥ t_{0}. Thus ∥u(t)∥ ≥ Ne^{ν(t−t_{0})}∥u(t_{0})∥ for constants N, ν > 0. This yields u(t_{0}) = 0, and hence u = 0. (ii) ⇒ (i): A) The assumption implies that I_{X} is invertible. Since I_{X} is closed its inverse I_{X}^{−1} is bounded and c := ∥I_{X}^{−1}∥ > 0. B) Fix 0 ≠ x ∈ X. The unique solvability of (2.1) for f = 0 in C_{X} yields u(t) := U(t, 0)x ≠ 0 for t ≥ 0. For each n ∈ ℕ choose a real continuous function φ_{n} on [0, ∞) such that 0 ≤ φ_{n} ≤ 1, φ_{n} = 1 on [0, n] and φ_{n} = 0 on [n + 1, ∞). Let f_{n}(t) := −φ_{n}(t)∥u(t)∥^{−1}u(t) for t ≥ 0. An easy computation shows that

u_{n}(t) := ∫_{t}^{∞} \frac{φ_{n}(ξ)}{∥u(ξ)∥}dξ u(t), t ≥ 0,

and f_{n} solve (1.2), i.e. I_{X}u_{n} = f_{n}. Hence, ∥u_{n}∥ ≤ c∥f_{n}∥ = c. Letting n → ∞ we obtain

∫_{t}^{∞} \frac{1}{∥u(ξ)∥}dξ ≤ \frac{c}{∥u(t)∥}, t ≥ 0.

The exponential boundedness of U yields ∥u(τ)∥ ≤ Ke^{α(τ−t)}∥u(t)∥, τ ≥ t ≥ 0, for constants K, α > 0 independent of x. By Lemma 2.6 there is a constant N ≥ 0 independent of x such that ∥u(t)∥ ≥ Ne^{\frac{1}{c} (t−s)}∥u(s)∥ for t ≥ s ≥ 0. C) It remains to show that U(t, s) is surjective for t ≥ s ≥ 0. Fix s ≥ 0. Let y ∈ X and set v(t) := U(t, s)y, t ≥ s. We choose a real continuous function φ on [0, ∞) with compact support such that |φ(t)| ≤ 1, suppφ ⊂ (s, ∞) and

∫_{s}^{∞} φ(ξ)dξ = 1.

Now consider the function w defined by

w(t) := v(t) ∫_{t}^{∞} φ(ξ)dξ, t ≥ s.

Then w ∈ C_{X}(s) and w is a solution of equation (2.1) with f(t) := −φ(t)v(t), t ≥ s. Extend f continuously to [0, ∞) by setting f(t) = 0 on [0, s]. Then there is a solution z ∈ C_{X} of equation (2.1) on [0, ∞). By our assumption z|_{[s,∞)} = w . Thus

y = w(s) = z(s) = U(s, 0)z(0) + ∫_{0}^{s} U(s, ξ)f(ξ)dξ = U(s, 0)z(0).

This proves the surjectivity of U(s, 0) for s ≥ 0. Together with U(t, 0) = U(t, s)U(s, 0) this yields the surjectivity of U(t, s) for t ≥ s ≥ 0. □

3. Exponential stability of bounded orbits

In Theorem 2.2 we saw that exponential stability of an evolution family U = (U(t, s))_{t≥s≥0} is characterized by the invertibility of the generator G_{0} of the evolution semigroup T on C_{0}. We now show that a condition on the approximate point spectrum of G_{0} implies exponential stability of all bounded orbits U(t, t_{0})x, t ≥ t_{0} ≥ 0. Recall that for an operator B on a Banach space Y the approximate point spectrum Aσ(B) of B is the set of all complex numbers λ such that for every ϵ > 0 there exists y ∈ D(B) with ∥y∥ = 1 and ∥(λ − B)y∥ ≤ ϵ.

As the whole spectrum σ(G_{0}) (see Corollary 2.4) the approximate point spectrum Aσ(G_{0}) is invariant under translations along the imaginary axis.

3.1 Lemma. Aσ(G_{0}) = iμ + Aσ(G_{0}) for all μ ∈ ℝ.

Proof. Consider the multiplication operator M_{μ} on C_{0} defined by (M_{μ}v)(t) := e^{iμt}v(t), t ≥ 0, v ∈ C_{0}. Then M_{μ} is an isometry and

\begin{aligned}(M_{μ}T (t)M_{−μ}v)(s) & = \{ \begin{matrix}e^{iμs}U(s, s − t)e^{−iμ(s−t)}v(s − t), & s ≥ t ≥ 0, \\ 0, & 0 ≤ s ≤ t, \end{matrix} \\ & = e^{iμt}(T (t)v)(s), \end{aligned}

v ∈ C_{0}, s ≥ 0. Thus M_{μ}T (t)M_{−μ} = e^{iμt}T (t), t ≥ 0, and hence M_{μ}G_{0}M_{−μ} = iμ+G_{0}. From this the assertion immediately follows. □

We now come to the main result of this section. A special case of it has been shown in [AuM, Theorems 7 and 7’]. Our proof follows the techniques of [AuM].

3.2 Theorem. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Banach space X such that Aσ(G_{0}) ∩ iℝ ≠ iℝ. Then every bounded orbit of U is exponentially stable. Precisely, if sup_{t≥t_{0}} ∥U(t, t_{0})x∥ < ∞ for fixed x ∈ X and t_{0} ≥ 0, then there exist constants N, ν > 0 independent of x and t_{0} such that

∥U(t, t_{0})x∥ ≤ Ne^{−ν(t−s)}∥U(s, t_{0})x∥, t ≥ s ≥ t_{0}.

Proof. Let x ∈ X and t_{0} ≥ 0 such that sup_{t≥t_{0}} ∥U(t, t_{0})x∥ < ∞. A) lim_{t→∞} ∥U(t, t_{0})x∥ = 0: Choose a real continuous function φ on [0, ∞) with compact support such that φ(t_{0} + 1) = 1 and φ|_{[0,t_{0}]} = 0. Define

v(t) := \{ \begin{matrix}φ(t)U(t, t_{0})x & for t > t_{0}, \\ 0 & for 0 ≤ t ≤ t_{0}. \end{matrix}

Then v ∈ C_{0} and

∥T (t + h)v − T (t)v∥

= max{ sup_{s≥t+h} ∥U(s, s − t − h)v(s − t − h) − U(s, s − t)v(s − t)∥,

sup_{t+h≥s≥t} ∥U(s, s − t)v(s − t)∥}

for t ≥ 0 and h > 0. From

U(s, s − t)v(s − t) = \{ \begin{matrix}φ(s − t)U(s, t_{0})x & for s − t > t_{0}, \\ 0 & for 0 ≤ s − t ≤ t_{0}, \end{matrix}

φ(0) = 0 and the uniform boundedness of U(t, t_{0})x, t ≥ t_{0}, we obtain

lim_{h↓0} sup_{t≥0} sup_{t+h≥s≥t} ∥U(s, s − t)v(s − t)∥ = 0.

On the other hand

sup_{s≥t+h} ∥U(s, s − t − h)v(s − t − h) − U(s, s − t)v(s − t)∥

= sup_{s≥t+h} ∥φ(s − t − h)U(s, t_{0})x − φ(s − t)U(s, t_{0})x∥

≤ sup_{s≥t+h} |φ(s − t − h) − φ(s − t)| sup_{s≥t_{0}} ∥U(s, t_{0})x∥.

By the uniform continuity of φ we have

lim_{h↓0} sup_{t≥0} sup_{s≥t+h} ∥U(s, s − t − h)v(s − t − h) − U(s, s − t)v(s − t)∥ = 0.

Thus t ↦ T (t)v is a uniformly continuous C_{0}-valued function on [0, ∞). Our assumption on G_{0} and Lemma 3.1 imply Aσ(G_{0}) ∩ iℝ = ∅. An application of a result of Batty and Vũ [BaV, Theorem 1] yields lim_{t→∞} T (t)v = 0. In particular,

0 = lim_{t→∞} ∥(T (t)v)(t + t_{0} + 1)∥ = lim_{t→∞} ∥U(t + t_{0} + 1, t_{0} + 1)v(t_{0} + 1)∥

= lim_{t→∞} ∥U(t + t_{0} + 1, t_{0})x∥.

B) ∥U(t, t_{0})x∥ ≤ Ne^{−ν(t−t_{0})}∥x∥, t ≥ t_{0}: Without loss of generality we may assume ∥x∥ = 1. Since 0 ∉ Aσ(G_{0}) there is a constant ν > 0 such that

∥G_{0}v∥ ≥ ν∥v∥ for v ∈ D(G_{0}).  (3.1)

Let u(t) := U(t, t_{0})x, t ≥ t_{0}, and t_{1} := sup{t ≥ t_{0} : U(t, t_{0})x ≠ 0}. The exponential boundedness of U implies that there are constants K, α > 0 (independent of x and t_{0}) such that ∥u(t)∥ ≤ Ke^{α(t−t_{0})}, t ≥ t_{0}. For n ∈ ℕ sufficiently large we choose a real continuous function ψ_{n} on [0, ∞) such that ψ_{n} has compact support contained in (t_{0}, t_{1}), 0 ≤ ψ_{n} ≤ 1 and

ψ_{n}(t) = 1 for t ∈ [t_{0} + \frac{1}{n}, min\{n, t_{1} − \frac{1}{n}\}].

Let

φ_{n}(t) := \{ \begin{matrix}0 & for t ∈ [0, t_{0}], \\ ∫_{t_{0}}^{t} ψ_{n}(ξ)∥u(ξ)∥^{−1}dξ & for t ∈ (t_{0}, t_{1}), \\ ∫_{t_{0}}^{t_{1}} ψ_{n}(ξ)∥u(ξ)∥^{−1}dξ & for t ∈ [t_{1}, ∞), \end{matrix}

u_{n}(t) := φ_{n}(t)U(t, t_{0})x, t ≥ 0, and f_{n}(t) := φ′_{n}(t)U(t, t_{0})x, t ≥ 0, where we set U(t, t_{0}) = 0 for 0 ≤ t < t_{0}. From A) and the definition of φ_{n} it follows that u_{n} and f_{n} satisfy the assumptions of Remark 1.2 c). Hence u_{n} ∈ D(G_{0}) and G_{0}u_{n} = −f_{n}. Thus, by (3.1),

1 = sup_{t≥t_{0}} ∥f_{n}(t)∥ = ∥f_{n}∥ ≥ ν∥u_{n}∥ = ν sup_{t≥t_{0}} ∥u_{n}(t)∥.

In particular,

ν^{−1} ≥ ∥u_{n}(t)∥ = |φ_{n}(t)|∥u(t)∥ = ∫_{t_{0}}^{t} ψ_{n}(ξ)∥u(ξ)∥^{−1}dξ ∥u(t)∥ for t ∈ [t_{0}, t_{1}).

Letting n → ∞ we obtain

∫_{t_{0}}^{t} ∥u(t)∥∥u(ξ)∥^{−1}dξ ≤ ν^{−1}.

By Lemma 2.3 there is a constant N only dependent on ν, K and α such that

∥U(t, t_{0})x∥ = ∥u(t)∥ ≤ Ne^{−ν(t−t_{0})}, t ≥ t_{0}.

C) Fix s ≥ t_{0} and set y := U(s, t_{0})x. Then sup_{t≥s} ∥U(t, s)y∥ < ∞ and from A) and B) we obtain

∥U(t, t_{0})x∥ = ∥U(t, s)y∥ ≤ Ne^{−ν(t−s)}∥y∥ = Ne^{−ν(t−s)}∥U(s, t_{0})x∥, t ≥ s. □

Denote by X_{0}(t_{0}) := {x ∈ X : lim_{t→∞} U(t, t_{0})x = 0} the stable subspace corresponding to U and t_{0} ≥ 0. From Theorem 3.2 we obtain the following properties of X_{0}(t_{0}).

3.3 Corollary. Under the conditions of Theorem 3.2 we have

\begin{aligned}X_{0}(t_{0}) & = \{x ∈ X : sup_{t≥t_{0}} ∥U(t, t_{0})x∥ < ∞\} \\ & = \{x ∈ X : ∥U(t, t_{0})x∥ ≤ Ne^{−ν(t−t_{0})}∥x∥\}, t_{0} ≥ 0, \end{aligned}

for fixed constants N, ν > 0, and X_{0}(t_{0}) is a closed linear subspace of X.

Remarks. a) Lemma 3.1 and Lemma 1.1 imply that the following assertions are equivalent:

(i) Aσ(G_{0}) ∩ iℝ ≠ iℝ.

(ii) 0 ∉ Aσ(G_{0}). (iii) Aσ(G_{0}) ∩ iℝ = ∅. (iv) There is a constant ν > 0 such that for any pair u, f ∈ C_{0} with u(t) =

∫_{0}^{t} U(t, ξ)f(ξ)dξ, t ≥ 0, one has ∥f∥ ≥ ν∥u∥.

b) An evolution family with an exponential dichotomy (see Definition 4.1) always satisfies Aσ(G_{0}) ∩ iℝ = ∅ (see the Remark after Theorem 4.3).

c) If B is strongly continuous and uniformly bounded function from [0, ∞) into the space of bounded linear operator on X, then there is a unique evolution family U_{B} = (U_{B}(t, s))_{t≥s≥0} satisfying the variation of constants formula

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U(t, ξ)B(ξ)U_{B}(ξ, s)xdξ, t ≥ s ≥ 0, x ∈ X

(see [RS2], [Sch] and the references therein). The generator of the evolution semigroup T_{B} on C_{0} induced by U_{B} is given by G_{0} + B(·). Suppose that G_{0} is invertible resp. 0 ∉ Aσ(G_{0}). Then G_{0} + B(·) has the same property for every perturbation B(·) such that ∥B(·)∥ = sup_{t≥0} ∥B(t)∥ is sufficiently small (cf. [Kat, IV.3.1]). In other words exponential stability of U resp. of all bounded orbits of U is robust with respect to small perturbations B(·) of U (see also [LMR2, Section 3.1], [Sch, Section 5.4]).

4. Exponential dichotomy

In this section we characterize the exponential dichotomy of an evolution family U by properties of the operators G_{0}, G_{X} and I_{X}, and of the operators G_{Z} and I_{Z} to be defined below. At first we recall the following definition.

4.1 Definition. An evolution family U = (U(t, s))_{t≥s≥0} on the Banach space X is said to have an exponential dichotomy if there exist bounded linear projections P (t), t ≥ 0, on X and constants N, ν > 0 such that

a) U(t, s)P (s) = P (t)U(t, s), t ≥ s ≥ 0,

b) the restriction U(t, s)_{|} : ker P (s) → ker P (t), t ≥ s ≥ 0, is an isomorphism (and we denote its inverse by U(s, t)_{|} : ker P (t) → ker P (s)),

c) ∥U(t, s)x∥ ≤ Ne^{−ν(t−s)}∥x∥ for x ∈ P (s)X, t ≥ s ≥ 0,

d) ∥U(s, t)_{|}x∥ ≤ Ne^{−ν(t−s)}∥x∥ for x ∈ ker P (t), t ≥ s ≥ 0.

In the following lemma we collect some properties of the family P (t), t ≥ 0. By L(Y, X) we denote the space of bounded linear operators between the Banach spaces Y and X.

4.2 Lemma. Let U be an evolution family having an exponential dichotomy with corresponding family of projections P (t), t ≥ 0, and constants N, ν > 0. Then the following holds:

a) M := sup_{t≥0} ∥P (t)∥ < ∞,

b) [0, t] ∋ s ↦ U(s, t)_{|} ∈ L(ker P (t), X) is strongly continuous for t > 0.

c) t ↦ P (t) is strongly continuous,

d) ∥U(t, s)P (s)∥ ≤ MNe^{−ν(t−s)} for t ≥ s ≥ 0,

e) ∥U(s, t)_{|}(I − P (t))∥ ≤ MNe^{−ν(t−s)} for t ≥ s ≥ 0.

Proof. a) can be shown as in [DaK, Lemma IV.1.1 and Lemma IV.3.2]. For sake of completeness we present the details. Fix t_{0} ≥ 0. Let P_{0} := P (t_{0}), P_{1} := Id − P (t_{0}) and X_{k} := P_{k}X, k = 0, 1. Set γ_{t_{0}} := inf{∥x_{0} + x_{1}∥ : x_{k} ∈ X_{k}, ∥x_{0}∥ = ∥x_{1}∥ = 1}. If x ∈ X and P_{k}x ≠ 0, k = 0, 1, then

\begin{aligned}γ_{t_{0}} & ≤ ∥\frac{P_{0}x}{∥P_{0}x∥} + \frac{P_{1}x}{∥P_{1}x∥}∥ = \frac{1}{∥P_{0}x∥}∥P_{0}x + \frac{∥P_{0}x∥}{∥P_{1}x∥}P_{1}x∥ \\ & = \frac{1}{∥P_{0}x∥}∥x + \frac{∥P_{0}x∥ − ∥P_{1}x∥}{∥P_{1}x∥}P_{1}x∥ ≤ \frac{2∥x∥}{∥P_{0}x∥}. \end{aligned}

As a consequence ∥P_{0}∥ ≤ 2γ_{t_{0}}^{−1}. It remains to show that there is a constant c > 0 (independent of t_{0}) such that γ_{t_{0}} ≥ c. For this fix x_{k} ∈ X_{k}, k = 0, 1, with ∥x_{0}∥ = ∥x_{1}∥ = 1. By the exponential boundedness of U we have ∥U(t, t_{0})(x_{0} + x_{1})∥ ≤ Ke^{α(t−t_{0})}∥x_{0} + x_{1}∥ for t ≥ t_{0} and constants K, α ≥ 0. Thus

∥x_{0} + x_{1}∥ ≥ K^{−1}e^{−α(t−t_{0})}∥U(t, t_{0})x_{0} + U(t, t_{0})x_{1}∥
≥ K^{−1}e^{−α(t−t_{0})}(N^{−1}e^{ν(t−t_{0})} − Ne^{−ν(t−t_{0})})
=: c_{t−t_{0}}, t ≥ t_{0},

and hence γ_{t_{0}} ≥ c_{t−t_{0}}. Obviously c_{m} > 0 for m sufficiently large. Thus 0 < c_{m} ≤ γ_{t_{0}}.

b) Fix t > 0, 0 ≤ s_{0} ≤ t and x ∈ ker P (t), and let (s_{n}) be a sequence in [0, t] converging to s_{0}. There is y ∈ ker P (0) such that U(t, 0)y = x. By the strong continuity of U we have

lim_{n} ∥U(s_{n}, t)_{|}x − U(s_{0}, t)_{|}x∥ = lim_{n} ∥U(s_{n}, 0)_{|}y − U(s_{0}, 0)_{|}y∥ = 0.

c) Note that

\begin{aligned}∥P (t)x − P (s)x∥ & ≤ ∥P (t)x − P (t)U(t, s)x∥ + ∥U(t, s)P (s)x − P (s)x∥ \\ & ≤ (sup_{r≥0} ∥P (r)∥) ∥x − U(t, s)x∥ + ∥U(t, s)P (s)x − P (s)x∥ \end{aligned}

for x ∈ X and t ≥ s ≥ 0. By the strong continuity of U we obtain that P (·) is strongly continuous from the right. In order to show strong continuity from the left set Q(·) := Id − P (·) and fix t > 0 and x ∈ X. For 0 ≤ s ≤ t we have Q(s)x = U(s, t)_{|}U(t, s)Q(s)x = U(s, t)_{|}Q(t)U(t, s)x. By b) the family (U(s, t)_{|})_{s∈[0,t]} ⊆ L(ker P (t), X) is strongly continuous and uniformly bounded. The strong continuity of U yields lim_{s↑t} Q(t)U(t, s)x = Q(t)x. Thus

lim_{s↑t} Q(s)x = lim_{s↑t} U(s, t)_{|}Q(t)U(t, s)x = Q(t)x,

i.e. Q(·)x and hence P (·)x is continuous from the left. Assertions d) and e) are immediate consequences of a). □

We come to our first main result. It characterizes evolution families with an exponential dichotomy by conditions on the operators G_{0}, G_{X} and I_{X}, respectively.

4.3 Theorem. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Banach space X. Then the following assertions are equivalent:

(i) U has an exponential dichotomy. (ii) The range R(G_{X}) of G_{X} coincides with C_{0} and X_{0}(0) is complemented in X. (iii) I_{X} is surjective and X_{0}(0) is complemented in X.

Proof. (i) ⇒ (ii): Let P (t), t ≥ 0, be the family of projections given by the exponential dichotomy. Then X_{0}(0) = P (0)X, and hence X_{0}(0) is complemented. If f ∈ C_{0} define v : [0, ∞) → X by

v(t) = ∫_{0}^{t} U(t, ξ)P (ξ)f(ξ)dξ − ∫_{t}^{∞} U(t, ξ)_{|}(Id − P (ξ))f(ξ)dξ.  (4.1)

An easy computation shows that v ∈ C_{X} and v is a solution of equation (1.2). By Lemma 1.1 we have G_{X}v = −f. Together with Remark 1.2 b) this implies R(G_{X}) = C_{0}.

(i) ⇒ (iii): We can use the same arguments as in the proof of (i) ⇒ (ii). Note that for f ∈ C_{X} the function v defined by (4.1) is also in C_{X} and satisfies (1.2). Hence by the definition of I_{X} we have v ∈ D(I_{X}) and I_{X}v = f, i.e. I_{X} is surjective. (ii) ⇒ (i): A) Let Z ⊆ X be a complement of X_{0}(0) in X, i.e. X = X_{0}(0) ⊕ Z. Set X_{1}(t) := U(t, 0)Z, t ≥ 0. Clearly,

\begin{aligned}U(t, s)X_{0}(s) & ⊆ X_{0}(t), t ≥ s, \\ U(t, s)X_{1}(s) & = X_{1}(t), t ≥ s. \end{aligned}  (4.2)

B) There are constants N, ν > 0 such that

∥U(t, 0)x∥ ≥ Ne^{ν(t−s)}∥U(s, 0)x∥ for x ∈ X_{1}(0) and t ≥ s ≥ 0.  (4.3)

In fact, let Y := {v ∈ D(G_{X}) : v(0) ∈ X_{1}(0)} endowed with the graph norm ∥v∥_{G_{X}} := ∥v∥ + ∥G_{X}v∥. Then Y is a closed subspace of the Banach space (D(G_{X}), ∥ · ∥_{G_{X}}), and hence Y is complete. By Remark 1.2 a) we have ker G_{X} = {v ∈ C_{X} : v(t) = U(t, 0)x for some x ∈ X_{0}(0)}. Since X = X_{0}(0) ⊕ X_{1}(0) and R(G_{X}) = C_{0} we obtain that G_{X} : Y → C_{0} is bijective and hence an isomorphism. Thus there is a constant ν > 0 such that

∥G_{X}v∥ ≥ ν∥v∥_{G_{X}} ≥ ν∥v∥ for v ∈ Y.  (4.4)

Let now 0 ≠ x ∈ X_{1}(0) and set u(t) := U(t, 0)x, t ≥ 0. By Remark 1.2 a) we have u(t) ≠ 0 for all t ≥ 0. For each n ∈ ℕ choose a real continuous function φ_{n} on [0, ∞) such that 0 ≤ φ_{n} ≤ 1, φ_{n} = 1 on [\frac{1}{n}, n] and φ_{n} = 0 on \{0\} ∪ [n + 1, ∞). Set f_{n}(t) := −φ_{n}(t)∥u(t)∥^{−1}u(t), t ≥ 0, and

u_{n}(t) := ∫_{t}^{∞} \frac{φ_{n}(ξ)}{∥u(ξ)∥}dξ u(t), t ≥ 0.

Then f_{n} ∈ C_{0} and u_{n} ∈ C_{X}. An easy computation shows that f_{n} and u_{n} satisfy (1.2). By Lemma 1.1 we have u_{n} ∈ D(G_{X}) and G_{X}u_{n} = −f_{n}. From (4.4) we obtain ∥f_{n}∥ ≥ ν∥u_{n}∥. Letting n → ∞ this yields

∫_{t}^{∞} \frac{1}{∥u(ξ)∥}dξ ≤ ν^{−1}∥u(t)∥^{−1}, t ≥ 0.

Now the exponential boundedness of U and Lemma 2.6 imply that there is a constant N ≥ 0 independent of x such that ∥u(t)∥ ≥ Ne^{ν(t−s)}∥u(s)∥, t ≥ s ≥ 0. C) X = X_{0}(t) ⊕ X_{1}(t), t ≥ 0: Let Y ⊆ C_{X} be as in B). Then D(G_{0}) = D(G_{X}) ∩ C_{0} ⊆ Y and (4.4) yields ∥G_{0}v∥ ≥ ν∥v∥ for v ∈ D(G_{0}). Thus 0 ∉ Aσ(G_{0}) ∩ iℝ and Corollary 3.3 implies that X_{0}(t) is closed for t ≥ 0. From (4.2), (4.3) and the closedness of X_{1}(0) we derive that X_{1}(t) is closed and X_{0}(t) ∩ X_{1}(t) = {0} for t ≥ 0. Finally, fix t_{0} > 0 and x ∈ X. Choose a real continuous function φ on [0, ∞) such that φ has compact support contained in [t_{0}, ∞) and ∫_{t_{0}}^{∞} φ(ξ)dξ = 1. Set v(t) := ∫_{t}^{∞} φ(ξ)dξ U(t, t_{0})x and f(t) := −φ(t)U(t, t_{0})x, t ≥ t_{0}. Then v is a solution of equation (2.1), and v ∈ C_{X}(t_{0}). Extend f continuously to [0, ∞) by setting f_{|[0,t_{0})} = 0. Then f ∈ C_{0} and by assumption there exists w ∈ C_{X} such that G_{X}w = −f. In view of Lemma 1.1 w is a solution of equation (1.2). In particular, w_{|[t_{0},∞)} satisfies (2.1). Thus

v(t) − w(t) = U(t, t_{0})(v(t_{0}) − w(t_{0})) = U(t, t_{0})(x − w(t_{0})), t ≥ t_{0}.

Since v − w_{|[t_{0},∞)} ∈ C_{X}(t_{0}) this implies x − w(t_{0}) ∈ X_{0}(t_{0}). On the other hand w(0) = w_{0} + w_{1} with w_{k} ∈ X_{k}(0), k = 0, 1. Then w(t_{0}) = U(t_{0}, 0)w_{0} + U(t_{0}, 0)w_{1} and by (4.2) we have U(t_{0}, 0)w_{k} ∈ X_{k}(t_{0}), k = 0, 1. Hence x = x − w(t_{0}) + w(t_{0}) ∈ X_{0}(t_{0}) + X_{1}(t_{0}). This proves C). D) Let P (t) be the projection from X onto X_{0}(t) with kernel X_{1}(t), t ≥ 0. Then (4.2) implies P (t)U(t, s) = U(t, s)P (s), t ≥ s ≥ 0. From (4.2) and (4.3) we obtain that U(t, s)_{|} : ker P (s) → ker P (t), t ≥ s ≥ 0 is an isomorphism. Finally, by (4.3), Theorem 3.2 and our assumption Aσ(G_{0}) ∩ iℝ ≠ iℝ there exist constants N, ν > 0 such that

∥U(t, s)x∥ ≤ Ne^{−ν(t−s)}∥x∥ for x ∈ P (s)X, t ≥ s ≥ 0,
∥U(s, t)_{|}x∥ ≤ Ne^{−ν(t−s)}∥x∥ for x ∈ ker P (t), t ≥ s ≥ 0.

Thus U has an exponential dichotomy. (iii) ⇒ (i): We use exactly the same arguments as in (ii) ⇒ (i). We only have to replace the operator G_{X} by I_{X}. □

Remark. Part C) of the proof shows that for an evolution family U with an exponential dichotomy the generator G_{0} of the evolution semigroup on C_{0} always satisfies 0 ∉ Aσ(G_{0}), and hence Aσ(G_{0}) ∩ iℝ = ∅ (see Section 3, Remark a)).

If X is a Hilbert space we only have to assume the closedness of the stable subspace.

4.4 Corollary. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Hilbert space H. Then the following assertions are equivalent:

(i) U has an exponential dichotomy. (ii) R(G_{H}) = C_{0} and H_{0}(0) is closed. (iii) I_{H} is surjective and H_{0}(0) is closed.

In the proof of Theorem 4.3, (ii) ⇒ (i), we saw that the invertibility of G_{X} restricted to a certain subspace of C_{X} plays a crucial role in order to prove the existence of an exponential dichotomy for a given evolution family U. In our next result we show that the existence of an exponential dichotomy can be even characterized by such an invertibility condition. Let us introduce the following notion. For a closed linear subspace Z of X let

C_{Z} := {f ∈ C_{X} : f(0) ∈ Z}.

Denote by G_{Z} the part of G_{X} in C_{Z}, i.e. D(G_{Z}) = D(G_{X}) ∩ C_{Z} and G_{Z}v = G_{X}v for v ∈ D(G_{Z}). Then G_{0} is the part of G_{Z} in C_{0}. In the same way let I_{Z} be the restriction of I_{X} to C_{Z}, i.e. D(I_{Z}) = D(I_{X}) ∩ C_{Z} and I_{Z}v = I_{X}v for v ∈ D(I_{Z}). Notice that the evolution semigroup T leaves C_{Z} invariant and that G_{Z} is the generator of the restriction of T to C_{Z}.

With this notation we obtain the following characterization of evolution families with exponential dichotomy. A similar result is shown in [BGK, Theorem 1.1] for the finite dimensional case (see also [Pal]). We point out that in contrast to Theorem 4.3 we do not have to assume that X_{0}(0) is complemented.

4.5 Theorem. Let U = (U(t, s))_{t≥s≥0} be an evolution family on the Banach space X and let Z be a closed linear subspace of X. Then the following assertions are equivalent:

(i) U has an exponential dichotomy with ker P (0) = Z. (ii) G_{Z} : D(G_{Z}) ⊆ C_{Z} → C_{0} is invertible. (iii) I_{Z} : D(I_{Z}) ⊆ C_{Z} → C_{X} is invertible.

Proof. (i) ⇒ (ii) : Let P (t), t ≥ 0, be a family of projections given by the exponential dichotomy such that ker P (0) = Z. Then P (0)X = X_{0}(0) and X = X_{0}(0) ⊕ Z. Fix f ∈ C_{0}. By Theorem 4.3 there is v ∈ D(G_{X}) such that G_{X}v = f. On the other hand u : [0, ∞) → X : t ↦ U(t, 0)P (0)v(0) belongs to C_{X} and G_{X}u = 0 (cf. Remark 1.2 a)). Then v − u ∈ D(G_{Z}) and G_{Z}(v − u) = G_{X}v = f. Hence G_{Z} : D(G_{Z}) → C_{0} is surjective. If w ∈ ker G_{Z} then w(t) = U(t, 0)w(0), t ≥ 0 (cf. Remark 1.2 a)). Since w ∈ C_{X} we have w(0) ∈ Z ∩ X_{0}(0) = {0} and hence w = 0, i.e. G_{Z} is injective.

(i) ⇒ (iii) can be shown by the same arguments. One only has to use Remark 1.4 a) instead of Remark 1.2 a). (ii) ⇒ (i) : Let G_{Z} : D(G_{Z}) → C_{0} be invertible. By Remark 1.2 b) we have C_{0} = R(G_{Z}) = R(G_{X}). The closedness of G_{Z} implies that G_{Z}^{−1} is bounded, and hence there exists ν > 0 such that ∥G_{Z}v∥ ≥ ν∥v∥ for all v ∈ D(G_{Z}). Since G_{0} is the part of G_{Z} in C_{0} we obtain 0 ∉ Aσ(G_{0}). Finally we show that X = X_{0}(0) ⊕ Z. Corollary 3.3 implies that X_{0}(0) is closed. Now let x ∈ X. If U(t, 0)x = 0 for some t > 0, then x ∈ X_{0}(0). Otherwise U(t, 0)x ≠ 0 for all t ≥ 0. Let φ : [0, ∞) → ℝ be continuously differentiable such that φ_{|[0,1]} = 1 and φ_{|[2,∞)} = 0. Set u(t) := φ(t)U(t, 0)x, t ≥ 0, and f(t) := φ′(t)U(t, 0)x, t ≥ 0. By Remark 1.2 c) we have u ∈ D(G_{X}) and G_{X}u = −f. On the other hand there exists v ∈ D(G_{Z}) such that G_{Z}v = −f. Thus u − v ∈ ker G_{X} and hence

(u − v)(t) = U(t, 0)(u(0) − v(0)) = U(t, 0)(x − v(0)), t ≥ 0.

Since u−v ∈ C_{X} this implies x−v(0) ∈ X_{0}(0). Thus x = x−v(0)+v(0) ∈ X_{0}(0)+Z. If y ∈ Z ∩ X_{0}(0), then w defined by w(t) := U(t, 0)y, t ≥ 0, belongs to C_{Z} ∩ ker G_{X} (see Remark 1.2 a)). Hence G_{Z}w = 0 and by the invertibility of G_{Z} we have w = 0. Thus 0 = w(0) = y, i.e. X_{0}(0) ∩ Z = {0}. This shows X = X_{0}(0) ⊕ Z. The assertion now follows from Theorem 4.3. (iii) ⇒ (i) is shown by the same arguments as (ii) ⇒ (i). □

Remark. As a special case of Theorem 4.5 the invertibility of G_{X} resp. I_{X} characterizes evolution families with an exponential dichotomy such that X_{0}(0) = P (0)X = {0}.

We conclude with an example of an evolution family U with non-trivial exponential dichotomy such that I_{X} is invertible. In particular this shows that invertibility of I_{X} is not equivalent to the exponential expansiveness of U (cf. Theorem 2.5).

4.6 Example. Let X = L^{1}[0, ∞). For 0 ≤ s ≤ t ≤ 1 set

U_{1}(t, s)f(ξ) := \{ \begin{matrix}0, & 0 ≤ ξ ≤ t − s, \\ f(ξ − t + s), & t − s < ξ, \end{matrix}

and for 1 ≤ s ≤ t set

U_{2}(t, s)f(ξ) := \{ \begin{matrix}e^{−(t−s)}f(ξ), & 0 ≤ ξ ≤ 1, \\ e^{(t−s)}f(ξ), & ξ > 1, \end{matrix}

f ∈ X. Then U = (U(t, s))_{t≥s≥0} defined by

U(t, s) := \{ \begin{matrix}U_{1}(t, s), & 0 ≤ s ≤ t ≤ 1, \\ U_{2}(t, 1)U_{1}(1, s), & s ≤ 1 ≤ t, \\ U_{2}(t, s), & 1 ≤ s ≤ t, \end{matrix}

is an evolution family on X with an exponential dichotomy. The corresponding family of projections P (t), t ≥ 0, is given by

P (t)f = χ_{[0,min{1,t}]}f, f ∈ X,

where χ_{C} denotes the characteristic function of a set C. In particular, {0} = P (0)X = X_{0}(0). Theorem 4.5 implies that I_{X} is invertible. Since ∥U(t, 1)P (1)∥ ≤ e^{−(t−1)}, t ≥ 1, the evolution family U cannot be exponentially expansive.

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