Fourier Multipliers in Stability and Control Theory

Y. Latushkin^{*}

Department of Mathematics University of Missouri Columbia, MO 65211, USA E-mail: yuri@math.missouri.edu

and

F. Räbiger

Department of Mathematics University of Tübingen D-72076 Tübinhen, Germany E-mail: frra@michelangelo.mathematik.uni-tuebingen.de

Stability for strongly continuous semigroups on Banach spaces is described in terms of L^{p}–Fourier multiplier properties of the resolvent of the generator. A discrete version of this theory is developed, that lead to a description of the spectra of the semigroups in terms of discrete L^{p}–Fourier multipliers. Applications to stability of linear control systems on Banach spaces are given.

Key Words: Semigroups asymptotics, operator-valued Fourier multipliers, inputoutput stability

1. INTRODUCTION

A strongly continuous semigroup on a Hilbert space X is uniformly exponentially stable provided the resolvent of its generator A belongs to the Hardy class H^{∞} = H^{∞}({Re λ > 0}; L(X)) of bounded analytic functions with values in the set L(X) of bounded linear operators, see, e.g., [9, Thm.V.1.11] or [7, Thm.5.1.5]. This theorem, generally, does not hold if X is a Banach space, see, e.g., [20].

In this paper we show, in particular, that on Banach spaces the H^{∞}- condition should be complemented by the condition that the resolvent re-

*Supported by the Research Board and Research Council of the University of Missouri.

stricted to the imaginary axis is a Fourier multiplier for L^{p}(ℝ; X) for some p ∈ [1, ∞). Recall, that a function ξ : ℝ → L(X) is called an L^{p}(ℝ; X)– Fourier multiplier if the operator M_{ξ} : f ↦ F^{−1}ξ(·)Ff has a bounded extension from a dense in L^{p}(ℝ; X) set to all of L^{p}(ℝ; X); here F is the Fourier transform.

Various connections of stability and hyperbolicity for strongly continuous semigroups and Fourier multipliers were noticed in the work by M.A. Kaashoek and S. M. Verduyn Lunel [13], L. Weis [21], M. Hieber [12], W. Arendt and S. Bu [2], and in [4]. In this paper we continue this work in a different direction, and describe the uniform exponential stability and exponential stability for a semigroup in terms of L^{p}(ℝ; X)–Fourier multiplier properties of the resolvent of its generator.

Using Datko–Pazy Theorem and a characterization of stability in terms of convolution with the semigroup, we prove for m = 0, 1, 2, … a formula for the m-th growth bound ω_{m}(A) in terms of the m-th spectral bound s_{m}(A) and L^{p}(ℝ; X)–Fourier multiplier properties of the function ξ_{m}(s) = (w−A)^{−m}(is−A)^{−1}, s ∈ ℝ, w > max{ω_{0}(A), 0}. Recall, that ω_{m}(A) gives the best exponential rate of growth for the solutions of ẋ = Ax that start in the domain D(A^{m}), while s_{m}(A) gives the minimal abscissa for the half-plane where the resolvent grows along vertical lines not faster than m-th power (see the definitions below and more information in [20, 23]). Although the case m > 0 brings new effects, our proof works for all m = 0, 1, 2, …, and is different from [4, 12] where the case m = 0 was considered. Via Plancherel’s Theorem, we recover the well known Hilbert space formula ω_{m}(A) = s_{m}(A) due to L. Gearhart and J. Prüss for m = 0 and to G. Weiss for m = 1, 2, …, see [17, 20, 23], and also I. Herbst [10].

We also develop a discrete version of this theory, that lead to a description of the spectra of the semigroups in terms of discrete L^{p}(ℤ; X)–Fourier multipliers, cf. [14, Thm.2.3] and a recent interesting development in [2, Thm.3.6]. For m = 0, 1, 2, … the corresponding discrete multiplier operator gives a unique periodic mild solution for the inhomogeneous equations with inhomogeneity that takes values in the domain D(A^{m}). In particular, for m = 0 our result gives the spectral mapping theorem for evolution semigroups from [14] and the celebrated Prüss’ Theorem [17]. However, for m = 1, 2, … we also prove that the multiplier operator is bounded if and only if D(A^{m}) ⊂ Rg (e^{2π} − I), where Rg (·) denotes the range of an operator. We do not know if this last condition ever appeared in the literature before. Using a connection to Kaashoek-Verduyn Lunel theory [13], the results of the current paper were recently generalized for the case of hyperbolicity of semigroups and m ∈ ℝ_{+} in [15].

Finally, we use L^{p}(ℝ; X)–Fourier multipliers to describe input–output stability for linear control systems on Banach spaces in terms of transfer function. Our main result here is that the control system is p-input-output stable (that is, the input-output operator is L^{p}-bounded) if and only if the corresponding transfer function is bounded on {Re λ > 0} and its restriction on iℝ is an L^{p}-Fourier multiplier. It is well-known that in the Hilbert space setting input-output stability is equivalent to external stability; the latter is defined as the boundedness of the transfer function in {Re λ > 0}, see [6, 19, 25]. Thus, the lack of Plancherel’s Theorem explains why this classical theorem cannot be, generally, extended to the Banach space setting. Finally, we prove that, under stabilizability and detectability assumptions, the exponential stability ω_{1}(A) < 0 of the nominal system is equivalent to the fact that an appropriate modification of the transfer function is an L^{p}-Fourier multiplier.

2. SPECTRUM AND DISCRETE FOURIER MULTIPLIERS

The discrete Fourier transform F_{d} maps a Banach space valued function f ∈ L^{1}([0, 2π]; X) to the sequence (x_{k})_{k∈ℤ} of its Fourier coefficients. Conversely, if x̄ = (x_{k})_{k=−N}^{N} is a finitely supported sequence on the Banach space X, then its inverse Fourier transform F_{d}^{−1}x̄ is the trigonometric polynomial (F_{d}^{−1}x̄)(τ) = ∑_{k=−N}^{N} e^{ikτ} x_{k}, τ ∈ [0, 2π]. By P we denote the set of all trigonometric polynomials f(τ) = ∑_{k=−N}^{N} x_{k}e^{ikτ}, x_{k} ∈ X, τ ∈ [0, 2π], N ∈ ℕ, with coefficients in X. A function ξ : ℤ → L(X) is called a Fourier multiplier for L^{p}([0, 2π]; X), 1 ≤ p < ∞, if the operator F_{d}^{−1}ξ(·)F_{d} on P defined by

F_{d}^{−1}ξ(·)F_{d} : ∑_{k=−N}^{N} e^{ik(·)}x_{k} ↦ ∑_{k=−N}^{N} e^{ik(·)}ξ(k)x_{k}  (1)

can be extended to a bounded operator on L^{p}([0, 2π]; X). Applying the operator F_{d}^{−1}ξ(·)F_{d} to the trigonometric monomials we have that a Fourier multiplier ξ(·) for L^{p}([0, 2π]; X) always satisfies sup_{k∈ℤ} ∥ξ(k)∥ < ∞. If X is a Hilbert space, then by Parseval’s identify the latter condition is necessary and sufficient for ξ(·) to be a Fourier multiplies for L^{2}([0, 2π]; X).

Let A be the generator of a strongly continuous semigroup {e^{tA}}_{t≥0} on the Banach space X. Recall that, for f ∈ L^{1}([0, 2π]; X), the function u ∈ C([0, 2π]; X) is a mild solution of the inhomogeneous equation

u̇(t) = Au(t) + f(t), 0 ≤ t ≤ 2π, if  (E)_{f}

u(t) = e^{tA}u(0) + ∫_{0}^{t} e^{(t−s)A}f(s)ds, 0 ≤ t ≤ 2π.  (M)_{u,f}

In what follows we show that the existence of a unique periodic mild solution u of (E)_{f} for f belonging to a certain class of functions can be expressed in terms of Fourier multipliers.

For m ∈ {1, 2, … } we denote by X_{m} the domain D(A^{m}) equipped with the norm ∥x∥_{m} = ∑_{k=0}^{n} ∥A^{k}x∥ and let X_{0} = X. Denote by Rg B the range, by D(B) the domain, by σ(B) the spectrum, by ρ(B) the resolvent set, and by R(λ, B) = (λ − B)^{−1} the resolvent of an operator B. Let E([0, 2π]; X) be the space L^{p}([0, 2π]; X), 1 ≤ p < ∞, or the space Cper ([0, 2π]; X) of 2π-periodic continuous functions. If m ∈ {0, 1, 2, … } and iℤ ⊂ ρ(A), we let ξ_{m}(k) = A^{−m}(A − ik)^{−1}, k ∈ ℤ, and consider the operator L_{m} = F_{d}^{−1}ξ_{m}(·)F_{d} defined in (1).

Theorem 2.1. Let m ∈ {0, 1, 2 … }. Then the following assertions are equivalent:

(i) iℤ ⊆ ρ(A) and D(A^{m}) ⊆ Rg (e^{2πA} − I);

(ii) For each f ∈ L^{p}([0, 2π]; X_{m}), 1 ≤ p < ∞, there is a unique mild solution u ∈ Cper ([0, 2π]; X) of (E)_{f};

(iii) For each f ∈ Cper ([0, 2π]; X_{m}) there is a unique mild solution u ∈ Cper ([0, 2π]; X) of (E)_{f};

(iv) iℤ ⊆ ρ(A) and the function ξ_{m}(k) = A^{−m}(ik − A)^{−1}, k ∈ ℤ, is a Fourier multiplier for L^{p}([0, 2π]; X), 1 ≤ p < ∞.

(v) iℤ ⊆ ρ(A) and ξ_{m}(k) = A^{−m}(ik − A)^{−1}, k ∈ ℤ, is a Fourier multiplier for Cper ([0, 2π]; X).

Proof. (iii)⇒(i): Define R : Cper ([0, 2π]; X_{m}) → Cper ([0, 2π]; X) by Rf = u if f ∈ Cper ([0, 2π]; X_{m}) and u ∈ Cper ([0, 2π]; X) satisfy (M)_{u,f}. An elementary computation shows that the operator R is closed. By assumption and the closed Graph Theorem R is bounded.

We claim that for each k ∈ ℤ the operator A − ik is uniformly injective on X. For that consider the part A_{m} of A in X_{m} with domain D(A_{m}) = D(A^{m+1}) and fix x ∈ D(A_{m}). Define f(k) = e^{ikt}(A−ik)x and u(t) = e^{ikt}x, t ∈ [0, 2π]. Then f ∈ Cper ([0, 2π]; X_{m}) and u ∈ Cper ([0, 2π]; X), and (M)_{u,f} holds. Thus Rf = u, and hence ∥x∥ = ∥u(0)∥ ≤ ∥Rf∥ ≤ ∥R∥∥f∥ = ∥R∥∥(A − ik)x∥_{m}. Therefore, there exists a constant c > 0 such that c∥x∥_{m} ≤ ∥(A − ik)x∥_{m} for all x ∈ D(A_{m}). Since the operators A on X and A_{m} on X_{m} are similar the claim follows.

Next we show that D(A^{m}) ⊆ Rg (e^{2πA} − I). Fix x ∈ D(A^{m}). Choose γ ∈ C[0, 2π] such that γ(0) = 0, γ(2π) = 1 and ∫_{0}^{2π} γ(s)ds = 1. Define a function f ∈ Cper ([0, 2π]; X_{m}) by f(t) = e^{tA}[(1 − γ(t))e^{2πA}x + γ(t)x], t ∈ [0, 2π]. By assumption, there exists u ∈ Cper ([0, 2π]; X) such that (M)_{u,f} holds. Then

(e^{2πA} − I)u(0) = e^{2πA}u(0) − u(2π) = ∫_{0}^{2π} e^{(2π−s)A}f(s)ds
= e^{2πA} ∫_{0}^{2π}[(1 − γ(s))e^{2πA}x + γ(s)x]ds
= e^{2πA}x = (e^{2πA} − I)x + x.

Thus x ∈ Rg (e^{2πA} − I).

From D(A^{m}) ⊆ Rg (e^{2πA} − I) we obtain that e^{2πA} − I has dense range. The spectral mapping theorem for the residual spectrum implies that A−ik has dense range for all k ∈ ℤ. Together with the uniform injectivity of A−ik this shows iℤ ⊆ ρ(A).

(i)⇒(iv),(v): Consider the convolution operator K on E([0, 2π]; X) defined by

\begin{aligned}(Kf)(τ) & = ∫_{0}^{2π} e^{sA}f(τ − s[mod 2π])ds \\ & = ∫_{0}^{τ} e^{sA}f(τ − s)ds + ∫_{τ}^{2π} e^{sA}f(τ − s + 2π)ds, τ ∈ [0, 2π]. \end{aligned}

Clearly, the operator K is bounded. If f = ∑_{k=−N}^{N} x_{k}e^{ik(·)} ∈ P, then a simple computation shows

Kf = L_{0}[(e^{2πA} − I)f(·)] = (e^{2πA} − I)[L_{0}f](·).  (2)

From iℤ ⊆ ρ(A) and the spectral mapping theorem for the point spectrum it follows that e^{2πA} − I is injective. Then S = (e^{2πA} − I)^{−1} with domain D(S) = Rg (e^{2πA} − I) is a closed operator. Our assumption implies that SA^{−m} is defined on the whole of X. Since SA^{−m} is closed it is bounded. Thus, (2) yields L_{m}f = SA^{−m}(Kf)(·) for f ∈ P, and the assertions follow.

(iv),(v)⇒(i): From (2) we obtain

K(A^{−m}f(·)) = (e^{2πA} − I)[L_{m}f](·) = L_{m}[(e^{2πA} − I)f(·)]  (3)

for f ∈ P. Since L_{m} has a bounded extension to E([0, 2π]; X), which we denote again by L_{m}, equation (3) holds for all f ∈ E([0, 2π]; X). Now let y ∈ D(A^{m}). Choose β ∈ C^{1}[0, 2π] such that

0 ≤ β(·) ≤ 1, β|_{[0, \frac{2π}{3} ]} ≡ 1 and β|_{[\frac{4π}{3} ,2π]} ≡ 0.  (4)

Define g ∈ E([0, 2π]; X) by g(τ) = β′(τ)e^{τA}A^{m}y, τ ∈ [0, 2π]. Then (3) yields:

(e^{2πA} − I)(L_{m}g)(τ) = K(β′(·)e^{(·)A}y)(τ)

= ∫_{0}^{τ} e^{sA}β′(τ − s)e^{(τ−s)A}yds + ∫_{τ}^{2π} e^{sA}β′(τ − s + 2π)e^{(τ−s+2π)A}yds

= (β(0) − β(τ))e^{τA}y + (β(τ) − β(2π))e^{(2π+τ)A}y

= (1 − β(τ))e^{τA}y + β(τ)e^{(2π+τ)A}y.

If E = Cper ([0, 2π]; X) this implies

(e^{2πA} − I)(L_{m}g)(2π) = e^{2πA}y = (e^{2πA} − I)y + y,  (5)

and hence y ∈ Rg (e^{2πA} − I). If E = L^{p}([0, 2π]; X) then the same argument works as soon as the following claim is proved: If (iv) holds, then Rg (L_{m}) ⊆ C([0, 2π]; X). To prove the claim, let g ∈ L^{p}([0, 2π]; X). Choose a sequence {g_{k}}_{k=1}^{∞} in P such that g_{k} → g in L^{p}([0, 2π]; X). Then u_{k} := L_{m}g_{k} → L_{m}g in L^{p}([0, 2π]; X), and u_{k} ∈ P for all k ∈ ℕ. Thus {u_{k}}_{k=1}^{∞} converges pointwise on a set Ω ⊆ [0, 2π] of full measure. Since g_{k} and u_{k} are polynomials, a simple calculation shows that

u_{k}(t) = e^{tA}u_{k}(0) + ∫_{0}^{t} A^{−m}e^{sA}g_{k}(t − s)ds, t ∈ [0, 2π], and therefore

∫_{0}^{t} A^{−m}e^{sA}g_{k}(t − s)ds → ∫_{0}^{t} A^{−m}e^{sA}g(t − s)ds uniformly for t ∈ [0, 2π].

(6)

Now fix a ∈ Ω∩(0, 2π). Then {u_{k}(a)}_{k=1}^{∞} converges in X, and we conclude from (6) that {e^{aA}u_{k}(0)}_{k=1}^{∞} converges in X as well. Let x = lim_{k} e^{aA}u_{k}(0). Then e^{tA}u_{k}(0) = e^{(t−a)A}e^{aA}u_{k}(0) → e^{(t−a)A}x uniformly for t ∈ [a, 2π]. Thus, by (6), h(t) := lim_{k} u_{k}(t) exists uniformly for t ∈ [a, 2π] and h ∈ C([a, 2π]; X). Since u_{k} → L_{m}g in L^{p}([0, 2π]; X) we obtain L_{m}g = h a.e. on [a, 2π]. Since Ω has full measure we can choose a ∈ Ω arbitrarily close to 0 from which the claim follows.

(i)⇒(ii),(iii): Since iℤ ⊆ ρ(A), the spectral mapping theorem for the point spectrum implies that e^{2πA} − I is injective. Let S = (e^{2πA} − I)^{−1} with domain D(S) = Rg (e^{2πA} − I). By our assumption, X_{m} ⊆ D(S). Now let f ∈ E([0, 2π]; X_{m}). Then ∫_{0}^{2π} e^{sA}f(2π − s)ds ∈ X_{m}. Let

u(0) = −S ∫_{0}^{2π} e^{sA}f(2π − s)ds and u(t) = e^{tA}u(0) + ∫_{0}^{t} e^{sA}f(t − s)ds.

Note that u(2π) = e^{2πA}u(0) − (e^{2πA} − I)u(0) = u(0). Hence, we have that u ∈ Cper ([0, 2π]; X), and (M)_{u,f} holds. In order to show that u is the only function from Cper ([0, 2π]; X) satisfying (M)_{u,f} we may assume that f = 0. Then u(0) = u(2π) = e^{2πA}u(0) or, in other words, (e^{2πA} − I)u(0) = 0. The injectivity of e^{2πA} − I then implies u(0) = 0, and hence u ≡ 0. The implication (ii)⇒(iii) follows from Cper ([0, 2π]; X) ⊂ L^{p}([0, 2π]; X).

Remark 2. 1. If m = 0 then (i) in Theorem 2.1 is equivalent to the fact that 1 ∈ ρ(e^{2πA}). Therefore, the equivalence (i)⇔(iii) for m = 0 gives a theorem by J. Prüss [17]. Also, the equivalence (i)⇔(iv),(v) for m = 0 was proved in [14], see also [3, Thm.2.31]. Note, that though the term “Fourier multiplier” has never been mentioned in [14], the function β as in (4) is used in [14], see also [3, Lem. 2.29]. ◇

Remark 2. 2. For m = 0 the proof of the implication (iv), (v)⇒(i) leads to a representation of (e^{2πA} − I)^{−1} in terms of the operator L_{0}. In fact, if one of the conditions in Theorem 2.1 holds, then formula (5) implies:

(e^{2πA} − I)^{−1}y = [L_{0}(β′(·)e^{(·)A}y)](2π) − y, y ∈ X,

where β is defined in (4). ◇

Remark 2. 3. We can explicitly relate the operator L_{0} to the operator F that appears in Greiner’s Spectral Mapping Theorem (see for example [20, Thm. 2.2.1]). To define the operator F, we assume that iℤ ⊂ ρ(A), set

F_{n}x = n^{−1} ∑_{N=0}^{n−1} ∑_{k=−N}^{N}(A − ik)^{−1}x, n ∈ ℕ, and let F x = lim_{n→∞} F_{n}x

for those x ∈ X where the limit exists. In other words, F x is the Cesaro sum of the series ∑(A−ik)^{−1}x for such x where the sequence {(A−ik)^{−1}x} is Cesaro summable. Computing Fourier coefficients of the function f(τ) = e^{τA}, τ ∈ [0, 2π], x ∈ X, and using Fejer’s Theorem, we have the following formula:

π(e^{2πA} + I)x = F (e^{2πA} − I)x for each x ∈ X.  (7)

It shows that F is defined on the range of e^{2πA} − I which is dense in X due to iℤ ⊂ ρ(A). Formula (2) for a polynomial f ∈ P shows that (Kf)(τ) is in the range of e^{2πA}−I for each τ ∈ [0, 2π]. Using (7) and (2), for every f ∈ P we have that F Kf = πKf + 2πL_{0}f. Greiner’s Spectral Mapping Theorem (see [20, Thm. 2.2.1]) says that for m = 0 assertion (i) in Theorem 2.1 is equivalent to the fact that iℤ ⊂ ρ(A) and F has a bounded extension from the range of e^{2πA} − I to X. Therefore, if one of the equivalent statements in Theorem 2.1 holds, then

F y = π(e^{2πA} + I)[L_{0}(β′(·)e^{(·)A}y)(2π) − y] for y ∈ X,

where β(·) is the function defined in (4). ◇

Remark 2. 4. Finally, we note how the operator L_{0} is related to the generator of the evolution semigroup, {E^{t}}_{t≥0}, on L^{p}([0, 2π]; X), associated to the given semigroup {e^{tA}}_{t≥0} on X. We define the evolution semigroup as follows, cf. [3]:

(E^{t}f)(τ) = e^{tA}f(τ − t[mod 2π]), τ ∈ [0, 2π], t ≥ 0, f ∈ L^{p}([0, 2π]; X).

Let d denote the operator of differentiation (with the periodic boundary conditions) and let A, (Af)(τ) = Af(τ), denote the multiplication operator by A (with the maximal domain) on L^{p}([0, 2π]; X). Then {E^{t}}_{t≥0} is the product of two commuting semigroups: the translation semigroup {e^{−td}}_{t≥0}, (e^{−td}f)(τ) = f(τ − t[mod 2π]), and the multiplication semigroup {e^{tA}}_{t≥0}, (e^{tA}f)(τ) = e^{tA}f(τ). Therefore, the generator Γ of the semigroup {E^{t}}_{t≥0} is the closure of the operator −d + A with the domain D(d)∩D(A). Note that the set P_{A} of all trigonometric polynomials f with coefficients in D(A) is the core for Γ. Since

(Γf)(τ) = ([−d + A]f)(τ) = ∑_{k=−N}^{N}(A − ik)x_{k}e^{ikτ} , τ ∈ [0, 2π],

for each polynomial f(τ) = ∑ x_{k}e^{ikτ}, x_{k} ∈ D(A), it is immediate that ΓLf = f if f ∈ P and LΓf = f if f ∈ P_{A}, and therefore L_{0} = Γ^{−1} ∈ L(L^{p}([0, 2π]; X)) if and only if for m = 0 assertion (i) in Theorem 2.1 is satisfied (cf. [3, Thm.2.30]). ◇

3. STABILITY AND FOURIER MULTIPLIERS

Let m ∈ {0, 1, 2, …} and recall the definitions of m-th growth bound ω_{m}(A) and m-th spectral bound s_{m}(A) (see, e.g., [20, 23]):

ω_{m}(A) = inf {ω ∈ ℝ : ∃M = M(ω) so that
∥e^{tA}x∥ ≤ Me^{ωt}∥x∥_{m}, x ∈ X_{m}, t ≥ 0},

(8)

s_{m}(A) = inf {ω ∈ ℝ : {Re λ > ω} ⊂ ρ(A) and

sup_{Re λ>ω}(1 + |λ|)^{−m}∥R(λ, A)∥ < ∞}.

Fix w > max{ω_{0}(A), 0}. Note that induction and the resolvent identity yield:

R^{m}(w, A)R(λ, A) = ∑_{k=0}^{m−1}(−1)^{k}(λ − w)^{−(k+1)}R^{m−k}(w, A)  (9)
+ (−1)^{m}(λ − w)^{−m}R(λ, A).

For m ∈ {0, 1, 2, …} elementary estimates and (9) imply:

s_{m}(A) = inf {ω ∈ ℝ :{Re λ > ω} ⊂ ρ(A)  (10)

and sup_{Re λ>ω} ∥(w − A)^{−m}R(λ, A)∥ < ∞}.

For λ ∈ ℂ we denote ξ_{m,λ}(s) = (w − A)^{−m}(is + λ − A)^{−1} and ξ_{m}(s) = ξ_{m,0}(s), s ∈ ℝ, and define operators 𝔾_{λ} and ℍ_{m,λ} as follows:

\begin{aligned}(𝔾_{λ}f)(t) & = ∫_{0}^{t} e^{(t−τ)(A−λ)}f(τ)dτ, \\ (ℍ_{m,λ}f)(t) & = (w − A)^{−m} ∫_{0}^{t} e^{(t−τ)(A−λ)}f(τ)dτ, t ≥ 0. \end{aligned}

Let 𝔾 = 𝔾_{0} and ℍ_{m} = ℍ_{m,0}. Note that 𝔾_{λ} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)) if and only if ℍ_{m,λ} ∈ L(L^{p}(ℝ_{+}; X)).

For m = 0, by a classical Datko–Pazy Theorem, ω_{0}(A) < 0 if and only if the orbit {e^{tA}x : t ∈ ℝ_{+}} belongs to L^{p}(ℝ_{+}; X), 1 ≤ p < ∞, for each x ∈ X, see, e. g., [16, Thm.4.41]. In turn, by the Datko-van Neerven Theorem this happens if and only if 𝔾 ∈ L(L^{p}(ℝ_{+}; X)), see [8, 20] or [3, Thm.1.44] and [12] for shorter proofs. A simple analytical continuation argument in Lemma 3.2 below shows that 𝔾 ∈ L(L^{p}(ℝ_{+}; X)) if and only if s_{0}(A) < 0 and f ↦ F^{−1}ξ_{0}(·)Ff is a bounded operator on L^{p}(ℝ; X). Note that for p = 2 and Hilbert space X this implies s_{0}(A) = ω_{0}(A).

For m = 1, 2, … the situation is quite different. For instance, Example 3.1 below shows that condition 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)), generally, does not imply the strict inequality ω_{m}(A) < 0. Nevertheless, in Theorem 3.2 we derive a formula for ω_{m}(A) in terms of s_{m}(A) and Fourier multiplier properties of the function ξ_{m}(·).

Lemma 3.1. If 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)), then ω_{m}(A) ≤ 0.

Proof. Fix x ∈ X_{m} and let f_{x}(t) = e^{tA}x for t ∈ [0, 1] and f_{x}(t) = 0 for t > 1. Then f_{x} ∈ L^{p}(ℝ_{+}; X_{m}) and for t ≥ 1 we have:

(𝔾f_{x})(t) = ∫_{0}^{1} e^{(t−τ)A}e^{τA}xdτ = e^{tA}x.

Thus the orbit t ↦ e^{tA}x belongs to L^{p}(ℝ_{+}; X) and by [20, Lemma 3.1.9] this implies lim_{t→∞} ∥e^{tA}x∥ = 0. Hence

ω(x) := inf{ω ∈ ℝ : sup_{t≥0} ∥e^{−ωt}e^{tA}x∥ < ∞} ≤ 0.

Since ω_{m}(A) = sup{ω(x) : x ∈ X_{m}}, we obtain ω_{m}(A) ≤ 0.

Theorem 3.1. The following assertions are equivalent:

(i) ω_{m}(A) < 0;

(ii) there exists ϵ > 0 such that 𝔾_{−ϵ} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)).

Proof. (i)⇒(ii). We prove that ω_{m}(A) < 0 implies 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)). Using this for A + ϵ instead of A, we have that (i) implies (ii) for all ϵ ∈ (0, −ω_{m}(A)).

Assume ω_{m}(A) < 0 and fix ω ∈ (ω_{m}(A), 0). If f ∈ L^{p}(ℝ_{+}; X_{m}) then let f̃(t) = f(t) for t ≥ 0 and f̃(t) = 0 for t < 0. By the definition of ω_{m}(A), we have for each t ≥ 0:

∥(𝔾f)(t)∥ ≤ ∫_{0}^{t} ∥e^{τA}f(t − τ)∥ds ≤ M(ω) ∫_{0}^{∞} e^{ωτ} ∥f̃(t − τ)∥_{m}dτ.

Using Minkowski’s inequality,

∥𝔾f∥_{L^{p}(ℝ_{+};X)} = [∫_{0}^{∞} ∥(𝔾f)(t)∥^{p}dt]^{\frac{1}{p}}

≤ M(ω) [∫_{0}^{∞} (∫_{0}^{∞} e^{ωτ} ∥f̃(t − τ)∥_{m}dτ)^{p} dt]^{\frac{1}{p}}

= M(ω) ∥∫_{0}^{∞} e^{ωτ} ∥f̃(· − τ)∥_{m}dτ∥_{L^{p}(ℝ_{+})}

≤ M(ω) ∫_{0}^{∞} ∥e^{ωτ} ∥f̃(· − τ)∥_{m}∥_{L^{p}(ℝ_{+})}dτ

= M(ω) ∫_{0}^{∞} e^{ωτ} dτ∥f∥_{L^{p}(ℝ_{+};X_{m})} = \frac{M(ω)}{|ω|}∥f∥_{L^{p}(ℝ_{+};X_{m})}.

(ii)⇒(i). This follows from Lemma 3.1 applied to A + ϵ instead of A.

The following example shows that, unlike the case of m = 0 in Datko-van Neerven Theorem [20, Thm.3.3.1], ϵ > 0 in condition (ii) of Theorem 3.1 cannot be dropped for m = 1.

Example 3.1. On X = ℓ^{1}(ℤ_{+}) we define the generator A : (x_{n})_{n∈ℤ_{+}} ↦ ((in − n^{−1})x_{n})_{n∈ℤ_{+}}, with D(A) = {(x_{n}) ∈ ℓ^{1}(ℤ_{+}) : (nx_{n}) ∈ ℓ^{1}(ℤ_{+})}, of the strongly continuous semigroup given by e^{tA} : (x_{n}) ↦ (e^{int}e^{−n^{−1}t}x_{n}). If \overline{x} = (δ_{kn})_{n∈ℤ_{+}}, the standard ort, then e^{tA}\overline{x} = (e^{ikt}e^{−k^{−1}t}δ_{kn})_{n∈ℤ_{+}} and ω(\overline{x}) = −k^{−1}. Therefore, ω_{1}(A) = sup\{ω(x) : x ∈ D(A)\} ≥ 0. To see that ω_{1}(A) ≤ 0, we check that the orbit t ↦ e^{tA}\overline{x} belongs to L^{1}(ℝ_{+}; X) for each \overline{x} ∈ D(A). Indeed, if \overline{x} = (x_{n})_{n∈ℤ_{+}} ∈ D(A) then

∫_{0}^{∞} ∥e^{tA}\overline{x}∥_{ℓ^{1}(ℤ_{+})}dt = ∑_{n≥0} ∫_{0}^{∞} e^{−n^{−1}t}|x_{n}|dt = ∑_{n≥0} n|x_{n}| < ∞.

Thus, we have ω_{1}(A) = 0.

However, the operator ℍ_{1} = A^{−1}𝔾 is bounded on L^{1}(ℝ_{+}; X). Indeed, for f(·) = (f(·, n))_{n∈ℤ_{+}} ∈ L^{1}(ℝ_{+}; ℓ^{1}(ℤ_{+})) we have:

(ℍ_{1}f)(t, n) = ∫_{0}^{t} \frac{n}{in^{2} − 1}e^{inτ} e^{−n^{−1}τ} f(t − τ, n)dτ, n ∈ ℤ_{+}, t ∈ ℝ_{+}.

Therefore,

∥ℍ_{1}f∥_{L^{1}(ℝ_{+};ℓ^{1}(ℤ_{+}))} = ∫_{0}^{∞} ∑_{n≥0} |(ℍ_{1}f)(t, n)|dt

≤ ∑_{n≥0} ∫_{0}^{∞} ∫_{0}^{t} |\frac{n}{in^{2} − 1}| e^{−n^{−1}τ} |f(t − τ, n)|dτdt

= ∑_{n≥0} |\frac{n}{in^{2} − 1}| ∫_{0}^{∞} e^{−n^{−1}τ} ∫_{τ}^{∞} |f(t − τ, n)|dtdτ

= ∑_{n≥0} |\frac{n}{in^{2} − 1}| ∫_{0}^{∞} e^{−n^{−1}τ} dτ∥f(·, n)∥_{L^{1}(ℝ_{+})}

≤ ∑_{n≥0} ∥f(·, n)∥_{L^{1}(ℝ_{+})} = ∥f∥_{L^{1}(ℝ_{+};ℓ^{1}(ℤ_{+}))}.

Thus, 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{1}); L^{p}(ℝ_{+}; X)). ◇

Remark 3. 1. By a theorem of J. van Neerven [20, Thm.4.6.4], if {e^{tA}}_{t≥0} is uniformly bounded and ω_{1}(A) ≥ 0, then there exists ε > 0 such that for each t > 0 there is an x ∈ X with ∥x∥_{X} = 1 so that

meas {τ ∈ ℝ_{+} : ∥e^{τA}x∥ ≥ ε} ≥ t.  (11)

An open question in [20] was whether this theorem could be improved to state that x in the assertion above could be taken in D(A) with ∥x∥_{D(A)} = 1. As is was noticed by J. van Neerven, our Example 3.1 shows that the answer to this question is negative. Indeed, suppose that 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{1}); L^{p}(ℝ_{+}; X)). A calculation shows that the L^{p}(ℝ_{+}; X)–norms of the orbits e^{(·)A}x with x ∈ D(A) and ∥x∥_{D(A)} = 1 are uniformly bounded. This contradicts the statement that for some ε > 0 and each t > 0 there is an x ∈ D(A) with ∥x∥_{D(A)} = 1 so that (11) holds. ◇

Next, we will characterize ω_{m}(A) in terms of Fourier multiplier properties of ξ_{m}. To recall the definition of the operator-valued Fourier multiplier for L^{p}(ℝ; X), see, e.g., [1, 11], we let S(ℝ) denote the Schwartz class of rapidly decaying smooth scalar functions, and let S(ℝ; X) denote the Schwartz class of X-valued functions. If x ∈ X and φ ∈ S(ℝ), we let x ⊗ φ ∈ S(ℝ; X) denote the function (x⊗φ)(t) = φ(t)x, t ∈ ℝ. Note that the lineal D_{S} = lin {x ⊗ φ : x ∈ X, φ ∈ S(ℝ)} is dense in L^{p}(ℝ; X). Let F and F^{−1} denote the Fourier transform and its inverse:

(Ff)(s) = ∫_{−∞}^{∞} e^{−ist}f(t) dt, (F^{−1}f)(t) = \frac{1}{2π} ∫_{−∞}^{∞} e^{ist}f(s) ds.

Definition 3.1. A function ξ : ℝ → L(X) is called a Fourier multiplier for L^{p}(ℝ; X) if the following assertions hold:

(a) ξ ∈ L^{∞}(ℝ; L(X));

(b) there exists a constant c > 0 such that

∥F^{−1}ξ(·)Ff∥_{L^{p}(ℝ;X)} ≤ c∥f∥_{L^{p}(ℝ;X)} for all f ∈ D_{S}.

Note that if (a) holds and f = x⊗φ ∈ D_{S} then Ff ∈ D_{S} and ξ(·)Ff(·) ∈ L^{1}(ℝ; X). Therefore, F^{−1}ξ(·)Ff ∈ C_{0}(ℝ; X). Define M_{ξ}f := F^{−1}ξ(·)Ff. If (b) holds, then M_{ξ}f ∈ L^{p}(ℝ; X) and, moreover, M_{ξ} : D_{S} → L^{p}(ℝ; X) can be extended to a bounded operator on L^{p}(ℝ; X). Thus, equivalently to Definition 3.1, a function ξ ∈ L^{∞}(ℝ; L(X)) is an Fourier multiplier for L^{p}(ℝ; X) if and only if M_{ξ} maps D_{S} to L^{p}(ℝ; X) and can be extended from D_{S} to L^{p}(ℝ; X) as a bounded operator.

Lemma 3.2. Let m ∈ {0, 1, 2, …} and assume that

sup_{Re λ≥0} ∥(w − A)^{−m}R(λ, A)∥ < ∞.  (12)

The following assertions are equivalent:

(i) 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X));

(ii) ξ_{m} is a Fourier multiplier on L^{p}(ℝ; X).

Moreover, if either (i) or (ii) holds, then ℍ_{m} = M_{ξ_{m}}.

Proof. Fix x ∈ X and φ ∈ S(ℝ). If Re λ > max{ω_{0}(A), 0}, then 𝔾_{λ} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)) and hence ℍ_{m,λ} ∈ L(L^{p}(ℝ_{+}; X)). Moreover, an easy application of Fubini’s theorem yields Fℍ_{m,λ}φ⊗x = ξ_{m,λ}(·)Fφ⊗x, and hence

ℍ_{m,λ}φ ⊗ x = F^{−1}ξ_{m,λ}(·)Fφ ⊗ x for Re λ > max{ω_{0}(A), 0}.  (13)

Now fix t ∈ ℝ. For Re λ ≥ 0 set

L(λ) = \begin{cases}(w − A)^{−m} ∫_{0}^{t} e^{−λ(t−τ)}e^{(t−τ)A}(φ ⊗ x)(τ)dτ & if t ≥ 0, \\ 0 & if t < 0, \end{cases}
= (H_{m,λ}φ ⊗ x)(t)

and

R(λ) = (2π)^{−1} ∫_{−∞}^{∞} e^{ist}ξ_{m,λ}(s)x(Fφ)(s)ds = (F^{−1}ξ_{m,λ}(·)Fφ ⊗ x)(t).

Due to the assumption, the functions L(·) and R(·) are continuous on {Re λ ≥ 0} and an easy application of Morera’s theorem shows that they are analytic on {Re λ > 0}. Hence, by (13) and the analyticity of L and R we have L(0) = R(0), i.e.

(ℍ_{m}φ ⊗ x)(t) = (F^{−1}ξ_{m}Fφ ⊗ x)(t).

From this and a density argument it follows easily that ℍ_{m} ∈ L(L^{p}(ℝ_{+}; X)) if and only if ξ_{m} is a Fourier multiplier for L^{p}(ℝ; X), and that ℍ_{m} = M_{ξ_{m}}.

Proposition 3.1. If 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)), m ∈ {0, 1, 2, …}, then the following holds:

(a) σ(A) ⊆ {Re λ < 0}; (b) sup_{s∈ℝ} ∥(w − A)^{−m}(is − A)^{−1}∥ < ∞.

Proof. (a). Assume that the assertion is not true. Then there exists λ ∈ {Re λ ≥ 0} belonging to the approximate point spectrum of A. Choose (x_{n}) in D(A) such that ∥x_{n}∥ = 1 and lim_{n}(λ − A)x_{n} = 0. Then

lim_{n} ((w − λ)^{−m} − (w − A)^{−m})x_{n} = 0.  (14)

Moreover,

lim_{n} [e^{t(A−λ)}x_{n} − x_{n}] = lim_{n} ∫_{0}^{t} e^{s(A−λ)}(A − λ)x_{n}ds = 0

uniformly for t in compact subsets in ℝ_{+}. By passing to a subsequence we may assume that

lim_{n} sup_{t∈[0,n]} ∥(e^{tA} − e^{λt})x_{n}∥ = 0.  (15)

By our assumption ℍ_{m} = 𝔾(w − A)^{−m} ∈ L(L^{p}(ℝ_{+}; X)). Set f_{n} = χ_{[0,1]}e^{λ·}x_{n}, n ∈ ℕ. Note that sup_{n} ∥f_{n}∥_{p} < ∞ and for t ≥ 1 we have

(ℍ_{m}f_{n})(t) = (w − A)^{−m} ∫_{0}^{t} e^{(t−s)A}f_{n}(s)ds

= (w − A)^{−m} ∫_{0}^{1} e^{λs}e^{(t−s)A}x_{n}ds

= (w − A)^{−m} ∫_{0}^{1} e^{λs}(e^{(t−s)A} − e^{λ(t−s)})x_{n}ds + e^{λt}(w − A)^{−m}x_{n}.

By (15) we obtain lim_{n} sup_{t∈[1,n]} ∥(ℍ_{m}f_{n})(t) − e^{λt}(w − A)^{−m}x_{n}∥ = 0. Thus,

by (14) we have

\overline{lim_{n}}∥ℍ_{m}f_{n}∥_{p} ≥ \overline{lim_{n}} (∫_{1}^{r} ∥e^{λs}(w − A)^{−m}x_{n}∥^{p}ds)^{1/p}

≥ \overline{lim_{n}} (∫_{1}^{r} ∥(w − λ)^{−m}e^{λs}x_{n}∥^{p} ds)^{\frac{1}{p}}

= \overline{lim_{n}}∥x_{n}∥ (∫_{1}^{r} |w − λ|^{−mp}e^{spRe λ}ds)^{\frac{1}{p}}

for r ≥ 1. Since Re λ ≥ 0 this implies \overline{lim_{n}}∥ℍ_{m}f_{n}∥_{p} = ∞ which is a contradiction.

(b). Assume that the assertion does not hold. There exist s_{n} ∈ ℝ and x_{n} ∈ D(A^{m+1}), n ∈ ℕ, such that

∥x_{n}∥ = 1 and lim_{n} ∥(w − A)^{m}(is_{n} − A)x_{n}∥ = 0.  (16)

If φ ∈ C_{c}^{1}(0, ∞), x ∈ D(A), then integration by parts shows that

𝔾(φ′ ⊗ x − φ ⊗ Ax) = φ ⊗ x.  (17)

Now choose 0 ≠ ψ ∈ C_{c}^{1}(0, ∞). Set a_{n} = max{n∥(w − A)x_{n}∥, n} and φ_{n}(t) = e^{is_{n}t}ψ (\frac{t}{a_{n}} ), n ∈ ℕ, t ≥ 0. From (17) we obtain

𝔾((φ′_{n} − is_{n}φ_{n}) ⊗ x_{n} − φ_{n} ⊗ (is_{n} − A)x_{n}) = φ_{n} ⊗ x_{n}.

Since ℍ_{m} = 𝔾(w − A)^{−m} ∈ L(L^{p}(ℝ_{+}; X)) we obtain

∥ψ∥_{L^{p}(ℝ_{+})}an^{\frac{1}{p}} = ∥φ_{n} ⊗ x_{n}∥_{L^{p}(ℝ_{+};X)}
≤ ∥ℍ_{m}∥(∥(φ′_{n} − is_{n}φ_{n}) ⊗ (w − A)^{m}x_{n}∥_{L^{p}(ℝ_{+};X)}
+ ∥φ_{n} ⊗ (w − A)^{m}(is_{n} − A)x_{n}∥_{L^{p}(ℝ_{+};X)})
= ∥ℍ_{m}∥(∥ψ′∥_{L^{p}(ℝ_{+})}(a_{n})^{−1+ \frac{1}{p}} ∥(w − A)^{m}x_{n}∥
+ ∥ψ∥_{L^{p}(ℝ_{+})}(a_{n})^{\frac{1}{p}} ∥(w − A)^{m}(is_{n} − A)x_{n}∥).

By the choice of a_{n} this implies

0 < ∥ψ∥_{L^{p}(ℝ_{+})} ≤ ∥ℍ_{m}∥(∥ψ′∥_{L^{p}(ℝ_{+})}n^{−1}+∥ψ∥_{L^{p}(ℝ_{+})}∥(w−A)^{m}(is_{n}−A)x_{n}∥).

Due to (16), the right hand side of the above inequality tends to 0 as n → ∞ which is a contradiction.

Remark 3. 2. If m ∈ {0, 1, 2, …} and 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)), then s(A) ≤ 0, where s(A) = sup{Re λ : λ ∈ σ(A)} is the spectral bound for A. Unless m = 0, we cannot conclude that s(A) < 0 (see Example 3.1). Since s_{m}(A) ≤ ω_{m}(A), from Lemma 3.1 we have that s_{m}(A) ≤ 0 provided 𝔾 ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)). By the Datko-van Neerven Theorem [20, Thm.3.3.1] for m = 0 we have that 𝔾 ∈ L(L^{p}(ℝ_{+}; X)) implies s_{0}(A)(≤ ω_{0}(A)) < 0. ◇

For m = 0, using the resolvent identity, we have that if s_{0}(A) ≤ 0 and sup_{s∈ℝ} ∥(is − A)^{−1}∥ < ∞, then s_{0}(A) < 0. Thus, the resolvent R(·, A) satisfies assumption (12). Recall that if ξ_{0} is a Fourier multiplier, then sup_{s∈ℝ} ∥(is − A)^{−1}∥ < ∞.

Corollary 3.1. ω_{0}(A) < 0 if and only if σ(A) ⊂ {Re λ < 0}, s_{0}(A) ≤ 0, and ξ_{0} is a Fourier multiplier for L^{p}(ℝ; X), 1 ≤ p < ∞.

Theorem 3.2. If m ∈ {0, 1, 2, …}, then

ω_{m}(A) = inf{ω > s_{m}(A) : ξ_{m,ω} is a Fourier multiplier for L^{p}(ℝ; X)}.

(18)

Proof. Denote the right-hand side of (18) by ω_{ξ} and set ω_{𝔾} := inf{ω ∈ ℝ : 𝔾_{ω} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X))}. We prove the following three inequalities.

ω_{m}(A) ≤ ω_{𝔾}. If ω > ω_{𝔾} then 𝔾_{ω} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)) and, by Lemma 3.1 for A − ω we have ω_{m}(A − ω) = ω_{m}(A) − ω ≤ 0.

ω_{𝔾} ≤ ω_{ξ}. If ω > ω_{ξ} then s_{m}(A − ω) < 0. Using (10), we have:

sup_{Re λ≥0} ∥(w − A)^{−m}R(λ, A − ω)∥ < ∞.  (19)

Apply the implication (ii)⇒(i) of Lemma 3.2 to A − ω instead of A. Now 𝔾_{ω} ∈ L(L^{p}(ℝ_{+}; X_{m}); L^{p}(ℝ_{+}; X)) since ξ_{m,ω} is a Fourier multiplier for L^{p}(R; X). Thus, ω ≥ ω_{𝔾}.

ω_{ξ} ≤ ω_{m}(A). If ω > ω_{m}(A), then s_{m}(A − ω) < 0 and (19) holds. Also, 𝔾_{ω} ∈ L(L^{p}(ℝ_{+}; X_{m}), L^{p}(ℝ_{+}; X)) by the proof of the implication (i)⇒(ii) in Theorem 3.1. Apply the implication (i)⇒(ii) of Lemma 3.2 to A − ω instead of A. Then ξ_{m,ω} is a Fourier multiplier for L^{p}(ℝ; X), and ω ≥ ω_{ξ}.

Remark 3. 3. If X is a Hilbert space, then Plancherel’s Theorem and Theorem 3.2 imply the well known results ω_{0}(A) = s_{0}(A) by Gearhart-Prüss and ω_{m}(A) = s_{m}(A) by G. Weiss (see [3, 10, 20, 24] and the literature therein). ◇

Remark 3. 4. For λ ∈ ℂ and m ∈ {0, 1, 2, …} we denote η_{m,λ}(s) = (w − is)^{−m}(is+λ−A)^{−1} and abbreviate η_{m}(s) = η_{m,0}(s), s ∈ ℝ. We claim that ξ_{m,ω} in Lemma 3.2 and Theorem 3.2 can be replaced by η_{m,ω}. To prove the claim, it suffices to see that ξ_{m} and η_{m} are Fourier multipliers for L^{p}(ℝ; X) only simultaneously. Passing to the functions x ⊗ φ, x ∈ X, φ ∈ S(ℝ), we have that the functions s ↦ (is − w)^{−(k+1)}R^{m−k}(w, A), k = 0, …, m − 1, are Fourier multipliers on L^{p}(ℝ; X). Using (9), the claim follows. ◇

There are many sufficient conditions that ensure that a function ξ : ℝ → L(X) is a Fourier multiplier for L^{p}(ℝ; X) (cf. Remark 4.1 below.)

Here we recall a convenient result called “Mikhlin+ε” Theorem (see [1, Cor.4.4] or [11, Prop.6.4]): Assume that ξ ∈ W_{2}^{∞}(ℝ; L(X)), the Sobolev space. If there exists an ε > 0 such that for the derivatives ξ^{(n)} we have sup_{s∈ℝ}(1 + |s|^{n+ε})∥ξ^{(n)}(s)∥_{L(X)} < ∞ for n = 0, 1, 2, then ξ is a Fourier multiplier for L^{p}(ℝ; X). This result, Theorem 3.2, and Remark 3.4 imply the following.

Corollary 3.2. Assume s_{0}(A) < 0. If there exists an ε < 0 such that

sup_{s∈ℝ}(1 + |s|^{n+ε})∥(is − A)^{−(n+1)}∥_{L(X)} < ∞ for n = 0, 1, 2,

then ω_{0}(A) < 0. Assume s_{1}(A) < 0. If there exist ϵ > 0 and ε > 0 such that

sup_{s∈ℝ}(1 + |s|^{n+ε})(1 + |s|)^{−1}(is − ϵ − A)^{−(n+1)}∥_{L(X)} < ∞ for n = 0, 1, 2,

then ω_{1}(A) < 0.

4. CONTROL THEORY AND FOURIER MULTIPLIERS

Let X, Y and U be Banach spaces, A be the generator of a strongly continuous semigroup on X, and assume that B ∈ L(U; X) and C ∈ L(X; Y). Consider the following control system (see [7] and the literature therein):

x(t) = e^{tA}x_{0} + ∫_{0}^{t} e^{(t−τ)A}Bu(τ)dτ, y(t) = Cx(t), t ≥ 0.  (20)

Let H(z) = C(z − A)^{−1}B, z ∈ ρ(A), denote its transfer function, and let 𝕃 denote the input-output operator, defined on Lloc^{p} (ℝ_{+}; U) by the formula

(𝕃u)(t) = C ∫_{0}^{t} e^{(t−τ)A}Bu(τ)dτ, t ≥ 0.

Recall that the control system (20) is called p-input-output stable if 𝕃 is a bounded operator from L^{p}(ℝ_{+}; U) to L^{p}(ℝ; Y), 1 ≤ p < ∞.

If iℝ ⊂ ρ(A), we denote ζ(s) = H(is) = C(is − A)^{−1}B, s ∈ ℝ.

Theorem 4.1. Assume that σ(A) ⊂ {Re λ < 0} and the transfer function H and its derivative H′ belong to the Hardy class H^{∞} = H^{∞}({Re λ > 0}; L(U; Y))). Then the following assertions are equivalent

(i) Control system (20) is p-input-output stable;

(ii) ζ is a Fourier multiplier from L^{p}(ℝ; U) to L^{p}(ℝ; Y),

and ∥M_{ζ}∥_{L(L^{p}(ℝ;U);L^{p}(ℝ;Y))} = ∥𝕃∥_{L(L^{p}(ℝ_{+};U);L^{p}(ℝ_{+};Y))}.

Proof. For λ ∈ ℂ and A_{λ} = A−λ we let ζ_{λ}(s) = C(is−A_{λ})^{−1}B, s ∈ ℝ, and let 𝕃_{λ} denote the input-output operator for A_{λ}. Fix u_{0} ∈ U, φ ∈ S(ℝ) with supp φ ⊂ (0, ∞), and define u = u_{0} ⊗ φ.

First, we claim that

(F𝕃_{λ}u)(s) = ζ_{λ}(s)(Fu)(s), s ∈ ℝ, for Re λ > ω_{0}(A).  (21)

Indeed, since Re λ > ω_{0}(A) and B and C are bounded operators, for each s ∈ ℝ and u_{0} ∈ U we have:

C(is − A_{λ})^{−1}Bu_{0} = C ∫_{0}^{∞} e^{(A_{λ}−is)σ}Bu_{0}dσ.

Moreover, due to Re λ > ω_{0}(A) and φ ∈ S(ℝ), the integrals below converge absolutely and Fubini’s theorem gives:

∫_{0}^{∞} ζ_{λ}(s)u_{0}φ(τ)e^{−isτ} dτ = C ∫_{0}^{∞} ∫_{0}^{∞} e^{(A_{λ}−is)σ}Bu_{0}φ(τ)e^{−isτ} dσdτ

= C ∫_{0}^{∞} ∫_{0}^{t} e^{(A_{λ}−is)(t−τ)}Bu_{0}φ(τ)e^{−isτ} dτdt

= ∫_{0}^{∞} e^{−ist}(𝕃_{λ}u)(t)dt.

Thus, claim (21) is proved.

Second, recall that ζ_{λ}(s) = H(λ+is), s ∈ ℝ, Re λ ≥ 0. Since H(·) ∈ H^{∞}, we have that ζ_{λ}(·) = H(λ + i(·)) ∈ L^{∞}(ℝ; L(U; Y)) for Re λ ≥ 0. Since Fu ∈ S(ℝ; U), we have ζ_{λ}(·)(Fu)(·) ∈ L^{1}(ℝ; Y) and F^{−1}ζ_{λ}Fu ∈ C_{0}(ℝ; X) for Re λ ≥ 0. Now (21) implies

(𝕃_{λ}u)(t) = (F^{−1}ζ_{λ}(·)Fu)(t), t ∈ ℝ, for Re λ > ω_{0}(A). (22)

Third, we claim that for each t ∈ ℝ the left- and right-hand sides of (22) define functions, analytic in {Re λ > 0}. Indeed, if

L(λ) := (𝕃_{λ}u)(t) = e^{−λt}C ∫_{0}^{t} e^{(t−τ)A}Bu_{0}e^{λτ} φ(τ)dτ

then

\frac{d}{dλ}L(λ) = C ∫_{0}^{t}(τ − t)e^{−λ(t−τ)}e^{A(t−τ)}Bu_{0}φ(τ)dτ.

If R(λ) := (F^{−1}ζ_{λ}(·)u_{0}Fφ)(t) for Re λ > 0, then its derivative

\frac{dR}{dλ} = (F^{−1}ζ′_{λ}(·)u_{0}Fφ)(t) = (F^{−1}H′(λ + i(·))u_{0}Fφ)(t).  (23)

By the assumption H′ ∈ H^{∞} we conclude that the function H′(λ + i(·)) belongs to L^{∞}(ℝ; L(U; Y)). Therefore, the improper integral in the inverse Fourier transform in the right-hand side of (23) converges, \frac{dR}{dλ} exists, and R(·) is analytic in {Re λ > 0}.

Using (22), by analytic continuation we have:

(𝕃u)(t) = (F^{−1}ζ(·)u_{0}Fφ)(t) = (M_{ζ}u)(t), t ∈ ℝ, u = u_{0} ⊗ φ ∈ D_{S}.

From a density argument it follows easily that (i) and (ii) are equivalent, and the equality for the norms.

Remark 4. 1. Recently, a significant progress has been made in finding sufficient conditions for operator valued functions to be L^{p}–Fourier multipliers, see [2, 5, 22] and literature therein. We formulate the following generalization of Bourgain’s and Mikhlin’s Theorems which is due to L. Weis [22]. Recall that a function ζ : ℝ → L(U, Y) is called R–bounded (randomized, or Rademacher, bounded) if there is a constant c such that for all n ∈ ℕ, s_{1}, …, s_{n} ∈ ℝ \ {0}, and u_{1}, …, u_{n} ∈ U we have

( ∫_{0}^{1} ∥ ∑_{j=0}^{n} r_{j}(τ)ζ(s_{j})u_{j}∥_{Y}^{p} dτ)^{1/p} ≤ c( ∫_{0}^{1} ∥ ∑_{j=0}^{n} r_{j}(τ)u_{j}∥_{U}^{p} dτ)^{1/p},

where r_{j}(τ) = sign sin(2^{j}πτ), τ ∈ [0, 1], are the Rademacher functions, and 1 ≤ p < ∞. Note that if this inequality holds for one p then it holds for all p. Also, if ζ is R–bounded, then it is bounded, and boundedness and R– boundedness are equivalent provided U and Y are Hilbert spaces. Assume that U and Y are UMD-spaces (for example, this holds for L^{q}– and many others function spaces). The result of L. Weis states that a differentiable function ζ : ℝ \ {0} → L(U; Y) is an L^{p}–Fourier multiplier provided [s ↦ ζ(s)] and [s ↦ sζ′(s)] are R–bounded. By a direct application of this result in Theorem 4.1 for the transfer function ζ(s) = C(is−A)^{−1}B we have that the control system (20) is p–input-output stable provided [s ↦ ζ(s)] and [s ↦ sζ′(s)] are R–bounded. Of course, if X = Y = U is an UMD-space, and B = C = I, then we have that ω_{0}(A) < 0 provided [s ↦ (is − A)^{−1}] and s ↦ s(is − A)^{−2} are R–bounded. ◇

Recall, that the control system (20) is called externally stable if H(·) ∈ H^{∞}. Recall that if ζ is an L^{p}-Fourier multiplier, then ζ ∈ L^{∞}(ℝ; L(U; Y))

(see Definition 3.1). Also, ∥M_{ζ}∥_{L(L^{p}(ℝ;L(U;Y))} ≥ ∥ζ∥_{L^{∞}(ℝ;L(U;Y))}. Therefore, Theorem 4.1 is related to a theorem of G. Weiss [25] stating that ∥𝕃∥ ≥ ∥ζ∥_{∞}. Using Plancherel’s Theorem, one can also easily deduce from Theorem 4.1 the following classical result: if p = 2 and U and Y are Hilbert spaces, then input-output stability is equivalent to external stability.

Recall, that the system (20) is called stabilizable (respectively, detectable) if there is an operator F ∈ L(X; U) (respectively, K ∈ L(Y; X)) such that ω_{0}(A + BF) < 0 (respectively, ω_{0}(A + KC) < 0). A well-known theorem due to R. Curtain [6] and R. Rebarber [19] says that if U and Y are Hilbert spaces then the control system (20) is uniformly exponentially stable (that is, ω_{0}(A) < 0) if and only if it is stabilizable, detectable, and externally stable. As it was shown in [4], if U and Y are Banach spaces, then the external stability in this assertion must be replaced by input-output stability (that is, by the condition 𝕃 ∈ L(L^{p}(ℝ_{+}; U); L^{p}(ℝ_{+}; Y))).

In the following result we relate the exponential stability (ω_{1}(A) < 0) to input-output stability with respect to classical solutions; we define the latter as the boundedness of the operator 𝕄_{A} acting from L^{p}(ℝ_{+}; U) to L^{p}(ℝ_{+}; Y) by the rule

(𝕄_{A}u)(t) = C ∫_{0}^{t}(w − A)^{−1}e^{(t−τ)A}Bu(τ)dτ, t ≥ 0, w > max{ω_{0}(A), 0}.

Theorem 4.2. Assume that control system (20) is stabilizable and detectable. The following assertions are equivalent.

(a) ω_{1}(A) < 0;

(b) there exists ϵ > 0 such that 𝕄_{A_{−ϵ}} ∈ L(L^{p}(ℝ_{+}; U); L^{p}(ℝ_{+}; Y)).

Proof. Consider the following operators:

(𝔾_{A+BF} f)(t) = ∫_{0}^{t} e^{(t−τ)(A+BF)}f(τ)dτ,
(𝔾_{A+KC}f)(t) = ∫_{0}^{t} e^{(t−τ)(A+KC)}f(τ)dτ.

Since ω_{0}(A + BF) < 0 and ω_{0}(A + KC) < 0 by assumptions, Datkovan Neerven Theorem implies that 𝔾_{A_{−ϵ}+BF} and 𝔾_{A_{−ϵ}+KC} belong to L(L^{p}(ℝ_{+}; X)) for sufficiently small ϵ > 0. Using variation of constant formulas for the semigroups {e^{t(A_{−ϵ}+KC)}}_{t≥0} and {e^{t(A_{−ϵ}+BF)}}_{t≥0}, respectively, it is easy to obtain the following identities:

𝔾_{A_{−ϵ}+KC} = 𝔾_{−ϵ} + 𝔾_{A_{−ϵ}+KC}KC𝔾_{−ϵ},  (24)

𝔾_{A_{−ϵ}+BF} = 𝔾_{−ϵ} + 𝔾_{−ϵ}BF 𝔾_{A_{−ϵ}+BF} ,  (25)

see [3, p.135] for more details. Note that 𝕄_{A_{−ϵ}} = Cℍ_{1,−ϵ}B, where ℍ_{1,−ϵ} = (w − A)^{−1}𝔾_{−ϵ}. If (a) holds, then ℍ_{1,−ϵ} ∈ L(L^{p}(ℝ_{+}; X)) by Theorem 3.1. Since B and C are bounded operators, (b) holds. To see that (b) implies (a), we multiply (24) by (w − A)^{−1}B and (25) by (w − A)^{−1}. Then

𝔾_{A_{−ϵ}+KC}(w − A)^{−1}B = ℍ_{1,−ϵ}B + 𝔾_{A_{−ϵ}+KC}K𝕄_{A_{−ϵ}},  (26)
(w − A)^{−1}𝔾_{A_{−ϵ}+BF} = ℍ_{1,−ϵ} + (ℍ_{1,−ϵ}B)F 𝔾_{A_{−ϵ}+BF} .  (27)

If (b) holds, then ℍ_{1,−ϵ}B is a bounded operator due to (26). Indeed, (w − A)^{−1}, B, K are bounded operators, 𝕄_{A_{−ϵ}} is bounded by (b), and 𝔾_{A_{−ϵ}+KC} is bounded due to the detectability assumption. As soon as we know that ℍ_{1,−ϵ}B is bounded, identity (27) shows that ℍ_{1,−ϵ} is bounded. Indeed, (w− A)^{−1} and F are bounded, and 𝔾_{A_{−ϵ}+BF} is bounded due to the stabilizability assumption. Now (a) holds by the implication (ii) ⇒ (i) in Theorem 3.1.

We remark that the condition 𝕄_{A} ∈ L(L^{p}(ℝ_{+}; U); L^{p}(ℝ_{+}; Y)) can be characterized in terms of the L^{p}–Fourier multiplier properties of the function [s ↦ (is − w)^{−1}C(is − A)^{−1}B], s ∈ ℝ, similarly to Theorem 3.2 and Remark 3.4.

Finally, we remark that the assumptions B ∈ L(U; X) and C ∈ L(X; Y) imposed in this section are rather restrictive. However, as it is recently shown in [18], the results of this section hold even if these assumptions are relaxed to allow that B : U → \overline{X} and C : \underline{X} → Y are bounded operators for a triple of Banach spaces \underline{X} ↪ X ↪ \overline{X} such that certain conditions of “regularity” are satisfied to cover the Weiss-regular or Pritchard–Salamon classes.

ACKNOWLEDGMENT

We thank M. Hieber and L. Weis for several discussions concerning Fourier multipliers, R. Chill for important remarks on the 1999 version of the manuscript, and J. van Neerven for a discussion that led to Remark 3.1. Also, the first author thanks R. Shvydkoy for many useful discussions.

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