Evolutionary Semigroups and Lyapunov Theorems in Banach

Spaces

Yuri Latushkin

University of Missouri, Columbia, Missouri 65211

and

Stephen Montgomery-Smith ^{1}

University of Missouri, Columbia, Missouri 65211

1. Introduction

Let us consider an autonomous differential equation v′ = Av in a Banach space E, where A is a generator of C_{0}-semigroup {e^{tA}}_{t≥0}. Denote, as usual, s(A) = sup{Re λ : λ ∈ σ(A)} and ω(A) = inf{ω : ∥e^{tA}∥ ≤ Me^{ωt}}.

A classical result of A. M. Lyapunov (see, e.g., [9]) shows that for any bounded operator A ∈ B(E) the spectrum σ(A) of A is responsible for the asymptotic behavior of the solution y(t) = e^{At}y(0) of the above equation. For example, if σ(A) is contained in the left half-plane, that is s(A) < 0, then the trivial solution is uniformly asymptotically stable, that is ω(A) < 0, and ∥e^{tA}∥ → 0 as t → ∞. This fact follows from the spectral mapping theorem (see, e.g., [21]):

σ(e^{tA})\{0} = exp tσ(A), t > 0,  (1)

which always holds for bounded A.

For unbounded A, equation (1) is not always true. Moreover, there are examples of generators A (see [21]) such that even s(A) < 0 does not guarantee ω(A) < 0 and ∥e^{tA}∥ → 0 as t → ∞. Since σ(A) does not characterize the asymptotic behavior of the solution v(·), we would like to find some other characterization that still does not involve solving the differential equation.

1This author was supported by the National Science Foundation under the grant DMS-9201357.

In this paper we solve precisely this problem in the following manner. Consider the space L_{p}(ℝ; E) of E-valued functions for 1 ≤ p < ∞, or the space C_{0}(ℝ; E) of continuous vanishing at ±∞ functions on ℝ, and the semigroup {e^{tB}}_{t≥0} of evolutionary operators

(e^{tB}f)(x) = e^{tA}f(x − t), t ≥ 0,  (2)

generated by the operator B that is the closure of −\frac{d}{dx} + A, x ∈ ℝ. It turns out that it is σ(B) in L_{p}(ℝ; E) (or in C_{0}(ℝ; E)) that is responsible for the asymptotic behavior of v(·) in E. For example, s(B) = ω(A), and s(B) < 0 in L_{p}(ℝ; E) or C_{0}(ℝ; E) implies ∥e^{tA}∥ → 0 as t → ∞ on E.

The order of the proofs is as follows. First, we consider the evolutionary semigroup {e^{tB}} in the space L_{p}([0, 2π]; E) or C([0, 2π]; E) of 2π-periodic functions. We prove that 1 ∉ σ(e^{2πA}) in E is equivalent to 1 ∉ σ(e^{2πB}) or 0 ∉ σ(B) in L_{p}([0, 2π]; E) or C([0, 2π]; E). The main part of the proof uses a modification of an idea due to C. Chicone and R. Swanson [6]. Next, using this result, we give a variant of the spectral mapping theorem for a semigroup {e^{tA}} in a Banach space E. This spectral mapping theorem is a direct generalization of L. Gearhart’s spectral mapping theorem for Hilbert spaces (see, e.g., [21, p. 95]), and is related to the spectral mapping theorem of G. Greiner [21, p. 94]. Finally, using a simple change of variables arguments, we prove that σ(e^{tA}) ∩ 𝕋 = ∅, t ≠ 0 in E, 𝕋 = {z : |z| = 1} is equivalent to σ(e^{tB}) ∩ 𝕋 = ∅, t ≠ 0 which in turn is equivalent to 0 ∉ σ(B) in L_{p}(ℝ; E) or C_{0}(ℝ; E).

We will also consider the well-posed nonautonomous equation v′ = A(x)v in E, and its associated evolutionary family {U(x, s)}_{x≥s}, which can be thought as a propagator of this equation, that is v(x) = U(x, s)v(s). We assume that U(· , ·) is strongly continuous and satisfies the usual ([31, p. 89]) algebraic properties of the propagator. Instead of the semigroup given by (2) we consider on L_{p}(ℝ; E) or C_{0}(ℝ; E) the evolutionary semigroup

(e^{tG}f)(x) = U(x, x − t)f(x − t), x ∈ ℝ, t ≥ 0.  (3)

We will show that σ(G) characterizes the asymptotic behavior of v(·) and prove the spectral mapping theorem for the semigroup {e^{tG}}.

We will be considering not only stability but also the exponential dichotomy (hyperbolicity) for the solutions of the equation v′ = A(x)v or evolutionary family {U(x, s)}. We say, that an evolutionary family {U(x, s)}_{x≥s} is (spectrally) hyperbolic if there exists a continuous in the strong sense, bounded, projection-valued function P : ℝ → B(E) such that: a) The norm of the restrictions U(x, s)| Im P (s) (resp. U(x, s)| Ker P (s)) exponentially decreases (resp. increases) with x − s, and b) Im U(x, s)| Ker P (s) is dense in Ker P (x). Note, that b) automatically follows from a) if the operators U(x, s) are invertible and defined for all (x, s) ∈ ℝ^{2}. This happens, in particular, if U(· , ·) is a norm-continuous propagator for the differential equation v′ = A(x)v with continuous and bounded A : ℝ → B(E). For this case the hyperbolicity of {U(x, s)} coincides with the exponential dichotomy (see, e.g., [9]) of the equation. However, generally a) does not imply b) (see [27]).

Exponential dichotomy in the theory of differential equations with bounded coefficients on E is a major tool used for proving instability theorems for nonlinear equations, and for showing existence and uniqueness of bounded solutions and Green’s functions, etc. (see, e.g. [7, 9]). The spectral mapping theorem for the semigroup (3) which is given here allows one to extend these ideas to the case of unbounded coefficients.

It turns out that the spectrum σ(e^{tG}) for nonperiodic A(·) plays the same role in the description of exponential dichotomy as the spectrum of the monodromy operator does in the usual Floquet theory for the periodic case. That is, the condition σ(e^{tG}) ∩ 𝕋 = ∅, t ≠ 0, or equivalently 0 ∉ σ(G), is equivalent to the (spectral) hyperbolicity of the evolutionary family {U(x, s)}_{x≥s}.

Showing that the hyperbolicity σ(e^{G}) ∩ 𝕋 = ∅ of the operator e^{G} implies the hyperbolicity of {U(x, s)} is a delicate matter. It turns out that the Riesz projection P for e^{G} on L_{p}(ℝ, E) or C_{0}(ℝ; E), that corresponds to σ(e^{G}) ∩ 𝔻, 𝔻 = {z : |z| < 1}, has the form (Pf)(x) = P (x)f(x). Here P (·) is a continuous in the strong sense, projectionvalued function, that defines the hyperbolicity of {U(x, s)}. Note that e^{G} = aR, where a is an operator of multiplication by the function a(x) = U(x, x − 1), that is (af)(x) = a(x)f(x), and R is a translation operator (Rf)(x) = f(x − 1), x ∈ ℝ. Therefore, e^{G} falls into the class of so-called weighted translation operators, which are well-understood in the case that E is Hilbert space and p = 2 (see [1, 2, 18, 27], and also [8, 23] and references therein). If U(· , ·) is norm-continuous, then P is an operator from a C^{∗}-algebra generated by R and the C^{∗}-algebra 𝔄_{nc} of operators of multiplication by the norm-continuous, bounded functions from ℝ to B(E). The techniques from the theory of weighted translation operators (see [1, 2, 18, 27]) allows one to conclude that P ∈ 𝔄_{nc}. This technique is not applicable to the case where {U(·, ·)} is only strongly-continuous, nor also to the case when E is not Hilbert space.

In this paper we present some new approaches, which allows one to derive the above result for any Banach space E and is new even for the Hilbert space case and when it is only known that U(·, ·) is strongly continuous. The main idea is to “discretize” the operator aR, that is to represent it by the family of operators π_{x}(a)S, x ∈ ℝ, acting on the “discrete” space l_{p}(ℤ; E). Here S : (v_{n})_{n∈ℤ} ↦ (v_{n−1})_{n∈ℤ} is the shift operator and π_{x}(a) : (v_{n})_{n∈ℤ} ↦ (a(x + n)v_{n})_{n∈ℤ} is a diagonal operator on l_{p}(ℤ; E). This idea goes back to the theory of regular representations of C^{∗}-algebras [26], and is related to works [1, 2, 13, 16, 17, 18]. As a result we prove that σ(aR)∩𝕋 = ∅ in L_{p}(ℝ; E) implies σ(π_{x}(a)S)∩𝕋 = ∅ in l_{p}(ℤ; E) for each x ∈ ℝ, and derive from this fact that P ∈ 𝔄, where 𝔄 is the set of bounded functions a : ℝ → B(E) which are continuous in strong operator topology on B(E).

We point out that the investigation of evolutionary operators (2)–(3) has a long history, probably starting from [14] (see also [10, 11, 19, 22]). Recently significant progress has been made in the papers [3, 4, 25, 27]. It is these papers that essentially motivated and influenced this present work.

Finally, the results of this article can be generalized to the case of the variational equation v′(t) = A(φ^{t}x)v(t) for a flow {φ^{t}} on a compact metric space X, or to the linear skew-product flow φ̂^{t} : X × E → X × E : (x, v) ↦ (φ^{t}x, Φ(x, t)v), t ≥ 0 (see [6, 12, 18, 29, 30] and references contained therein). Here Φ : X × ℝ_{+} → L(E) is a cocycle over φ^{t}, that is, Φ(x, t+s) = Φ(φ^{t}x, s)Φ(x, t). Let us recall (see [29, 30]) that part of the purpose of the theory of linear skew-product flows was to be able to handle the equation v′ = A(t)v in the case when A(·) is almost-periodic .

To answer the question when φ̂^{t} is hyperbolic (or Anosov), instead of (3) one considers the semigroup of so called weighted composition operators (see [6, 15, 18]) on L_{p}(X; μ; E):

(T^{t}f)(x) = (\frac{dμ ∘ φ^{−t}}{dμ} )^{1/p}Φ(φ^{−t}x, t)f(φ^{−t}x), x ∈ X, t ≥ 0. (4)

Here μ is a φ^{t}-quasi-invariant Borel measure on X. As above, the condition σ(T^{t}) ∩ 𝕋 = ∅ is equivalent to the spectral hyperbolicity of the linear skew-product flow φ̂^{t}. The spectral hyperbolicity coincides with the usual hyperbolicity if Φ(x, t), x ∈ X, t ≥ 0 are invertible or compact operators. A detailed investigation of weighted composition operators and their connections with the spectral theory of linear skewproduct flows and other questions of dynamical system theory may be found in [18] (see also [27]).

We will use the following notations: 𝔻 = {z : |z| < 1}; 𝕋 = {z : |z| = 1}; “|” denotes the restriction of an operator; D(·) = D_{F} (·) denotes the domain of an operator in a space F; σ(·) = σ(·; F) denotes the spectrum; σ_{ap}(·) = σ_{ap}(·; F) denotes the approximative point spectrum; σ_{r}(·) = σ_{r}(·; F) denotes the residual spectrum; and ρ(·) = ρ(·; F) denotes the resolvent set of an operator on F. For an operator A in E we denote by A the operator of multiplication by A in a space of E-valued functions: (Af)(x) = Af(x), f : ℝ → E.

The authors would very much like to thank C. Chicone for help and suggestions, and R. Rau for many illuminating discussions.

The authors also would like to thank the referee for the suggestion to shorten the proof of Theorem 2.5 and Remark below.

2. Autonomous Case

Let A be a generator of a C_{0}-semigroup {e^{tA}}_{t≥0} on a Banach space E. The semigroup is called hyperbolic if σ(e^{tA}) ∩ 𝕋 = ∅ for t ≠ 0. In this section we will characterize the hyperbolicity of the semigroup {e^{tA}} in terms of evolutionary semigroup {e^{tB}}_{t≥0} and its generator B. This semigroup acts by the rule (e^{tB}f)(x) = e^{tA}f(x − t) on functions f with values in E. In Subsection 2.1 we consider {e^{tB}} acting on the space L_{p}([0, 2π]; E) and C([0, 2π]; E). In Subsection 2.3 {e^{tB}} acts on L_{p}(ℝ; E) and C_{0}(ℝ; E). Subsection 2.2 is devoted to a spectral mapping theorem for {e^{tA}}_{t≥0} on E which generalizes the spectral mapping theorem of L. Gearhart for Hilbert space.

2.1. Periodic Case

Let F denote one of the spaces L_{p}([0, 2π], E), 1 ≤ p < ∞ or C([0, 2π], E) of 2π-periodic E-valued functions f, f(0) = f(2π). Consider the evolutionary semigroup {e^{tB}}_{t≥0} acting on F, defined by the rule

(e^{tB}f)(x) = e^{tA}f([x − t](mod 2π)), x ∈ [0, 2π].

Of course, [0, 2π] here was chosen for convenience, and for a semigroup (e^{tB}f)(x) = e^{tA}f ([x − t](mod t_{0})) the proofs below remain the same for any t_{0} > 0.

Note that e^{tB} in F is a product of two commuting semigroups (U^{t}f)(x) = f([x − t](mod 2π)) and (e^{tA}f)(x) = e^{tA}f(x). Hence the generator B is the closure of the operator

(B_{0}f)(x) = −\frac{d}{dx}f(x) + Af(x),  (5)

where B_{0} is defined on the core D(B_{0}) of B (see [21, p. 24]). Moreover, D_{F} (B_{0}) = D_{F} (−d/dx) ∩ D_{F} (A), where the derivative d/dx is taken in the strong sense in E, and D_{F} (B_{0}) = {f : [0, 2π] → D(A)|f ∈ F is absolutely continuous, \frac{d}{dx}f ∈ F, and Af ∈ F \}.

Since Be^{ik·}f(·) = e^{ik·}(B−ik)f(·), k ∈ ℤ, for the operator (L_{k}f)(x) = e^{ikx}f(x) one has BL_{k} = L_{k}(B − ik). Therefore, the spectrum σ(B) in F is invariant under translations by i.

We will need the following Lemma. Lemma 2.1. If 1 ∈ σ_{ap}(e^{2πA}) in E, then 0 ∈ σ_{ap}(B) in F.

Proof. Fix m ∈ ℕ, m ≥ 2. Since 1 ∈ σ_{ap}(e^{2πA}), we can choose v ∈ E such that ∥v∥_{E} = 1 and ∥v − e^{2πA}v∥_{E} < \frac{1}{m}. Note also that ∥e^{2πA}v∥_{E} ≥ 1 − \frac{1}{m}.

Let α : [0, 2π] → [0, 1] be any smooth function with bounded derivative such that α(x) = 0 for x ∈ [0, \frac{2π}{3}] and α(x) = 1 for x ∈ [\frac{4π}{3}, 2π]. Define a function g : [0, 2π] → E by the the formula

g(x) = [1 − α(x)]e^{(2π+x)A}v + α(x)e^{xA}v, x ∈ [0, 2π].  (6)

Note that g(0) = g(2π) = e^{2πA}v. Obviously, g ∈ F. Also,

(e^{tB}g)(x) = [1 − α(x − t)]e^{(2π+x)A}v + α(x − t)e^{xA}v,

g ∈ D_{F} (B), and

(Bg)(x) = α′(x)e^{xA}[e^{2πA}v − v], x ∈ [0, 2π].  (7)

Let us denote a = max{|α′(x)| : x ∈ [0, 2π]} and b = max{∥e^{xA}∥ : x ∈ [0, 2π]}. Note, that ∥e^{2πA}v∥ = ∥e^{(2π−x)A}e^{xA}v∥ ≤ b∥e^{xA}v∥ for any x ∈ [0, 2π].

First let us suppose that F = L_{p}([0, 2π]; E). Then

∥Bg∥_{L_{p}([0,2π];E)} ≤ (2π)^{1/p} \frac{ab}{m}.

On the other hand,

∥g∥_{L_{p}([0,2π];E)}^{p} ≥ ∫ 2π_{\frac{4π}{3}} ∥e^{xA}v∥^{p}dx ≥ \frac{2π}{3} b^{−p}∥e^{2πA}v∥^{p} ≥ \frac{2π}{3} b^{−p}(1 − \frac{1}{m})^{p}.

Finally,

∥Bg∥_{L_{p}([0,2π];E)} ≤ (2π)^{1/p} \frac{ab}{m} ≤ 3^{1/p}ab^{2}∥g∥_{L_{p}([0,2π];E)} · \frac{1}{m − 1}. (8)

Since this holds for all m, it follows that 0 ∈ σ_{ap}(B).

Now suppose that F = C([0, 2π], E). Then

∥Bg∥_{C([0,2π],E)} ≤ \frac{ab}{m}, ∥g∥_{C([0,2π],E)} ≥ ∥g(0)∥_{E} = ∥e^{2πA}v∥ ≥ 1 − \frac{1}{m}(9)

and hence

∥Bg∥_{C([0,2π];E)} ≤ \frac{ab}{m − 1}∥g∥_{C([0,2π];E)}.  (10)

Since this is true for all m, it follows that 0 ∈ σ_{ap}(B). □

Theorem 2.2. Let F be one of the spaces L_{p}([0, 2π], E), 1 ≤ p < ∞ or C([0, 2π], E). Then the following are equivalent:

1) 1 ∈ ρ(e^{2πA}) in E;

2) 1 ∈ ρ(e^{2πB}) in F;
3) 0 ∈ ρ(B) in F.

Proof. 1) ⇒ 2). Note that (e^{2πB}f)(x) = e^{2πA}f(x). Hence σ(e^{2πA}; E) = σ(e^{2πB}; F). Note also that σ_{r}(e^{2πA}; E) = σ_{r}(e^{2πB}; F).

2) ⇒ 3) follows from the spectral inclusion theorem e^{2πσ(B)} ⊂ σ(e^{2πB}) (see [24, p. 45]).

3) ⇒ 1). Assume 0 ∈ ρ(B; F) but 1 ∈ σ(e^{2πA}; E) = σ_{ap}(e^{2πA}; E) ∪ σ_{r}(e^{2πA}; E). By Lemma 2.1 it follows that 1 ∈ σ_{r}(e^{2πA}; E), and hence 1 ∈ σ_{r}(e^{2πB}; F). By the spectral mapping theorem for the residual spectrum ([24, Theorem 2.5 (ii)]) it follows that ik ∈ σ_{r}(B; F) for some k ∈ ℤ. Since σ(B; F) is invariant under translations by i, we have that 0 ∈ σ(B; F), contradicting 3). □

2.2. Spectral Mapping Theorem for Banach Spaces

As it is well-known (see, e.g., [21, p. 82–89]), that in general, the inclusion e^{tσ(A)} ⊂ σ(e^{At}), t ≠ 0 for a C_{0}-semigroup {e^{tA}}_{t≥0} on a Banach space E is improper. In particular, iℤ ⊂ ρ(A) is implied by but does not imply 1 ∈ ρ(e^{2πA}). For Hilbert space E, however, the following spectral mapping theorem of L. Gearhart (see [21, p. 95]) is true: 1 ∈ ρ(e^{2πA}) if and only if iℤ ⊂ ρ(A) and sup_{k∈ℤ} ∥(A − ik)^{−1}∥ < ∞. We will now give a direct generalization of this result to any arbitrary Banach space E. This generalization is related (but independent) to G. Greiner’s spectral mapping theorem [21, p. 94] that involves Césaro summability of the series ∑_{k}(A − ik)^{−1}v, v ∈ E.

Theorem 2.3. Let {e^{tA}} be any C_{0}-semigroup on a Banach space E and let F be one of the spaces L_{p}([0, 2π], E), 1 ≤ p < ∞ or C([0, 2π], E). Then the following are equivalent:

1) 1 ∈ ρ(e^{2πA});

2) iℤ ⊂ ρ(A) and there is a constant C > 0 such that

∥ ∑_{k}(A − ik)^{−1}e^{ikx}v_{k}∥_{F} ≤ C∥ ∑_{k} e^{ikx}v_{k}∥_{F}  (11)

for any finite sequence {v_{k}} ⊂ E.

Proof. Consider the evolutionary semigroup {e^{tB}}_{t≥0} on F from the previous subsection. Consider a finite sequence {v_{k}} ⊂ E. Assume that (A − ik)^{−1} exists for all k ∈ ℤ. Define functions f, g ∈ F by the rule

f(x) = ∑_{k}(A − ik)^{−1}e^{ikx}v_{k}, g(x) = ∑_{k} e^{ikx}v_{k}, x ∈ [0, 2π]. (12)

Since (A − ik)^{−1} : E → D(A), one has Bf = g. Indeed

(Bf)(x) = \frac{d}{dt}e^{tA}f([x − t](mod 2π))|_{t=0}

= ∑_{k}[A(A − ik)^{−1}e^{ikx}v_{k} − ik(A − ik)^{−1}e^{ikx}v_{k}] = g.

1) ⇒ 2). If 1 ∈ ρ(e^{2πA}), then the inclusion iℤ ⊂ ρ(A) follows from the spectral inclusion theorem e^{2πσ(A)} ⊂ σ(e^{2πA}). In accordance with part 1) ⇒ 3) of Theorem 2.2, the operator B has bounded inverse B^{−1} on F provided that 1 ∈ ρ(e^{2πA}). Denote C = ∥B^{−1}∥, and consider functions (12). Then ∥f∥_{F} = ∥B^{−1}g∥_{F} ≤ C∥g∥_{F}, and (11) is proved.

2) ⇒ 1). First, we show that 2) implies 0 ∉ σ_{ap}(B). Indeed, the functions of type g in (12) are dense in F. If we let u_{k} = (A − ik)^{−1}v_{k}, then we note that the functions of type f are also dense in F. Now (11) implies ∥Bf∥_{F} = ∥g∥_{F} ≥ C^{−1}∥f∥_{F}, and 0 ∉ σ_{ap}(B).

Assume that 2) is fulfilled, but 1 ∈ σ(e^{2πA}) = σ_{r}(e^{2πA})∪σ_{ap}(e^{2πA}). If 1 ∈ σ_{ap}(e^{2πA}) in E then, by Lemma 2.1, 0 ∈ σ_{ap}(B), in contradiction to the previous paragraph. On the other hand, 1 ∈ σ_{r}(e^{2πA}) implies, by the spectral mapping theorem for residual spectrum, that ik ∈ σ_{r}(A) for some k ∈ ℤ, contradicting iℤ ⊂ ρ(A). □

Remark. We note, that 1) ⇒ 2) can be also seen directly. Indeed, assuming 1), let us denote ϕ(s) = (e^{2πA} − I)^{−1}e^{sA}, s ∈ [0, 2π]. Then the convolution operator

(Kf)(x) = ∫_{0}^{2π}ϕ(s)f(x − s) ds

is a bounded operator on F. But

(A − ik)^{−1} = ∫_{0}^{2π}e^{−iks}(e^{2πA} − I)^{−1}e^{sA} ds, k ∈ ℤ,

are Fourier coefficients of ϕ : [0, 2π] → B(E). Inequality (11) can be viewed as the condition of boundedness of K, which gives 2).

Let us show now that Theorem 2.3 is really a direct generalization of L. Gearhart’s Theorem, mentioned above. Indeed, for Hilbert space E and p = 2, Parseval’s identity implies:

∥ ∑_{k}(A − ik)^{−1}e^{ik·}v_{k}∥_{L_{2}([0,2π];E)} = (2π ∑_{k} ∥(A − ik)^{−1}v_{k}∥_{E}^{2})^{1/2}

∥ ∑_{k} e^{ik·}v_{k}∥_{L_{2}([0,2π];E)} = (2π ∑_{k} ∥v_{k}∥_{E}^{2})^{1/2}.

Clearly, (11) is equivalent to the condition sup{∥(A−ik)^{−1}∥ : k ∈ ℤ} < ∞.

We conclude this subsection by giving four more statements equivalent to 1) and 2) in Theorem 2.3:

3) iℤ ⊂ ρ(A) and there is a constant C > 0 such that

∥Bf∥_{L_{1}([0,2π];E)} ≥ C^{−1}∥f∥_{C([0,2π];E)}

for all f ∈ C([0, 2π]; E) such that Bf ∈ L_{1}([0, 2π]; E);

4) iℤ ⊂ ρ(A) and there is a constant C > 0 such that

∥Bf∥_{C([0,2π];E)} ≥ C^{−1}∥f∥_{L_{1}([0,2π];E)}

for all f ∈ L_{1}([0, 2π]; E) such that Bf ∈ C([0, 2π]; E);

5) iℤ ⊂ ρ(A) and there is a constant C > 0 such that

∥ ∑_{k}(A − ik)^{−1}e^{ikx}v_{k}∥_{C([0,2π];E)} ≤ C∥ ∑_{k} e^{ikx}v∥_{L_{1}([0,2π];E)}

for any finite sequence {v_{k}} ⊂ E;

6) iℤ ⊂ ρ(A) and there is a constant C > 0 such that

∥ ∑_{k}(A − ik)^{−1}e^{ikx}v_{k}∥_{L_{1}([0,2π];E)} ≤ C∥ ∑_{k} e^{ikx}v∥_{C([0,2π];E)}

for any finite sequence {v_{k}} ⊂ E.

2.3. Real Line

Consider now the evolutionary semigroup {e^{tB}}_{t≥0},

(e^{tB}f)(x) = e^{tA}f(x − t)  (13)

acting on the space F = L_{p}(ℝ; E), 1 ≤ p < ∞, or F = C_{0}(ℝ; E). Formula (5) is valid and the identities

e^{tB}e^{iξ·}f(·) = e^{iξ·}e^{−iξt}e^{tB}f(·),
Be^{iξ·}f(·) = e^{iξ·}(B − iξ)f(·), ξ ∈ ℝ,

show that σ(e^{tB}) in F is invariant under rotations centered at the origin, and that σ(B), σ_{ap}(B) and σ_{r}(B) in F are invariant under translations parallel to iℝ.

First we state a simple lemma. Let F_{s} be one of the spaces F_{s} = l_{p}(ℤ; E), 1 ≤ p < ∞ or F_{s} = c_{0}(ℤ; E) (of sequences (v_{n})_{n∈ℤ} such that v_{n} → 0 as n → ±∞). Let S be a shift operator on F_{s}, that is S : (v_{n}) ↦ (v_{n−1}). For an operator a on E we will denote by D_{a} the diagonal operator on F_{s}, acting by the rule D_{a} : (v_{n})_{n∈ℤ} ↦ (av_{n})_{n∈ℤ}. Lemma 2.4. The following are equivalent:

1) σ(a) ∩ 𝕋 = ∅ in E;

2) σ(D_{a}S) ∩ 𝕋 = ∅ in F_{s}.

Proof. We will give the proof for the case when F_{s} = l_{p} = l_{p}(ℤ; E). The case F_{s} = c_{0}(ℤ; E) can be considered similarly.

1) ⇒ 2). Since σ(a) ∩ 𝕋 = ∅, there exists a Riesz projection p̂ for a in E that corresponds to the part of the spectrum σ(a) ∩ 𝔻. Define q̂ = I − p̂, and consider in F complimentary projections Dp̂ and Dq̂. Since D_{a}SDp̂ = D_{a}Dp̂S, the decomposition F_{s} = Im Dp̂ ⊕ Im Dq̂ is D_{a}S-invariant. For the spectral radius

r(·) = lim_{n→∞} ∥(·)^{n}∥^{\frac{1}{n}}

one has r(p̂ap̂) < 1 and r([q̂aq̂]^{−1}) < 1 in E. Hence, r(D_{a}SDp̂) < 1, r([D_{a}SDq̂]^{−1}) < 1, and σ(D_{a}S) ∩ 𝕋 = ∅ in F_{s}.

2) ⇒ 1). Assume that I − D_{a}S is invertible in l_{p}, but for any ϵ > 0 there is a vector v ∈ E such that ∥v∥_{E} = 1 and ∥v − av∥_{E} < ϵ. Fix q > 0 such that

|1 − e^{±q}| < ϵ,

and define a sequence (v_{n}) ∈ l_{p} by v_{n} = e^{−q|n|}v, n ∈ ℤ. Then

I − D_{a}S : (v_{n})_{n∈ℤ} ↦ (e^{−q|n|}(v − av) + (e^{−q|n|} − e^{−q|n−1|})av)_{n∈ℤ}.

A direct calculation shows that

∥(I − D_{a}S)(v_{n})∥_{l_{p}} ≤ (1 + ∥a∥) · ϵ · ∥(v_{n})∥_{l_{p}},

contradicting the invertibility of I − D_{a}S in l_{p}.

Let us show now that I −a has a dense range in E, provided I −D_{a}S is an operator onto l_{p}. Indeed, for any u ∈ E consider a sequence (u_{n}) ∈ l_{p} defined by u_{0} = u and u_{n} = 0 for n ≠ 0. Find a sequence (v_{n}) ∈ l_{p} such that (I − D_{a}S)(v_{n}) = (u_{n}), that is v_{n} − av_{n−1} = u_{n} for n ∈ ℤ. But then for k ∈ ℕ one has

u = ∑_{n=−k}^{k}(v_{n} − av_{n−1})

= v_{k} − av_{−k−1} + (y_{k} − ay_{k}),

where

y_{k} = ∑_{n=−k}^{k−1} v_{n}.

Therefore, Im(I − a) ∋ y_{k} − ay_{k} → u, since v_{k} → 0 and av_{−k−1} → 0 as k → ∞. □

Theorem 2.5. Let F be one of the spaces L_{p}(ℝ; E), 1 ≤ p < ∞ or C_{0}(ℝ; E), and let t > 0. Then the following are equivalent:

1) σ(e^{tA}) ∩ 𝕋 = ∅ in E;

2) σ(e^{tB}) ∩ 𝕋 = ∅ in F;
3) 0 ∈ ρ(B) in F.

Proof. 2) ⇒ 3) follows from the spectral inclusion theorem for {e^{tB}}.

3) ⇒ 2) we will prove for F = L_{p}(ℝ; E); the arguments for F = C_{0}(ℝ; E) are similar.

Since σ(e^{tB}) is invariant under the rotations with the center at origin, it suffices to prove that 3) implies 1 ∈ ρ(e^{tB}). Also, to confirm our previous notations, we will consider only the case t = 2π. The proof stays the same for any t.

The idea is to apply Theorem 2.2, to show that 1 ∈ ρ(e^{2πB}) is implied by 0 ∈ ρ(B′). Here the operator B′ = −\frac{d}{ds} − \frac{d}{dx} +A acts on L_{p}([0, 2π]× ℝ; E), s ∈ [0, 2π], x ∈ ℝ. Indeed, by formula (5) one has B = −\frac{d}{dx} +A. Hence, B′ on L_{p}([0, 2π]; L_{p}(ℝ; E)) is the generator of the evolutionary semigroup for the semigroup {e^{tB}} on L_{p}(ℝ; E). But the change of variables u = [s − x](mod 2π), v = x shows that ρ(B′) = ρ(−\frac{d}{dv} + A) = ρ(B). Let us now make this argument more formal.

Consider the semigroups

(e^{tB′}h)(s, x) = e^{tA}h([s − t](mod 2π), x − t), t > 0, s ∈ [0, 2π], x ∈ ℝ,
(e^{tB}h)(s, x) = e^{tA}h(s, x − t), t > 0, s ∈ [0, 2π], x ∈ ℝ,

and an invertible isometry J,

(Jh)(s, x) = h([s + x](mod 2π), x),

acting on the space

L_{p}([0, 2π] × ℝ; E) = L_{p}([0, 2π]; L_{p}(ℝ; E)).

Since e^{tB} acting on L_{p}([0, 2π]; L_{p}(ℝ; E)) is actually the operator of multiplication by the operator e^{tB} in L_{p}(ℝ; E), one has:

σ(e^{tB}) = σ(e^{tB}) and σ(B; L_{p}([0, 2π]; L_{p}(ℝ; E))) = σ(B; L_{p}(ℝ; E)).

Also,

(Je^{tB′}h)(s, x) = e^{tA}h([s + x − t](mod 2π), x − t) =
(e^{tB}Jh)(s, x),

and hence one has Je^{tB′} = e^{tB}J and JB′ = BJ. Therefore,

σ(e^{tB}) = σ(e^{tB′}) and σ(B) = σ(B′) in L_{p}([0, 2π]; L_{p}(ℝ; E)).

Thus 3) implies 0 ∈ ρ(B) and 0 ∈ ρ(B′).

The semigroup {e^{tB′}} acts on L_{p}([0, 2π]; L_{p}(ℝ; E)) by the rule

(e^{tB′}f)(s) = e^{tB}f([s − t](mod 2π)),

where f(s) = h(s, ·) ∈ L_{p}(ℝ; E) for almost all s ∈ [0, 2π]. Hence, {e^{tB′}} on the space L_{p}([0, 2π]; L_{p}(ℝ; E)) is the evolutionary semigroup for the semigroup {e^{tB}} on L_{p}(ℝ; E). Now one can apply the part 3) ⇒ 1) of the Theorem 2.2 and conclude that 1 ∈ ρ(e^{2πB}) on L_{p}(ℝ; E).

1) ⇔ 2) we will prove for F = L_{p}(ℝ; E); the arguments for F = C_{0}(ℝ; E) are similar.

Let us denote, for brevity, a = e^{2πA} and (Rf)(x) = f(x − 2π) on L_{p}(ℝ; E). Thus e^{2πB} = aR. Consider the invertible isometry

j : L_{p}(ℝ; E) → l_{p}(ℤ; L_{p}([0, 2π]; E)) :
f ↦ (f_{n}), f_{n}(s) = f(s + 2πn), n ∈ ℤ, s ∈ [0, 2π).

Let S : (f_{n}) ↦ (f_{n−1}) be a shift operator on l_{p}(ℤ; L_{p}([0, 2π]; E)). Then jaR = D_{a}Sj and σ(aR) = σ(D_{a}S). Therefore, 2) is equivalent to σ(D_{a}S)∩𝕋 = ∅. By Lemma 2.4 this in turn is equivalent to σ(a)∩𝕋 = ∅ in L_{p}(ℝ; E). □

Note, that 3) ⇒ 1) in the above theorem can also be derived directly by constructing a function g in a similar manner as in the proof of Lemma 2.1. This proof will be given elsewhere.

3. Non-autonomous Case

Consider a non-autonomous differential equation v′(x) = A(x)v(x), x ∈ ℝ in E. We will assume that this equation is well-posed. This means that there exists an evolutionary family {U(x, s)}_{x≥s} (propagator) for the equation, that is v(x) = U(x, s)v(s), x ≥ s. Recall the definition of evolutionary family (see, e.g., [27, 31]). Definition 3.1. A family {U(x, s)}_{x≥s} of bounded in E operators U(x, s) is called an evolutionary family if the following conditions are fulfilled:

(i) for each v ∈ E the function (x, s) ↦ U(x, s)v is continuous for x ≥ s; (ii) U(x, s) = U(x, r)U(r, s), U(x, x) = I, x ≥ r ≥ s; (iii) ∥U(x, s)∥ ≤ Ce^{β(x−s)}, x ≥ s for some constants C, β > 0.

The evolutionary family {U(x, s)} generates an evolutionary semigroup {e^{tG}}_{t≥0} acting on the space F = L_{p}(ℝ; E), 1 ≤ p < ∞ or F = C_{0}(ℝ; E) by the rule

(e^{tG}f)(x) = U(x, x − t)f(x − t), x ∈ ℝ.  (14)

In Subsection 3.1 we will prove the spectral mapping theorem σ(e^{tG})\{0} = e^{tσ(G)}, t ≠ 0 for {e^{tG}}. We will achieve this by applying a simple change of variables argument (cf. the proof of Theorem 2.5) to deduce this from Theorem 2.5. In Subsection 3.2 we will prove that the hyperbolicity σ(e^{tG}) ∩ 𝕋 = ∅ of the semigroup in F is equivalent to the so-called spectral hyperbolicity of the family {U(x, s)}. Spectral hyperbolicity is a generalization of the notion of exponential dichotomy (see, e.g., [9]) for the equation v′(x) = A(x)v(x) with bounded A : ℝ → B(E).

3.1. The Spectral Mapping Theorem for Evolutionary Semigroup

Let G be the generator of the evolutionary semigroup {e^{tG}}_{t≥0} acting on the space F = L_{p}(ℝ; E), 1 ≤ p < ∞ or F = C_{0}(ℝ; E) by equation (14). Theorem 3.1. The spectrum σ(G) is invariant under translations along the imaginary axis, and the following are equivalent:

1) 0 ∈ ρ(G) on F;

2) σ(e^{tG}) ∩ 𝕋 = ∅ on F, t > 0.

Proof. For any ξ ∈ ℝ it is true that e^{tG}e^{iξ·}f(·) = e^{iξ(·−t)}e^{tG}f(·) and Ge^{iξ·} = e^{iξ·}(G − iξ). Hence σ(e^{tG}) is invariant under rotations centered at the origin, and σ(G) is invariant under translations along the imaginary axis.

2) ⇒ 1) follows from the spectral inclusion theorem for {e^{tE}}.

1) ⇒ 2). We will first consider the case when F = L_{p}(ℝ; E), 1 ≤ p < ∞.

The idea of the proof is almost identical to the proof of 3) ⇒ 2) from Theorem 2.5. If U(·, ·) is a smooth propagator for the equation v′ = A(x)v, then G = −\frac{d}{dx} + A(x). Consider the evolutionary semigroup for \{e^{tG}\}, that is the semigroup with the generator B = −\frac{d}{ds} + G on L_{p}(ℝ; L_{p}(ℝ; E)). Theorem 2.5 shows that 1 ∈ ρ(e^{tG}) is implied by 0 ∈ ρ(B), where B = −\frac{d}{ds} − \frac{d}{dx} +A(x), s, x ∈ ℝ by formula (5). The change of variables u = s−x, v = x shows that ρ(B) = ρ(−\frac{d}{dv} +A(v)) = ρ(G). Let us now make this argument more formal.

Consider the semigroups {e^{tB}}_{t≥0} and {e^{tG}}_{t≥0} acting on the space L_{p}(ℝ × ℝ; E) = L_{p}(ℝ; L_{p}(ℝ; E)) by

(e^{tB}h)(s, x) = U(x, x − t)h(s − t, x − t), (s, x) ∈ ℝ^{2}, t > 0,
(e^{tG}h)(s, x) = U(x, x − t)h(s, x − t).

Note that e^{tG} in L_{p}(ℝ; L_{p}(ℝ; E)) is the operator of multiplication by e^{tG}, that is (e^{tG}f)(s) = e^{tG}f(s), where f(s) = h(s, ·) ∈ L_{p}(ℝ; E). Similarly, (Gf)(s) = Gf(s) = Gh(s, ·) for f(s) = h(s, ·) ∈ D_{L_{p}(ℝ;E)}(G) for almost all s ∈ ℝ.

Consider an isometry J on L_{p}(ℝ × ℝ; E) defined by (Jh)(s, x) = h(s + x, x). Then for h ∈ L_{p}(ℝ × ℝ; E) one has:

(e^{tG}Jh)(s, x) = U(x, x − t)h(s + x − t, x − t) = (Je^{tB}h)(s, x). (15)

Also (15) implies

GJh = JBh, h ∈ D(B) and J^{−1}Gh = BJ^{−1}h, h ∈ D(G).

Therefore, σ(G) = σ(B) on L_{p}(ℝ × ℝ; E).

Note that G on L_{p}(ℝ × ℝ; E) has bounded inverse (G^{−1}f)(s) = G^{−1}f(s), s ∈ ℝ, provided G has bounded inverse G^{−1} on L_{p}(ℝ; E). Hence 1) implies 0 ∈ ρ(B).

Let us apply the part 3) ⇒ 1) of Theorem 2.5 to the semigroups {e^{tG}} and {e^{tB}}. To this end we note that (e^{tB}f)(s) = e^{tG}f(s − t) for f : ℝ → L_{p}(ℝ; E) : s ↦ h(s, ·). Hence 0 ∈ ρ(B) on L_{p}(ℝ; L_{p}(ℝ; E)) implies σ(e^{tG}) ∩ 𝕋 = ∅, t ≠ 0 on L_{p}(ℝ; E).

The proof for the case F = C_{0}(ℝ; E) is identical, and uses exactly the same semigroups and isometries on C_{0}(ℝ; F) = C_{0}(ℝ × ℝ; E). □

3.2. Hyperbolicity

Let {U(x, s)}_{x≥s} be an evolutionary family on a Banach space E. In this subsection we relate the (spectral) hyperbolicity of the evolutionary family and the hyperbolicity of the evolutionary semigroup {e^{tG}} on the space L_{p} = L_{p}(ℝ; E) in the case when the Banach space E is separable. The case F = C_{0}(ℝ; E) (without the assumption of separability) and the case of a Hilbert space E and p = 2 was considered in [27, 28].

Definition 3.2. An evolutionary family {U(x, s)}_{x≥s} is called (spectrally) hyperbolic if there exists a projection-valued, bounded function P : ℝ → B(E) such that the function ℝ ∋ x ↦ P (x)v ∈ E is continuous for every v ∈ E and for some constants M, λ > 0 and all x ≥ s the following conditions are fulfilled:

a) P (x)U(x, s) = U(x, s)P (s):

b) ∥U(x, s)v∥ ≤ Me^{−λ(x−s)}∥v∥ if v ∈ Im P (s), ∥U(x, s)v∥ ≥ M^{−1}e^{λ(x−s)}∥v∥ if v ∈ Ker P (s);

c) Im(U(x, s)| Ker P (s)) is dense in Ker P (x).

This notion generalizes the notion of exponential dichotomy (see, e.g., [9]) for the solutions of differential equation v′(x) = A(x)v(x), x ∈ ℝ, with bounded and continuous A : ℝ → B(E). In this case the evolutionary family (propagator) {U(x, s)}_{(x,s)∈ℝ^{2}} consists of invertible operators, the function (x, s) ↦ U(x, s) is norm-continuous, and P (·) from Definition 3.2 is also a bounded, norm-continuous function P : ℝ → B(E).

The second inequality in b) implies that the restriction U(x, s)| Ker P (s) is uniformly injective as an operator from Ker P (s) to Ker P (x) (that is ∥U(x, s)v∥ ≥ c∥v∥ for some c > 0 and all v ∈ Ker P (s)). Thus condition c) implies that U(x, s)| Ker P (s) is invertible as an operator from Ker P (s) to Ker P (x). Obviously, if U(x, s) is invertible in E or dim Ker P (x) ≤ d < ∞, condition c) in Definition 3.2 is redundant. The inequality dim Ker P (x) ≤ d < ∞ holds, for example, if the U(x, s) are compact operators in E ([R. Rau, private communication]). Generally, of course, b) does not imply c).

¿From now on we will assume that the Banach space E is separable. As we will see below, the spectral hyperbolicity of the evolution-

ary family {U(x, s)} is equivalent to the hyperbolicity σ(e^{tG}) ∩ 𝕋 = ∅, t > 0 of the evolutionary semigroup {e^{tG}}_{t≥0} in L_{p}(ℝ; E). Therefore, by Theorem 3.1 the spectral hyperbolicity of the evolutionary family {U(x, s)}_{x≥s} is also equivalent to the condition σ(G) ∩ iℝ = ∅. That is why we used the term spectral hyperbolicity in Definition 3.2. A remarkable observation by R. Rau [27] shows that generally the condition c) in Definition 3.2 cannot be dropped: there exists an evolutionary family that satisfies conditions a) and b) but σ(e^{tG}) ∩ 𝕋 ≠ ∅ for the associated evolutionary semigroup.

If the operator T = e^{G} is hyperbolic in L_{p}(ℝ; E), that is σ(T)∩𝕋 = ∅, we let P denote the Riesz projection for T, corresponding to the part σ(T) lying inside the unit disk 𝔻, and set Q = I − P. Lemma 3.2. If σ(e^{tG}) ∩ 𝕋 = ∅, t > 0, then P has a form (Pf)(x) = P (x)f(x), where P : ℝ → B(E) is a bounded projection-valued function such that the function ℝ ∋ x ↦ P (x)v ∈ E is (strongly) measurable for each v ∈ E.

Proof. We will show first that

χP = Pχ  (16)

for any scalar function χ ∈ L_{∞}(ℝ; ℝ). Then we will derive that (Pf)(x) = P (x)f(x) from (16).

Note that the decomposition L_{p}(ℝ; E) = Im P ⊕Im Q is T -invariant. Denote T_{P} = PT P = T | Im P, T_{Q} = QT Q = T | Im Q. Note that σ(T_{P}) ⊂ 𝔻, and T_{Q} is invertible with σ(T_{Q}^{−1}) ⊂ 𝔻 in Im Q. Hence for some λ, M > 0 and all n ∈ ℕ, the following inequalities hold:

∥T_{P}^{n}f∥_{L_{p}} ≤ Me^{−λn}∥f∥_{L_{p}}, f ∈ Im P,  (17)
∥T_{Q}^{n}f∥_{L_{p}} ≥ M^{−1}e^{λn}∥f∥_{L_{p}}, f ∈ Im Q.  (18)

We show first that Im P = {f ∈ L_{p}(ℝ; E) : T^{n}f → 0 as n → ∞}. Indeed, f ∈ Im P implies that T^{n}f → 0 by (17). Conversely, if T^{n}f → 0, then for f = Pf + Qf, the inequality (18) implies

∥Qf∥ ≤ Me^{−λn}∥T_{Q}^{n}Qf∥ ≤ Me^{−λn}{∥T^{n}f∥ + ∥T_{P}^{n}f∥} → 0,

and hence f ∈ Im P.

Consider on L_{p} = L_{p}(ℝ; E) the operator χ of multiplication by χ(·) ∈ L_{∞}(ℝ; ℝ). Note that (T^{n}χf)(x) = χ(x − n)(T^{n}f)(x). Hence for f ∈ Im P

∥T^{n}χf∥_{L_{p}} ≤ ∥χ∥_{L_{∞}}∥T^{n}f∥_{L_{p}} → 0 as n → ∞,

and so χf ∈ Im P. Thus, to prove (16), it suffices to show that f ∈ Im Q implies χf ∈ Im Q.

Fix f ∈ Im Q. Recall that T_{Q} is invertible on Im Q. Let f_{n} = T_{Q}^{−n}f, and define functions g_{n}(x) = χ(x + n)f_{n}(x), n = 0, 1, …. Decompose g_{n} = Pg_{n} + Qg_{n}. Since the decomposition L_{p}(ℝ; E) = Im P ⊕ Im Q is T -invariant, one has:

χf = T^{n}g_{n}, Pχf = T_{P}^{n}Pg_{n}, Qχf = T_{Q}^{n}Qg_{n}, n = 0, 1, ….

Now (17)–(18) imply:

∥Pχf∥ ≤ Me^{−λn}∥Pg_{n}∥ ≤ Me^{−λn}{∥g_{n}∥ + ∥Qg_{n}∥}
≤ Me^{−λn}{∥χ∥_{L_{∞}}∥f_{n}∥ + Me^{−λn}∥Qχf∥}
≤ Me^{−λn}{∥χ∥_{L_{∞}}Me^{−λn}∥f∥ + Me^{−λn}∥Qχf∥} → 0,

and hence χf ∈ Im Q. Thus (16) is proved.

In order to define P (·) such that (Pf)(x) = P (x)f(x), fix m ∈ ℤ and let χ_{m}(x) = 1 if x ∈ [m, m + 1), and χ_{m}(x) = 0 otherwise. Let {e_{n}}_{n∈ℤ} be a linearly independent set with dense span E_{0}.

Consider the function f ∈ L_{p}(ℝ; E), defined by f(x) = χ_{m}(x)e_{n}. Since P is bounded on L_{p}(ℝ; E), it is true that Pf ∈ L_{p}(ℝ; E). For x ∈ [m, m + 1) define a vector P (x)e_{n} ∈ E as P (x)e_{n} = (Pf)(x). For v = ∑_{n=1}^{k} d_{n}e_{n} ∈ E_{0} set P (x)v = ∑_{n=1}^{k} d_{n}P (x)e_{n}.

Let Δ be a measurable subset in [m, m + 1), and let χ_{Δ} be its characteristic function. Now (16) implies that

∫_{Δ} ∥P (x)v∥_{E}^{p} dx = ∫_{ℝ} ∥χ_{Δ}(Pχ_{m}v)(x)∥_{E}^{p} dx = ∫_{ℝ} ∥(Pχ_{Δ}v)(x)∥_{E}^{p} dx
≤ ∥P∥_{B(L_{p}(ℝ;E))}^{p} ∫_{Δ} ∥v∥_{E}^{p} dx.

Therefore, ∥P (x)v∥_{E} ≤ ∥P∥_{B(L_{p})}∥v∥_{E} for a.e. x ∈ ℝ and all v ∈ E_{0}. Hence, P (x) can be extended to a bounded operator on E, such that ∥P (x)∥ ≤ ∥P∥ for a.e. x ∈ ℝ. That the function x ↦ P (x)v is a measurable function for all v ∈ E_{0} (and, hence, for all v ∈ E) follows from the fact that the function x ↦ (Pf)(x) is measurable.

To show that (Pf)(x) = P (x)f(x) we can assume that f is a simple function, f = ∑ χ_{Δ_{k}}v_{k}, where Δ_{k} ⊂ [m_{k}, m_{k}+1), m_{k} ∈ ℤ, and v_{k} ∈ E_{0}. Then (16) implies:

(Pf)(x) = ∑ χ_{m_{k}}(x)(Pχ_{Δ_{k}}v_{k})(x) = ∑ χ_{Δ_{k}}(x)P (x)v_{k} = P (x)f(x).

□

Let us stress that the function P (·) above is only defined on a set ℝ_{0} ⊂ ℝ such that mes(ℝ \ ℝ_{0}) = 0. In Theorem 3.4 below we will establish that, in fact, this function P (·) can be extended to all of ℝ as a continuous function (in the strong operator topology in B(E)). To prove this fact we will need a few definitions and Lemma.

Let 𝔄 be the set of all operators a in L_{p}(ℝ; E) of the form (af)(x) = a(x)f(x), where the function a : ℝ → B(E) is bounded and the function ℝ ∋ x ↦ a(x)v ∈ E is continuous for each v ∈ E. For a ∈ 𝔄 and x ∈ ℝ let us define an operator π_{x}(a) on l_{p}(ℤ; E) by the rule

π_{x}(a) = diag{a(x + n)}_{n∈ℤ} : (v_{n})_{n∈ℤ} ↦ (a(x + n)v_{n})_{n∈ℤ}. (19)

Finally, let S : (v_{n})_{n∈ℤ} ↦ (v_{n−1})_{n∈ℤ} be a shift operator on l_{p}(ℤ; E).

Let us denote: T = e^{G}, a(x) = U(x, x − 1), (Rf)(x) = f(x − 1). Then T = aR. For λ ∈ 𝕋 set b = λI − aR, and for x ∈ ℝ set π_{x}(b) = λI − π_{x}(a)S. Lemma 3.3. If σ(T) ∩ 𝕋 = ∅ in L_{p}(ℝ; E) then σ(π_{x}(a)S) ∩ 𝕋 = ∅ in l_{p}(ℤ; E) for all x ∈ ℝ. Moreover, for all λ ∈ 𝕋 the following estimate holds:

∥[π_{x}(b)]^{−1}∥_{B(l_{p}(ℤ;E))} ≤ ∥b^{−1}∥_{B(L_{p}(ℝ;E))}, x ∈ ℝ.  (20)

Proof. First, for any ξ ∈ ℝ one has:

π_{x}(a)SL = e^{−iξ}Lπ_{x}(a)S,

where L is the operator (v_{n}) ↦ (e^{iξn}v_{n}). Hence σ(π_{x}(a)S) is invariant under rotations centered at the origin. Thus it suffices to prove the Lemma for the special case λ = 1, that is, to show that if b = I − aR is invertible in L_{p}(ℝ; E) then for each x_{0} ∈ ℝ that the operator π_{x_{0}}(b) = I − π_{x_{0}}(a)S is invertible in l_{p}(ℤ; E), and that estimate (22) is valid for this b and x = x_{0}.

Further, it suffices to prove the Lemma only for x_{0} = 0. Indeed, let us denote â(x) = a(x + x_{0}), x ∈ ℝ for any fixed x_{0} ∈ ℝ. Obviously,

π_{x_{0}}(I − aR) = I − π_{x_{0}}(a)S = I − π_{0}(â)S = π_{0}(I − âR).

Consider the invertible isometry J_{x_{0}} on L_{p}(ℝ; E), defined as (J_{x_{0}}f)(x) = f(x+x_{0}). Clearly, I−âR = J_{x_{0}}(I−aR)J_{x_{0}}^{−1}. Hence, the operator I−âR is invertible if and only if the operator b = I − aR is invertible, and ∥(I − âR)^{−1}∥ = ∥b^{−1}∥. Therefore, the estimate (20) for x = x_{0} follows from the estimate (20) for x = 0.

Thus our purpose is to prove if b = I − aR is invertible in L_{p}(ℝ; E), then the operator π_{0}(b) = I − π_{0}(a)S is invertible in l_{p}(ℤ; E), and

∥[π_{0}(b)]^{−1}∥_{B(l_{p}(ℤ;E))} ≤ ∥b^{−1}∥_{B(L_{p}(ℝ;E))}.  (21)

We first show that for any (v_{n}) ∈ l_{p}(ℤ; E) the following estimate holds:

∥(v_{n})∥_{l_{p}(ℤ;E)} ≤ ∥b^{−1}∥_{B(L_{p}(ℝ;E))}∥π_{0}(b)(v_{n})∥_{l_{p}(ℤ;E)}.  (22)

Let us fix a sequence (v_{n})_{n∈ℤ} ∈ l_{p}(ℤ; E), a natural number N > 1, and ϵ > 0.

Recall that the function ℝ ∋ x ↦ a(x)v ∈ E is continuous for each v ∈ E. Choose δ < 1 such that

∥[a(x + n) − a(n)]v_{n−1}∥_{E} < ϵ, ∀x ∈ [0, δ], n = −N, …, N. (23)

Define f ∈ L_{p}(ℝ; E) by f(x) = v_{n} for x ∈ [n, n + δ], |n| ≤ N, and f(x) = 0 otherwise. Since b is an invertible operator in L_{p}(ℝ; E), it follows that:

∥b^{−1}∥_{B(L_{p})}^{p}∥bf∥_{L_{p}}^{p} ≥ ∥f∥_{L_{p}}^{p} = ∑_{n=−N}^{N} ∫_{n}^{n+δ} ∥v_{n}∥^{p} dx = δ ∑_{n=−N}^{N} ∥v_{n}∥^{p}. (24)

On the other hand, using (23), one has:

∥bf∥_{L_{p}}^{p} = ∫_{ℝ} ∥f(x) − a(x)f(x − 1)∥_{E}^{p} dx

= ∑_{n=−N+1}^{N} ∫_{n}^{n+δ} ∥v_{n} − a(x)v_{n−1}∥_{E}^{p} dx + ∫_{−N}^{−N+δ} ∥v_{−N}∥_{E}^{p} dx

+ ∫_{N+1}^{N+1+δ} ∥a(x)v_{N}∥_{E}^{p} dx

≤ ∑_{n=−N+1}^{N} ∫_{0}^{δ} ∥v_{n} − a(n)v_{n−1} − [a(x + n) − a(n)]v_{n−1}∥_{E}^{p} dx

+ δ∥v_{−N}∥_{E}^{p} + δ max_{x∈ℝ}∥a(x)∥^{p} · ∥v_{N}∥_{E}^{p}

≤ δ ∑_{n=−N}^{N}(∥v_{n} − a(n)v_{n−1}∥_{E} + ϵ)^{p} + δ∥v_{−N}∥_{E}^{p} + δ max_{x∈ℝ}∥a(x)∥^{p}∥v_{N}∥_{E}^{p}.

Combining this inequality with (24), one has:

∑_{n=−N}^{N} ∥v_{n}∥^{p} ≤ ∥b^{−1}∥_{B(L_{p})}^{p}{ ∑_{n=−N}^{N}(∥v_{n} − a(n)v_{n−1}∥_{E} + ϵ)^{p}

+ ∥v_{−N}∥_{E}^{p} + max_{x∈ℝ}∥a(x)∥^{p}∥v_{N}∥_{E}^{p}}.

If ϵ → 0 and N → ∞, then

∥(v_{n})∥_{l_{p}(ℤ;E)} ≤ ∥b^{−1}∥_{B(L_{p})}( ∑_{n=−∞}^{∞} ∥v_{n} − a(n)v_{n−1}∥_{E}^{p})^{1/p}

and (22) is proved.

Note, that (22) is sufficient to show (21) provided the operator π_{0}(b) is invertible, and so it only remains to show that π_{0}(b) is an operator onto l_{p}(ℤ; E).

Fix any (v_{n}) ∈ l_{p}(ℤ; E) and let f(x) = U(x, n − 1)v_{n−1}, x ∈ [n − 1/2, n + 1/2), n ∈ ℤ. Property (iii) from the Definition 3.1 of the evolutionary family {U(x, s)} implies that

∥U(x, n − 1)∥ ≤ Ce^{β(x−n+1)} ≤ Ce^{\frac{3}{2} β} for x ∈ [n − 1/2, n + 1/2).

Hence f ∈ L_{p}(ℝ; E). Since the operator b, defined by (bg)(x) = g(x) − U(x, x−1)g(x−1), is invertible in L_{p}(ℝ; E), it follows that there exists a unique function g ∈ L_{p}(ℝ; E) such that

g(x) − U(x, x − 1)g(x − 1) = f(x)  (25)

for almost all x ∈ ℝ. Since

∥g∥_{L_{p}}^{p} = ∑_{n∈ℤ}∫_{−n−1/2}^{−n+1/2} ∥g(s)∥^{p} ds = ∫_{−1/2}^{1/2} ( ∑_{n∈ℤ}∥g(s + n)∥^{p}) ds < ∞,

the sequence (g(s + n))_{n∈ℤ} belongs to l_{p}(ℤ; E) for all s ∈ Ω for some subset Ω ⊂ (−1/2, 1/2) of full measure. For each s ∈ Ω, let us define a function h_{s} by the rule:

h_{s}(x) = \begin{cases}g(x), & if n − \frac{1}{2} ≤ x ≤ n + s, \\ U(x, n + s)g(n + s), & if n + s ≤ x < n + \frac{1}{2}, \end{cases} n ∈ ℤ.

Clearly, h_{s} ∈ L_{p}(ℝ; E) for each s ∈ Ω because (g(s + n))_{n∈ℤ} ∈ l_{p}(ℤ; E), and

∥U(x, n + s)∥ ≤ Ce^{β(1/2−s)} for x ∈ [n + s, n + 1/2).

We note that h_{s} is a solution of equation (25). Indeed, for x ∈ [n−1/2, n+s], equation (25) implies h_{s}(x)−U(x, x−1)h_{s}(x−1) = f(x). For x ∈ [n + s, n + 1/2), one has:

h_{s}(x) − U(x, x − 1)h_{s}(x − 1) =

U(x, n + s)[g(n + s) − U(n + s, n − 1 + s)g(n − 1 + s)] = U(x, n + s)f(n + s) = U(x, n + s)U(n + s, n − 1)v_{n−1} = f(x).

But equation (25) has only one solution g in L_{p}(ℝ; E). Hence g = h_{s} for all s ∈ Ω.

Fix s < 0, s ∈ Ω. The function h_{s}(·) is defined for x = n, n ∈ ℤ. Moreover, the sequence (h_{s}(n))_{n∈ℤ} belongs to l_{p}(ℤ; E). Indeed, h_{s}(n) = U(n, n + s)g(n + s), the sequence (g(n + s))_{n∈ℤ} ∈ l_{p}(ℤ; E), and ∥U(n, n + s)∥ ≤ Ce^{−βs} by (iii) from Definition 3.1.

Set u_{n} = h_{s}(n)+v_{n}. Since h_{s}(·) satisfies the equation (25) for x = n, n ∈ ℤ, we have:

π_{0}(b)(u_{n}) = u_{n} − U(n, n − 1)u_{n−1} = h_{s}(n) + v_{n} − U(n, n − 1)h_{s}(n − 1) − U(n, n − 1)v_{n−1} = v_{n} + h_{s}(n) − U(n, n − 1)h_{s}(n − 1) − f(n) = (v_{n}).

This identity proves that π_{0}(b) is an operator onto L_{p}(ℤ; E). □

Theorem 3.4. Let {U(x, s)}_{x≥s} be an evolutionary family in a separable Banach space E, and let {e^{tG}}_{t≥0} be the evolutionary semigroup acting on L_{p}(ℝ; E), 1 ≤ p < ∞ by the rule (e^{tG}f)(x) = U(x, x−t)f(x−t). The following conditions are equivalent:

1) {U(x, s)}_{x≥s} is (spectrally) hyperbolic in E;
2) σ(e^{tG}) ∩ 𝕋 = ∅, t ≠ 0, in L_{p}(ℝ; E);
3) 0 ∈ ρ(G) in L_{p}(ℝ; E).

Moreover, the Riesz projection P that corresponds to the part σ(e^{G})∩𝔻 of the spectrum of the hyperbolic operator e^{G} is related to a strongly continuous, projection-valued function P : ℝ → B(E) that satisfies Definition 3.2 by the formula (Pf)(x) = P (x)f(x), x ∈ ℝ, f ∈ L_{p}(ℝ; E).

Proof. 2) ⇔ 3) was proved in Theorem 3.1.

1) ⇒ 2). Without loss of generality assume t = 1. ¿From the projection-valued function P (·) from Definition 3.2, let us define a projection P in L_{p}(ℝ; E) by the rule (Pf)(x) = P (x)f(x). Denote Q = I − P. For T = e^{G}, condition a) of Definition 3.2 implies PT = T P. Set T_{P} = PT P and T_{Q} = QT Q. Then b) implies σ(T_{P}) ⊂ 𝔻 in Im P = {f ∈ L_{p}(ℝ; E) : f(x) ∈ Im P (x)}. Also b) and c) imply that the operator T_{Q}, which can be written as (T_{Q}f)(x) = Q(x)U(x, x−1)Q(x−1)f(x−1), is an invertible operator, and σ(T_{Q}^{−1}) ⊂ 𝔻 in Im Q = Ker P. Hence, σ(T) ∩ 𝕋 = ∅.

2) ⇒ 1). Let 𝔅 be a Banach algebra with a norm ∥ · ∥_{1} consisting of the operators d on L_{p}(ℝ; E) of the form

d = ∑_{k=−∞}^{∞} a_{k}R^{k}, a_{k} ∈ 𝔄, ∥d∥_{1} := ∑_{k=−∞}^{∞} ∥a_{k}∥_{B(L_{p}(ℝ;E))} < ∞.

We first show that if b = λ−T is invertible in L_{p}(ℝ; E) for all λ ∈ 𝕋, then (λ − T)^{−1} ∈ 𝔅. This fact will be proved in several steps.

First, without loss of generality let λ = 1. Since σ(T) ∩ 𝕋 = ∅, by Lemma 3.2 the Riesz projection P has a form (Pf)(x) = P (x)f(x), where the function ℝ_{0} ∋ x ↦ P (x)v ∈ E is a bounded, measurable (in the strong sense in E) function for each v ∈ E, where ℝ_{0} is a set of full measure in ℝ. Recall also that Q(x) = I − P (x). Decompose b = I − T = (P − T_{P}) ⊕ (Q − T_{Q}). Since σ(T_{P}) ⊂ 𝔻 and σ(T_{Q}^{−1}) ⊂ 𝔻, one has that b^{−1} = (P − T_{P})^{−1} ⊕ (Q − T_{Q})^{−1}, where

(P − T_{P})^{−1} = ∑_{k=0}^{∞} T_{P}^{k}; (Q − T_{Q})^{−1} = [−T_{Q}(Q − T_{Q}^{−1})]^{−1} = − ∑_{k=−∞}^{−1} T_{Q}^{k}. (26)

Notice that T_{Q}^{−1} = (QaRQ)^{−1} = (QaQ(· − 1)R)^{−1} = R^{−1}(QaQ(· − 1))^{−1} = [Q(· + 1)a(· + 1)Q(·)]^{−1}R^{−1}, and that T_{P} = aRP = aP (· − 1)R. Hence both operators T_{P}^{k} and T_{Q}^{k} can be written as a_{k}R^{k} for some multiplication operators a_{k}. The Neumann series in (26) converge absolutely. Therefore,

b^{−1} = ∑_{k=−∞}^{∞} a_{k}R^{k}, ∑_{k=−∞}^{∞} ∥a_{k}∥_{B(L_{p}(ℝ;E))} < ∞,  (27)

for each k ∈ ℤ the function a_{k} : ℝ_{0} → B(E) is bounded, and the function ℝ_{0} ∋ x ↦ a_{k}(x)v ∈ E is measurable for each v ∈ E.

Our next aim is to show that the a_{k} from (27) belong to 𝔄, that is, the function x ↦ a_{k}(x)v ∈ E extends to a continuous function from ℝ for each v ∈ E. To this end let us define for a_{k} from (27) and all x ∈ ℝ_{0} the operator π_{x}(a_{k}) in l_{p}(ℤ; E) as in (19). Denote:

π_{x}(b^{−1}) = ∑ π_{x}(a_{k})S^{k}, π_{x}(a_{k}) = diag{a_{k}(x + n)}_{n∈ℤ} for x ∈ ℝ_{0}. (28)

Identities bb^{−1} = b^{−1}b = I in L_{p}(ℝ; E) imply that π_{x}(b) · π_{x}(b^{−1}) = π_{x}(b^{−1}) · π_{x}(b) = I in l_{p}(ℤ; E) for x ∈ ℝ_{0}. Since the operator b is invertible in L_{p}(ℝ; E) by assumption, for each x ∈ ℝ the operator π_{x}(b) is invertible in l_{p}(ℤ; E) by Lemma 3.3. Hence

π_{x}(b^{−1}) = [π_{x}(b)]^{−1} for x ∈ ℝ_{0}.  (29)

Recall that the function ℝ ∋ x ↦ a(x)v ∈ E is continuous for each v ∈ E. Also, the function ℝ ∋ x ↦ ∥a(x)∥ ∈ ℝ_{+} is bounded. Hence for each (v_{n}) ∈ l_{p}(ℤ; E) the function ℝ ∋ x ↦ π_{x}(b)(v_{n}) ∈ l_{p}(ℤ; E) is continuous. By Lemma 3.3 ∥[π_{x}(b)]^{−1}∥_{B(l_{p})} are uniformly bounded for x ∈ ℝ. This implies that the function ℝ ∋ x ↦ [π_{x}(b)]^{−1}(v_{n}) ∈ l_{p}(ℤ; E) is continuous for each (v_{n}) ∈ l_{p}(ℤ; E). Indeed,

|| ([π_{x}(b)]^{−1} − [π_{x_{0}}(b)]^{−1}) (v_{n})||_{l_{p}(ℤ;E)} =
||[π_{x}(b)]^{−1} · [π_{x}(b) − π_{x_{0}}(b)] · [π_{x_{0}}(b)]^{−1}(v_{n})||_{l_{p}(ℤ;E)}

for any x, x_{0} ∈ ℝ, and the function ℝ ∋ x ↦ [π_{x}(b)]^{−1}(v_{n}) ∈ l_{p}(ℤ; E) is continuous at x = x_{0}.

Fix k_{0} ∈ ℤ, x_{0} ∈ ℝ, and v ∈ E. Define (ṽ_{n}) ∈ l_{p}(ℤ; E) as ṽ_{−k_{0}} = v and ṽ_{n} = 0 for n ≠ −k_{0}. Consider a sequence x_{m} → x_{0}, x_{m} ∈ ℝ_{0}. We will show that {a_{k_{0}}(x_{m})v}_{m∈ℕ} is a Cauchy sequence in E and will define

a_{k_{0}}(x_{0})v = lim_{m→∞} a_{k_{0}}(x_{m})v. Then the function ℝ ∋ x ↦ a_{k_{0}}(x)v ∈ E

becomes a continuous function, and a_{k_{0}} ∈ 𝔄.

Note, that the π_{x_{m}}(b^{−1}) are defined by the formula (28) since x_{m} ∈ ℝ_{0}. For the sequence (ṽ_{n}) one has the following estimate:

∥[π_{x_{m′}}(b^{−1}) − π_{x_{m′′}}(b^{−1})](ṽ_{n})∥_{l_{p}}^{p}

= ∑_{n∈ℤ} ∥ ∑_{k∈ℤ}[a_{k}(x_{m′} + n) − a_{k}(x_{m′′} + n)]ṽ_{n−k}∥_{E}^{p}

≥ ∥ ∑_{k}[a_{k}(x_{m′}) − a(x_{m′′})]ṽ_{−k}∥_{E}^{p}

= ∥[a_{k_{0}}(x_{m′}) − a_{k_{0}}(x_{m′′})]v∥_{E}^{p}.  (30)

Since x_{m} ∈ ℝ_{0}, formula (29) is applicable. Then the sequence

π_{x_{m}}(b^{−1})(ṽ_{n}) = [π_{x_{m}}(b)]^{−1}(ṽ_{n})

is a Cauchy sequence in l_{p}(ℤ; E) since the function ℝ ∋ x ↦ [π_{x}(b)]^{−1}(ṽ_{n}) ∈ l_{p}(ℤ; E) is continuous. In accordance with (30) the sequence {a_{k_{0}}(x_{m})v}_{m∈ℕ} is a Cauchy sequence in E, and a_{k_{0}} ∈ 𝔄.

Since the a_{k} from (27) are continuous, we have proved that (λI − T)^{−1} ∈ 𝔅 for all λ ∈ 𝕋.

The rest of the proof is standard (cf. [2, 18, 27]). Indeed, consider the absolutely convergent Fourier series f : λ ↦ λI − aRλ^{0} with the coefficients from 𝔅. For each λ ∈ 𝕋, the operator f(λ) = b is invertible in 𝔅. Hence the function 𝕋 ∋ λ ↦ [f(λ)]^{−1} ∈ 𝔅 is expandable (see, e.g., [5]) into an absolutely convergent Fourier series

[f(λ)]^{−1} = (λI − aR)^{−1} = ∑_{k=−∞}^{∞} d_{k}λ^{k}, ∑_{k} ∥d_{k}∥ < ∞, d_{k} ∈ 𝔅.

By the integral formula (see, e.g., [9, p. 20]) for the Riesz projection P, we conclude that P = d_{−1} ∈ 𝔅. Hence for some a_{k} ∈ 𝔄 one has that

P = ∑_{k=−∞}^{∞} a_{k}R^{k}, where ∑_{k=−∞}^{∞} ∥a_{k}∥ < ∞.

We will show that a_{k} = 0 for k ≠ 0, and P = a_{0} ∈ 𝔄. Indeed, by (16), χP = Pχ for any bounded continuous scalar function χ. Then

χP − Pχ = ∑_{k} a_{k}(·)[χ(·) − χ(· − k)]R^{k} = 0.

Then by picking x_{0} and χ such that χ(x_{0}) ≠ χ(x_{0} − k) for k ≠ 0, it follows that a_{k}(x_{0}) = 0, k ≠ 0. □

As we have mentioned above, for the space C_{0}(ℝ; E), part 1) ⇔ 2) of Theorem 3.4 was proved in [27].

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