A SPECTRAL BOUND FOR ASYMPTOTICALLY

NORM-CONTINUOUS SEMIGROUPS

Mark D. Blake ^{∗}

Abstract

We introduce a new growth bound for C_{0}-semigroups giving information about the absence of norm-continuity of the semigroup and we give a corresponding spectral bound. For semigroups on general Banach spaces we prove an inequality between these bounds and we give a version of the spectral mapping theorem in terms of the new growth bound. For semigroups on Hilbert space we show that the bounds are equal and hence obtain new characterizations of asymptotically normcontinuous semigroups and semigroups norm-continuous for t > 0 in terms of the resolvent of the infinitesimal generator. In the last section we prove that versions of the spectral mapping theorem holds for three different definitions of the essential spectrum and give nice relationships between the new growth bound and the essential growth bound of the semigroup.

1 Introduction

Let T (t) be a C_{0}-semigroup on a Banach space X with generator A. It is well known that if T (t) is eventually norm-continuous then the spectral mapping theorem holds (cf. [14] A-III, Theorem 6.6). In [13] the wider class of semigroups that are norm-continuous at infinity is introduced and it is shown that for these semigroups a spectral mapping theorem holds for the boundary spectrum. In [18] a similar result is given for another generalization of eventually norm-continuous semigroups - semigroups that are essentially norm-continuous. Such results are of particular interest as when such a spectral mapping theorem holds the spectral bound and growth bound of the semigroup coincide and hence the asymptotic behaviour of solutions of the associated Cauchy problem is determined by the spectrum of the generator.

∗I am very grateful to Charles Batty for helpful discussions concerning this paper. This work was supported by a grant from the EPSRC.

In this paper we study these continuity properties via the introduction of new bounds, δ(T), the growth bound of the local variation of T (t) which gives information about the absence of norm-continuity, and an associated spectral bound s_{0}^{∞}(A), involving the behaviour of the resolvent away from the real axis. We relate these bounds to essentially norm-continuous and essentially norm-measurable semigroups, and show that semigroups that are norm-continuous at infinity (for which we shall use the alternative terminology asymptotically norm-continuous) have a particularly simple characterization in terms of δ(T) (Proposition 3.5). By reformulating the spectral mapping theorem in terms of δ(T), we obtain a very general version (Theorem 3.6) that has the versions given in [14], [13] and [18] as immediate corollaries.

When the underlying space is a Hilbert space we find that δ(T) = s_{0}^{∞}(A) and that the classes of essentially norm-continuous semigroups, essentially norm-measurable semigroups and asymptotically norm-continuous semigroups coincide (Theorem 4.4). It follows that these properties are all equivalent to the purely spectral condition s_{0}^{∞}(A) < s(A). Using the same techniques we obtain sufficient conditions on the resolvent for a semigroup on a Hilbert space to be norm-continuous for t > α (Theorem 4.5). A new characterization of norm-continuity for t > 0 on Hilbert space follows.

In the final section we investigate the relationship between continuity and compactness properties of the semigroup. We show that essentially compact semigroups are asymptotically norm-continuous (Theorem 5.1), and we prove a spectral mapping theorems for the Browder essential spectrum, the upper Fredholm essential spectrum and the Fredholm essential spectrum (Theorem 5.6). These results enable us to prove that the essential growth bound and the essential spectral bound of a semigroup are nicely related to δ(T) (Corollary 5.8). Together with the results of the previous chapter these give a characterization of essentially compact semigroups on Hilbert space in terms of the resolvent and essential spectrum of the generator (Corollary 5.9).

2 Laplace Transform Results

In this section we shall consider results that do not require T to be a semigroup. We shall assume only that the one-parameter families T that we consider should be strongly measurable, exponentially bounded functions from ℝ_{+} into B(X). We shall define the type of such a family, written ω_{0}(T) by

ω_{0}(T) := inf{ω ∈ ℝ : there exists M_{ω} such that ∥T (t)∥ ≤ M_{ω}e^{ωt} for all t ≥ 0}.

We shall denote the Laplace transform by L,

L (T) (λ)x := ∫_{0}^{∞} e^{−λt}T (t)x dt for all x ∈ X,

and λ ↦ L (T) (λ) is a holomorphic function from Re(λ) > ω_{0}(T) into B(X). For N ≥ 0 we shall denote the Nth derivative of the L (T) by L (T)^{(N)}. For 1 ≤ p < ∞ we shall denote by L^{p}(ℝ, X) the space of strongly measurable functions f : ℝ → X with ∥f∥_{p} := {∫_{−∞}^{∞} ∥f(t)∥^{p} dt}^{1/p} < ∞.. We shall identify L^{p}(ℝ_{+}, X) as a subspace of L^{p}(ℝ, X) in the obvious manner by regarding a function in L^{p}(ℝ_{+}, X) as taking the value 0 on (−∞, 0). We shall denote by F the Fourier transform on L^{1}(ℝ, X) defined by

(Ff)(s) := ∫_{−∞}^{∞} e^{−ist}f(t) dt (∗)

Then by the Riemann-Lebesgue lemma F maps L^{1}(ℝ, X) into the space of continuous X-valued functions vanishing at infinity. We shall also denote by F the Fourier transform on L^{2}(ℝ, X) defined by (∗) for f ∈ L^{1}(ℝ, X) ∩ L^{2}(ℝ, X). If X is a Hilbert space then we also have the Hilbert space valued Plancherel Theorem that \frac{1}{\sqrt{2π}} F extends to an isometry on L^{2}(ℝ, X).

In [18], Definition 2.6, essentially norm-continuous and essentially normmeasurable C_{0}-semigroups are introduced, generalizing both eventually normcontinuous semigroups and essentially compact semigroups. Here we modify these definitions to make them applicable to the more general situation of exponentially bounded families T : ℝ_{+} → B(X) of operators.

Definition 2.1 Let T : ℝ_{+} → B(X) be an exponentially bounded family of operators on a Banach space X, and let β > 0. T (t) is said to be essentially norm-continuous (of type β) if for each α such that 0 < α < β there exist families of operators S_{1}, S_{2} : ℝ_{+} → B(X) (depending on α) such that

T (t) = S_{1}(t) + S_{2}(t)

with S_{1} right-continuous on (0, ∞) in the operator norm topology and ω_{0}(S_{2}) ≤ ω_{0}(T) − α.

T is said to be essentially norm-measurable (of type β) if for each 0 < α < β there exist families of operators S_{1}, S_{2} : ℝ_{+} → B(X) such that

T (t) = S_{1}(t) + S_{2}(t)

with S_{1} norm-measurable on (0, ∞) and ω_{0}(S_{2}) ≤ ω_{0}(T) − α.

Definition 2.2 Let T : ℝ_{+} → B(X) be an exponentially bounded family of operators on a Banach space X. Define the function f_{T} : [0, ∞) → [0, ∞) by

f_{T} (t) := lim sup_{h→0+} ∥T (t + h) − T (t)∥ .

Then we may define

δ(T) := inf{ω ∈ ℝ : there exists M_{ω} such thatf_{T} (t) ≤ M_{ω}e^{ωt}}

and we have δ(T) ≤ ω_{0}(T).

Notice that if T : ℝ_{+} → B(X) is a family of operators on a Banach space X with ω_{0}(T) > −∞ then we can define T′(t) := e^{−ω_{0}(T)t}T (t) and we have ω_{0}(T′) = 0, δ(T′) = δ(T) − ω_{0}(T), and T′(t) is essentially norm-continuous (resp. essentially norm-measurable) of type β if and only if T (t) also has this property. Hence by replacing T (t) by T′(t) if necessary we will usually be able to assume that ω_{0}(T) = 0.

Definition 2.3 Let T : ℝ_{+} → B(X) be an exponentially bounded, strongly measurable family of operators on a Banach space X. For N ∈ ℕ, γ > ω_{0}(T), define

ϕ_{γ}^{T} (N) := (lim sup_{|s|→∞} ∥\frac{1}{N!}L (T)^{(N)} (γ + is)∥)^{\frac{1}{N+1}} .

Theorem 2.4 Let T be a strongly measurable, exponentially bounded family of operators on a Banach space X, and let β > 0. Then we have (1)⇒(2)⇒(3)⇒(4), and (3)⇒(4’), where

(1) T (t) is essentially norm-continuous of type β.

(2) δ(T) ≤ ω_{0}(T) − β.

(3) T (t) is essentially norm-measurable of type β.

(4) There is a constant C > 0 such that given γ > ω_{0}(T) there exists r_{γ} > 0 such that ∥L (T) (γ + is)∥ < C whenever |s| > r_{γ}. (4’) Let γ > ω_{0}(T). For each 0 < α < β there exists N_{α} such that ϕ_{γ}^{T} (N_{α}) ≤ (γ + α)^{−1}.

Proof.

(1)⇒(2). Suppose that T (t) is essentially norm-continuous of type β, so if 0 < α < β then there exist S_{1}, S_{2} such that T (t) = S_{1}(t) + S_{2}(t) where S_{1} is right norm-continuous for t > 0 and ω_{0}(S_{2}) ≤ ω_{0}(T) − α. Then as S_{1} is right norm-continuous,

δ(T) = δ(S_{1} + S_{2}) = δ(S_{2}) ≤ ω_{0}(S_{2}) ≤ ω_{0}(T) − α.

As 0 < α < β was arbitrary, we get that δ(T) ≤ ω_{0}(T) − β. (2)⇒(3). Without loss of generality we may assume that ω_{0}(T) = 0, so, assuming (2), we have that if 0 < α < β there is a constant C > 0 such that for all t > 0 we have  lim sup_{h→0+} ∥T (t + h) − T (t)∥ ≤ Ce^{−αt}. So for each t ≥ 0 there exists a half open interval U_{t} = [t, x_{t}) with ∥T (s) − T (t)∥ < 2Ce^{−αt} for all s ∈ U_{t}. We may also assume that the intervals have length less than d where e^{−αd} > \frac{1}{2}. As [0, ∞) is Lindelöf in the Sorgenfrey topology (see [4] Example 3.8.14 p248) generated by the half open intervals [a, b), there is a countable family \{U_{t_{i}} : i ∈ ℕ\} that covers it. Let V_{t_{i}} := U_{t_{i}} \ ⋃_{j<i} U_{t_{j}} so that V_{t_{i}} ⊆ U_{t_{i}}, the V_{t_{i}} are disjoint and have union [0, ∞), and define S(t) = _{∞ ∑ 0} χ^{V_{t_{i}}} (t) T (t^{i}).

Then if s ∈ [0, ∞), there is t_{i} such that s ∈ V_{t_{i}}, and

∥T (s) − S(s)∥ = ∥T (s) − T (t_{i})∥
≤ 2Ce^{−αt_{i}}
= 2Ce^{−αs}e^{−α(t_{i}−s)}
≤ 4Ce^{−αs}.

S(t) is a limit of step functions, so it is norm-measurable. So putting S_{1}(t) = S(t), and S_{2}(t) = T (t) − S(t) we have T (t) = S_{1}(t) + S_{2}(t) with S_{1} norm-measurable, and ω_{0}(S_{2}) ≤ −α. As 0 < α < β was arbitrary, it follows that T (t) is essentially norm-measurable of type β.

(3)⇒(4). Suppose that T (t) is essentially norm-measurable of type β > 0 and, without loss of generality, that ω_{0}(T) = 0. Let 0 < α < β. Then we have that there exist S_{1}, S_{2} and K > 0 such that T (t) = S_{1}(t) + S_{2}(t) with S_{1} norm-measurable and

∥S_{2}(t)∥ ≤ Ke^{−αt}.

Let γ > 0. Then as S_{1} is norm-measurable with ω_{0}(S_{1}) = 0, we have that the function t ↦ e^{−γt}S_{1}(t) is in L^{1}((0, ∞), B(X)). Hence by the Riemann-Lebesgue lemma there exists r_{γ} > 0 such that ∥F (e^{−γ.}S_{1}(.)) (s)∥ < \frac{K}{α} whenever |s| > r_{γ}. Then if x ∈ X,

∥L (T) (is + γ)x∥ ≤ ∥∫_{0}^{∞} e^{−(γ+is)t}T (t)x dt∥

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