Prelim Semigroup Notes

Harry Gaebler

1 C0-semigroups

Unless otherwise stated, we will assume throughout that X is a complex Banachspace, with L (X) denoting the Banach space of continuous linear operators onX.

Definition. (Semigroups) By a semigroup, we mean a one-parameter family ofcontinuous linear operators, {T (t)}t≥0 ⊂ L (X), such that,

1. T (0) = I, where I is the identity map on X.

2. For all s, t ≥ 0, we have T (s+ t) = T (s) ◦ T (t).

Additionally, if one has limt→0+ T (t)x = x for every x ∈ X, then the semigroupis said to be “strongly continuous,” or C0, for short.

To every C0 semigroup, we associate another linear map called its generator,which describes how the semigroup is affecting initial data in X.

Definition. (Infinitesimal Generator) Given a C0-semigroup, {T (t)}t≥0, its in-finitesimal generator is the map A : D(A)→ X, where

D(A) ={x ∈ X

∣∣∣ limt→0+

T (t)x− xt

exists}

and Ax is defined to be the value of this found limit for x ∈ D(A).

It is quite clear that D(A) is a subspace of X, and further, that A is a (notnecessarily continuous) linear map. C0-semigroups and their generators enjoysome rather pleasant basic properties, which we will now elucidate.

Lemma. (Continuity of Orbits) Given a C0-semigroup, {T (t)}t≥0, we have thatthe orbit map t 7→ T (t)x is continuous for each x ∈ X.

Proof. Apply the semigroup property and then strong continuity.

Lemma. (Exponential Bounds) Given a C0-semigroup, {T (t)}t≥0, there existsconstants M ≥ 1 and ω ∈ R such that for every t ≥ 0, ||T (t)|| ≤Meωt.

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Proof. Suppose for a contradiction that for every n ∈ N, there exists a pointtn ∈ [0, 1

n ] such that ||T (tn)|| ≥ n. Now, defining

{T (tn) | n ∈ N} ⊂ L (X)

we see that for each x ∈ X, supn∈N ||T (tn)x|| ≤ sup[0,1] ||T (t)x|| < ∞ by thecontinuity of the orbit map. Then, by uniform boundedness, it follows thatsupn∈N ||T (tn)|| <∞, but this is an obvious contradiction.

Specifically, we are now afforded an n0 ∈ N such that ||T (t)|| < n0 for everyt ∈ [0, 1

n0]. Then, fixing t∗ ≥ 0 arbitrarily, we may write t∗ = k 1

n0+ r for some

k ∈ N0 and r ∈ [0, 1n0

), which yields that,

||T (t∗)|| =∣∣∣∣∣∣T(k 1

n0+ r)∣∣∣∣∣∣ ≤Mk+1 M = sup

{||T (t)||

∣∣∣ t ∈ [0, 1

n0

]}= Meln(Mk) ≤Meωt∗ ω = n0 ln(M)

and since M ≥ 1 and ω ∈ R do not depend on t∗, we are done.

The next lemma rounds out our basic “toolbox” of techniques for manipu-lating C0-semigroups and their generators.

Lemma. (Toolbox) Given a C0-semigroup, {T (t)}t≥0, and its generator A :D(A)→ X, we may conclude the following:

1. For every x ∈ X and t ≥ 0, we have

(a) 1h

∫ t+ht

T (s)xds→ T (t)x as h→ 0+

(b)∫ t

0T (s)xds ∈ D(A) and A(

∫ t0T (s)xds) = T (t)x− x

2. For every x ∈ D(A) and t ≥ 0, we have

(a) T (t)x ∈ D(A) and ddtT (t)x = AT (t)x = T (t)Ax

(b) For every 0 ≤ s ≤ t, we have∫ tsT (τ)Axdτ =

∫ tsAT (τ)xdτ = T (t)x−

T (s)x

3. A is densely-defined and closed.

Proof. Continuity of the orbit map implies 1 (a), and 1 (b) is a straightforwardapplication of 1 (a). Next, 2 (a) follows from the semigroup property, and 2 (b)is simply the fundamental theorem of calculus. Lastly, 3 requires just a littlebit of unpacking. First, fix x ∈ X, and define,

xn = n

∫ n−1

0

T (t)xdt (n ∈ N)

so that xn ∈ D(A) by 1 (b), and xn → x by 1 (a) as n → ∞, meaning that Ais densely-defined. Then, supposing

xn → y ∈ X and Axn → z

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it follows by 2 (b) that for h > 0, we have

T (h)xn − xnh

=1

h

∫ h

0

T (t)Axndt

and since ||T (t)Axn − T (t)z|| ≤ Meωh||Axn − z|| → 0 as n → ∞, it follows byuniform convergence on [0, h] that,

T (h)y − yh

=1

h

∫ h

0

T (t)zdt

so taking h→ 0+, it follows by 1 (a) that y ∈ D(A) and Ay = z, meaning thatA is a closed operator, as required.

The main idea behind semigroup theory is that it allows us to solve the so-called “Abstract Cauchy Problem” (ACP), which is the initial value problem,{

u′(t) = Au(t) t ≥ 0

u(0) = u0 u0 ∈ D(A)

for some linear map A and unknown function u : [0,∞) → X. We are alwayshoping to show that A generates a C0-semigroup, in which case the ACP iswell-posed, and the solution is given via a “semigroup representation,”

u(t) = T (t)u0 (t ≥ 0)

Recall from ODE theory that we should have T (t) ≈ etA, and it is easy to seethat A ∈ L (X) will generate {etA}t≥0. Conversely, it is straightforward to showthat given an exponential semigroup, its generator must be the exponentiatedbounded operator. More generally, the collection of exponential semigroups con-stitute a subset of strongly continuous semigroups which are said to be uniformlycontinuous.

Definition. (UC Semigroups) A strongly continuous semigroup, {T (t)}t≥0, iscalled uniformly continuous if limt→0+ ||T (t)− I|| = 0.

The following lemma is an important general fact, and we will use it herein the context of showing that the exponential semigroups and UC semigroupscoincide.

Lemma. (Von Neumann) If B1, B2 ∈ L (X) where B1 is an invertible map,and ||B2|| < ||B−1

1 ||−1, then B1 −B2 ∈ L (X) is an invertible operator.

Proof. We’ll split this proof into two steps. The first step is to consider the casewhere B1 ≡ I and ||B2|| < 1. In this case,

(I −B2)

n∑j=0

Bj2 =

n∑j=0

Bj2 −n∑j=0

Bj+12 = I −Bn+1

2 (n ∈ N)

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and taking n→∞, it follows that (I−B2)∑∞j=0B

j2 = I, and as this sum is also

clearly a left inverse of I − B2, we have that I − B2 is invertible and∑∞j=0B

j2

is its inverse. We next consider the general case,

B1 −B2 = B1(I −B−11 B2)

and since ||B−11 B2|| < 1, the LHS must be invertible, so B1 − B2 is invertible

with inverse given by, (B1 −B2)−1 = (I −B−11 B2)−1B−1

1 .

Theorem. (EXP ↔ UC) The collection of exponential semigroups exactlycoincides with the collection of UC semigroups, that is

{{etA}t≥0 | A ∈ L (X)} = {UC semigroups}

Proof. The ⊆ containment is obvious, and for the ⊇ containment, suppose that{T (t)}t≥0 is UC and note that for a sufficiently small r > 0, we have∣∣∣∣∣∣I − 1

r

∫ r

0

T (t)dt∣∣∣∣∣∣ ≤ 1

r

∫ r

0

||I − T (t)||dt < 1

so by the Von Neumann lemma, it follows that∫ r

0T (t)dt must be invertible.

Then, fixing x ∈ X and h ∈ (0, r), we have

T (h)x− xh

=[ ∫ r

0

T (t)dt]−1[ ∫ r

0

T (t)dt]T (h)x− x

h

=[ ∫ r

0

T (t)dt]−1[ 1

h

∫ r+h

r

T (t)xdt− 1

h

∫ h

0

T (t)xdt]

so letting G : D(G) → X generate {T (t)}t≥0, one has X ⊆ D(G), and hence,G = [

∫ r0T (t)dt]−1[T (r)− I] ∈ L (X).

Our results about UC semigroups now allow us to characterize the generatorsof arbitrary C0-semigroups, and this is hugely important because it tells us whenthe ACP is well-posed.

Definition. (Resolvent and Spectrum) The resolvent set of a closed operator,A : D(A)→ X, is the set given by

ρ(A) = {λ ∈ C | λ−A : D(A)→ X is bijective}

and its complement, σ(A) = C \ ρ(A), is said to be the spectrum of A. Notethat we do not assume the density of D(A) in X - only the closedness of A isneeded for a reasonable spectral theory. For each λ ∈ ρ(A), we may form theoperator,

R(λ,A) = (λ−A)−1

and since this operator is defined on X and closed (by the closedness of A),the closed graph theorem implies its boundedness. Lastly, R(λ,A) is called theresolvent operator of A at the point λ.

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We will return later to the spectral theory of semigroups and their generators,but in fact, this definition is the last necessary prerequisite for giving a completecharacterization of C0-semigroup generators.

Theorem. (Hille-Yosida for Contraction Semigroups) Given a linear map, A,we can say that A will generate a contraction semigroup on X iff it is closed,densely-defined, and for every λ ∈ (0,∞), we have ||R(λ,A)|| ≤ λ−1.

Proof. For necessity, suppose that A generates a strongly continuous (contrac-tion) semigroup, {T (t)}t≥0, so ||T (t)|| ≤ 1 for every t ≥ 0. The toolbox lemmaalready tells us that A is closed and densely-defined, so we need only check thethird condition. To that end, fix λ > 0, and consider the operator,

Rλx :=

∫ ∞0

e−λtT (t)xdt (x ∈ X)

which is clearly bounded in operator norm by λ−1, so we need only prove λ−Ato be a bijection onto X with inverse Rλ. For x ∈ X, consider

T (h)Rλx−Rλxh

=1

h

∫ ∞h

e−λ(t−h)T (t)xdt− 1

h

∫ ∞0

e−λtT (t)xdt

=eλh − 1

h

∫ ∞h

e−λtT (t)xdt− 1

h

∫ h

0

e−λtT (t)xdt

and since the RHS converges to λRλx− x as h→ 0+, it follows that for x ∈ X,

(λ−A)Rλx = λRλx−ARλx = x

and conversely, since the toolbox lemma shows that A and T (t) commute whenrestricted to D(A), it follows that for x ∈ D(A), we must have,

Rλ(λ−A)x = λRλx−RλAx = λRλx−ARλx = x

where we have also used the fact that A is a closed operator, and this completesthe proof of necessity.

For sufficiency, we begin with a closed, densely-defined operator, A : D(A)→ X,whose resolvent set contains the positive real line, and whose resolvent operatorat the point λ > 0 is bounded in operator norm by λ−1. Then, setting

Aλ = λAR(λ,A) = λ2R(λ,A)− λ (λ > 0)

we have that,

1. Since A and R(λ,A) commute for x ∈ D(A), we may write,

||λR(λ,A)x− x|| = ||AR(λ,A)x|| ≤ ||Ax||λ

and by density, this implies limλ→∞ λR(λ,A)x = x for every x ∈ X.

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2. Since Aλ ∈ L (X), it must generate the UC semigroup {etAλ}t≥0, and as

||etAλ || = ||et(λ2R(λ,A)−λ)|| ≤ e−tλetλ

2||R(λ,A)|| ≤ 1

this semigroup is also a contraction.

Now, fixing t ≥ 0, we consider x ∈ D(A) with λ, µ > 0 to write,

||etAλx− etAµx|| =∣∣∣∣∣∣ ∫ 1

0

d

ds

[etsAλet(1−s)Aµ

]xds∣∣∣∣∣∣

=∣∣∣∣∣∣ ∫ 1

0

etsAλet(1−s)Aµ(tAλ − tAµ)xds∣∣∣∣∣∣ ≤ t||Aλx−Aµx||

and since the RHS converges to zero as λ, µ→∞ (by 1.), the density of D(A)in X allows us to define,

T (t)x := limλ→∞

etAλx (x ∈ X)

and in particular, the one-parameter family {T (t) | t ≥ 0} ⊂ L (X) must be acontraction semigroup, as strong continuity results from T (t)x being the uniformlimit of continuous functions on any bounded interval. Assuming G : D(G)→ Xgenerates this semigroup, we’ll let x ∈ D(A), and consider

T (h)x− xh

=1

h( limλ→∞

ehAλx− x)

=1

hlimλ→∞

∫ h

0

etAλAλxdt

=1

h

∫ h

0

T (t)Axdt uniform convergence on [0, h]

and this implies A ⊆ G as operators, and since 1 ∈ ρ(A) ∩ ρ(G), we have

(1−G)[D(A)] = (1−A)[D(A)] = X

meaning that D(A) = R(1, G)[X] = D(G), which completes the proof.

The above theorem is the workhorse of basic semigroup theory in the sensethat a full characterization of strongly continuous semigroup generators can nowbe (more or less) obtained through rescaling/renorming arguments alone.

Theorem. (Full Hille-Yosida) Given a linear map, A, we can say that A willgenerate a strongly continuous semigroup, {T (t)}t≥0, satisfying ||T (t)|| ≤Meωt

for each t ≥ 0, iff A is closed, densely-defined, and for every <(λ) ∈ (ω,∞), wehave that ||R(λ,A)n|| ≤M(<(λ)− ω)−n for each n ∈ N.

The full Hille-Yosida result is often difficult to apply as the resolvent boundmust be checked for each n ∈ N. Of more practical importance is the so-called

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Lumer-Philips Theorem, which relates the Hille-Yosida theorem for contractionsto the concept of a dissipative operator.

Briefly speaking, writing X ′ for the dual space of X, the complex version ofHahn-Banach shows that for each x ∈ X,

F (x) = {x′ ∈ X ′ | ||x′||2X′ = x′(x) = ||x||2}

must be nonempty, and then, given an operator, A : D(A) → X, dissipativityis expressed by the condition,

∀x ∈ D(A) ∃x′ ∈ F (x) s.t. <[x′(Ax)] ≤ 0

which may be equivalently expressed as ||(λ−A)x|| ≥ λ||x|| for every x ∈ D(A)and λ > 0. This equivalence is not trivial (sufficiency requires Banach-Alaoglu),but it leads to the following characterization of C0-semigroup generators.

Theorem. (Lumer-Philips) Let A be a densely-defined operator. If A generatesa contraction semigroup, then it must be dissipative, and for each λ > 0, we musthave (λ− A)[D(A)] = X. Conversely, if A is dissipative and there is a λ0 > 0such that (λ0 −A)[D(A)] = X, then A must generate a contraction semigroup.

As an example, assume X = L2([0, 1],R) is equipped with the usual L2 innerproduct, and consider the subset,

D(A) = {u ∈ H1([0, 1],R) | u(1) = 0}

which is dense in X, with Au = u′. Next, fixing u ∈ D(A), we’ll have

||u||2 = 〈u, u〉 = x′(u)

for some x′ ∈ X ′, where ||x′|| = ||u||, so it follows that x′ ∈ F (u), and then,

x′(Au) = 〈Au, u〉 =

∫ 1

0

u′udx

= −1

2u(0)2 ≤ 0 integration by parts

and since <[x′(Au)] ≤ 0, we have that A is dissipative. Lastly, for a fixed λ0 > 0,one can show that the equation,

(λ0 −A)u = f

is solvable for any f ∈ X by some u ∈ D(A), so it follows by Lumer-Philips thatA generates a contraction semigroup on X. In hindsight, this result should notbe terribly surprising, given that A is a derivative operator.

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2 Asymptotics of Semigroup Solutions

The previous section was concerned with introducing the concept of a stronglycontinuous semigroup, and investigating the well-posedness of the ACP by char-acterizing semigroup generators. We will now explore a different question: givena well-posed ACP, what is the long-term behavior of the semigroup solution?

Definition. (Growth Bound) Let {T (t)}t≥0 be a strongly continuous semigroupwith ||T (t)|| ≤Meωt for each t ≥ 0. Since strongly continuous semigroups maybe uniquely associated with their generators, we suppose A generates {T (t)}t≥0,and we define the growth bound of the semigroup to be the number,

ω(A) = inf{ω ∈ R | ∃Mω s.t. ||T (t)|| ≤Mωeωt ∀t ≥ 0}

and in the event that this set has no greatest lower bound in R, we allow thecase ω(A) = −∞.

When presented with a well-posed ACP, our goal is usually to show that thegrowth bound of the semigroup solution is strictly negative, as this will implythe exponential decay of the solution.

Lemma. Suppose that η : [0,∞) → R is subadditive and bounded on compactintervals, then we must have,

inft>0

η(t)

t= limt→∞

η(t)

t

where the existence of the limit is part of the claim.

Proof. Fixing p > 0, we may, for any t∗ > p, write t∗ = kp+ r for some positivenatural number k, and r ∈ [0, p). Then,

η(t∗)

t∗=η(kp+ r)

kp+ r≤ η(kp) + η(r)

kp+ rsubadditivity

≤ kη(p) + η(r)

kp+ rsubadditivity, again

≤ kη(p)

kp+ r+

η(r)

kp+ r≤ η(p)

p+η(r)

kp

and since η(r) ≤ sup{|η(x)| | x ∈ [0, p]} <∞, it follows that,

lim supt∗→∞

η(t∗)

t∗≤ η(p)

pas k →∞ when p, r stay fixed/bdd

and as this inequality holds for an arbitrary p > 0, we may write,

lim supt∗→∞

η(t∗)

t∗≤ infp>0

η(p)

p≤ lim inf

t∗→∞

η(t∗)

t∗

which proves the claim (and note that this limit/inf may equal −∞).

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There are several different ways to represent the growth bound of a semi-group, and the above lemma links them together.

Theorem. (Representation of the Growth Bound) Let {T (t)}t≥0 be the stronglycontinuous semigroup generated by A, with growth bound ω(A). Then, ω(A) =

inft>0log ||T (t)||

t = limt→∞log ||T (t)||

t = log r(T (t∗))t∗

for every t∗ > 0, where r(T (t∗))denotes the spectral radius of T (t∗).

Proof. First, note that log ||T (t)|| : [0,∞) → R is subadditive and bounded oncompact intervals, so by the above lemma, we have

inft>0

log ||T (t)||t

= limt→∞

log ||T (t)||t

and calling this (extended) real number ν, it follows that for a fixed t∗ > 0 andfor any ω > ω(A),

ν ≤ ||T (t∗)||t∗

≤ logMeωt∗

t∗=

logM + ωt∗t∗

=logM

t∗+ ω

implying (by the arbitrariness of t∗) that ν ≤ ω for any ω > ω(A), and hence,that ν ≤ ω(A) as well. For the reverse inequality, we may fix µ > ν, and find at∗ > 0 such that,

ν ≤ log ||T (t)||t

< µ (t ≥ t∗)

but this implies that ||T (t)|| ≤ eµt for t ≥ t∗, and since t 7→ ||T (t)|| is boundedon [0, t∗], we may find an M∗ ≥ 1 such that ||T (t)|| ≤M∗eµt for all t ≥ 0, whichshows that ω(A) ≤ µ. Now, as ν ≤ ω(A) ≤ µ, the reverse inequality will followby the arbitrariness of µ > ν, and lastly, the spectral radius representation isderived by an application of Gelfand’s formula.

Given the multitude of ways to represent the growth bound of a stronglycontinuous semigroup, it should come as no surprise that there are multiplereprentations for the negativity of this quantity. This condition is called (uni-form) stability of the semigroup.

Theorem. (Uniform Stability) Let {T (t)}t≥0 be the strongly continuous semi-group generated by A. Then, the following condtions are equivalent.

1. ω(A) < 0

2. limt→∞ ||T (t)|| = 0

3. There exists t > 0 such that ||T (t)|| < 1

4. There exists t > 0 such that r(T (t)) < 1

Proof. First, if ω(A) < 0, then we may find ω ∈ (ω(A), 0) such that ||T (t)|| ≤Meωt and since the RHS converges to zero as t → ∞, we have that 1 =⇒ 2.

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Next, 2 =⇒ 3 is trivial, and supposing we may find t∗ > 0 so that ||T (t∗)|| < 1,the growth bound theorem tells us that

log r(T (t∗))

t∗≤ log ||T (t∗)||

t∗< 0

which implies 1 > elog r(T (t∗))

t∗ = elog r(T (t∗))1t∗ = r(T (t∗))

1t∗ , so raising both sides

to the t∗ power, it follows that 3 =⇒ 4. Lastly, if t∗ is such that r(T (t∗)) < 1,then the growth bound theorem tells us that

ω(A) =log r(T (t∗))

t∗< 0

so we have that 4 =⇒ 1, which completes the proof.

We have already seen some uniformly stable semigroups. For instance, anyUC-semigroup will clearly satisfy condition 3, and more generally, this conditionholds for any so-called “eventually norm continuous” semigroup (e.g. it is UCafter some finite time). The above theorem is not an exhuastive list of uniformstability criteria, as we have omitted a related condition called the Datko-PazyTheorem, but it is more than sufficient for a proof of the Gearhart-Pruss The-orem, which is our next goal.

Gearhart-Pruss characterizes the uniform stability of a strongly continuoussemigroup on a Hilbert space. We’ll need to introduce several preliminary ideasbefore attempting a proof, and we return first to the notion of the resolvent andspectrum of a closed operator.

Lemma. (Resolvent Properties) Given a closed operator, A : D(A) → X, thefollowing properties hold:

1. The resolvent set, ρ(A), is open in C, and for each λ0 ∈ ρ(A), one has

R(λ,A) =

∞∑j=0

(λ0 − λ)jR(λ0, A)j+1

for all λ ∈ C with |λ0 − λ| ≤ 1||R(λ0,A)|| .

2. The map λ 7→ R(λ,A) is analytic, and moreover, we have,

dn

dλnR(λ,A) = (−1)nn!R(λ,A)n+1 (n ∈ N)

3. Given λ0 ∈ C and a sequence (λ)n ⊂ ρ(A) which converges to λ0, we havethat λ0 ∈ σ(A) iff limn→∞ ||R(λn, A)|| =∞.

Proof. Fix λ0 ∈ ρ(A) and consider that for any λ ∈ C, we have,

λ−A = λ0 −A+ λ− λ0 = [I − (λ0 − λ)R(λ0, A)]︸ ︷︷ ︸inverse will be in L (X)

(λ0 −A)

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which shows that if |λ0−λ| < 1||R(λ0,A)|| , then the RHS is invertible, and further,

must be a bijection from D(A) onto X, meaning that λ ∈ ρ(A). Hence, ρ(A) isopen in C, and we may again apply Von Neumann to conclude that,

R(λ,A) = R(λ0, A)

∞∑j=0

(λ0 − λ)jR(λ0, A)j =

∞∑j=0

(λ0 − λ)jR(λ0, A)j+1

for each λ in the prescribed open ball centered at λ0. Now, since the resolventoperator has a local power series representation, it must be an analytic function,and differentiating through the sum yields the desired formula for the nth-orderderivative of R(λ,A). Finally, consider a point λ0 ∈ C and a sequence (λn)n ⊂ρ(A) which converges to λ0. Assuming λ0 ∈ σ(A), the first claim of this lemmashows that,

dist(λ0, λn) ≥ 1

||R(λn, A)||(n ∈ N)

so rearranging and observing that dist(λ0, λn) converges to zero as n→∞,

∞ = lim infn→∞

1

dist(λ0, λn)≤ lim inf

n→∞||R(λn, A)||

and this implies limn→∞ ||R(λn, A)|| =∞, as desired. For the converse implica-tion, assume that the resolvent norms tend to∞, but suppose for a contradictionthat λ0 ∈ ρ(A). Then, we have,

{λn | n ∈ N}

is closed and bounded, and hence compact in C. Now, as the resolvent map iscontinuous, it must be bounded on this set, which yields a contradiction.

The next preliminary concept is that of the so-called “adjoint semigoup.” Ingeneral, given {T (t)}t≥0, the adjoint semigroup is the collection of operators,

{T ∗(t) ∈ L (X ′) | t ≥ 0} T ∗(t) := [T (t)]∗

which is necessarily a semigroup, but is not necessarily strongly continuous. Asan example, the semigroup of left translations on L1(R) is strongly continuous,but its adjoint consists of the right translations of L∞(R), which is not a stronglycontinuous semigroup. Specifically, if the underlying space is reflexive, then theadjoint semigroup must be strongly continuous, and what’s more, its generator,A∗, must be the adjoint of the original semigroup generator, A, and we will alsohave that σ(A∗) = σ(A).

Lastly, recall that in the Hille-Yosida theorem for contraction semigroups, werepresented the resolvent of A at the point λ in terms of the Laplace Transformof the semigroup. This yields two important ideas,

1. We can represent the resolvent along a vertical line in the complex planein terms of the Fourier Transform of the semigroup.

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2. Representation of the resolvent in terms of the semigroup begs the questionof whether the semigroup has a representation in terms of the resolvent.

Both 1 and 2 arise naturally in the proof Gearhart-Pruss, and specifically, 2 isthe notion of “semigroup inversion.” This area is rather technical, and we willsimply need one result, whose proof can be found in [2].

Lemma. (Semigroup Inversion) Let {T (t)}t≥0 be the strongly continuous semi-group generated by A. Then, for any x ∈ D(A2), we have that

T (t)x =j − 1

tj−1

1

2πilimn→∞

∫ ω+in

ω−ineztR(z,A)jxdz

will hold for any j ∈ N and t > 0, whenever ω > ω(A). Additionally, the integralconverges absolutely for any fixed positive t, and as a function of t, it convergesuniformly on compact subintervals of (0,∞).

Finally, we note that whenever A generates a strongly continuous semigrouponX, thenD(A2) must be dense inX as well, and this brings us to the Gearhart-Pruss Theorem. As a notational convention, we set C+ = {z ∈ C | <(z) > 0}.

Theorem. (Gearhart-Pruss) Suppose that A generates the strongly continuoussemigroup, {T (t)}t≥0, on the Hilbert space, H . Then, {T (t)}t≥0 is a uniformlystable semigroup iff C+ ⊆ ρ(A) and M = supλ∈C+

||R(λ,A)|| <∞.

Proof. Assume first that ω(A) < 0, and pick ω ∈ (ω(A), 0). Then,

C+ ⊂ {λ ∈ C | <(λ) > ω} ⊆ ρ(A)

and also, there exists an Mω ≥ 1 such that for every <(λ) > ω and n ∈ N,

||R(λ,A)n|| ≤ Mω

(<(λ)− ω)n

Thus, taking λ ∈ C+ and n = 1, we arrive at the estimate,

||R(λ,A)|| ≤ Mω

<(λ)− ω≤ Mω

−ω

which implies that M = supλ∈C+||R(λ,A)|| ≤ Mω

−ω <∞. Now, for the converseimplication, we first note that by the third claim from the “resolvent properties”lemma, we must have C+ ∪ iR ⊆ ρ(A), and moreover, the uniform estimate onthe norm of the resolvent operator will extend to this set by continuity. Second,setting ω > |ω(A)|+ 1, we may rescale our strongly continuous semigroup by,

T−ω(t) = e−ωtT (t)

so that {T−ω(t)}t≥0 is a uniformly stable, strongly continuous semigroup gen-erated by A − ω. We can now represent the resolvent operator of our original

12

semigroup along the vertical line ω + is ⊂ C+ in terms of the resolvent of ourrescaled semigroup, that is, for a fixed x ∈H and for s ∈ R,

R(ω + is, A)x =

∫ ∞0

e−(ω+is)tT (t)xdt

=

∫ ∞0

e−istT−ω(t)xdt = R(is, A− ω)x

specifically, setting T−ω(t) = 0 for all t < 0, we may then realize R(ω+is, A)x asthe Fourier Transform of the function T−ω(t)x ∈ L2(R,H ), so by Plancharel’sTheorem, we have,∫

R||R(ω + is, A)x||2ds = 2π

∫R||T−ω(t)x||2dt ≤ L · ||x||2

where we may find L > 0 by the uniform stability of the rescaled semigroup.Next, applying the resolvent identity, we observe,

R(is, A) = R(ω + is, A) + (is− (ω + is))R(is, A)R(ω + is, A)

= [I − ωR(is, A)]R(ω + is, A)

so for any x ∈H , we must have ||R(is, A)x|| ≤ (1 +Mω)||R(ω+ is, A)x||, andthis yields the estimate,∫

R||R(is, A)x||2ds ≤ (1 +Mω)2

∫R||R(ω + is, A)x||2ds

≤ (1 +Mω)2 · L · ||x||2

Now, since ||T (t)|| = ||T ∗(t)|| for all t ≥ 0, since C+ ∪ iR ⊆ ρ(A) ∩ ρ(A∗) (asthe spectra of A and A∗ coincide), and as the adjoint resolvent operator mustalso be uniformly bounded in norm on this set by M , we may perform the sameestimate for the adjoint semigroup, that is, for each y ∈H ,∫

R||R(is, A∗)||2ds ≤ (1 +Mω)2 · L · ||y||2

Finally, applying the inversion formula in the case where j = 2, we may fixx ∈ D(A2), y ∈H , and for any t > 0, write that,

|〈tT (t)x, y〉| =∣∣∣⟨ 1

2πilimn→∞

∫ n

−ne(ω+is)tR(ω + is, A)2xds, y

⟩∣∣∣=

1

2πlimn→∞

∣∣∣ ∫ n

−ne(ω+is)t〈R(ω + is, A)2x, y〉ds

∣∣∣where viewing the inner product as a continuous linear functional on H lets usfirst move the limit outside the inner product, and then move the inner product

13

under the integral. By Cauchy’s Theorem, we may then derive the upper bound,

≤ 1

2πlim supn→∞

∣∣∣ ∫ n

−neist〈R(is, A)2x, y〉ds

∣∣∣+

1

2πlim supn→∞

∣∣∣ ∫ ω

0

e(r+in)t〈R(r + in,A)2x, y〉dr∣∣∣

+1

2πlim supn→∞

∣∣∣ ∫ ω

0

e(r−in)t〈R(r − in,A)2x, y〉dr∣∣∣ = (#)

and noting that for any 0 6= λ ∈ C+ ∪ iR and x ∈ D(A2) ⊆ D(A), we must have

||R(λ,A)x|| = 1

|λ|||λR(λ,A)x|| = 1

|λ|||R(λ,A)Ax− x|| ≤ 1

|λ|

(M ||Ax||+ ||x||

)which means that we may upper bound (#) by,

(#) ≤ 1

2πlim supn→∞

∫ n

−n|〈R(is, A)2x, y〉|ds+ lim sup

n→∞

ωMeωt||y||(M ||Ax||+ ||x||)πn2

=1

2πlim supn→∞

∫ n

−n|〈R(is, A)x,R(−is, A∗y〉|ds

([R(is, A)]∗ = R(−is, A∗)

)≤ 1

2π

∣∣∣∣∣∣||R(is, A)x|| · ||R(−is, A∗)y||∣∣∣∣∣∣

1

≤ 1

2π

∣∣∣∣∣∣||R(is, A)x||∣∣∣∣∣∣

2·∣∣∣∣∣∣||R(−is, A∗)y||

∣∣∣∣∣∣2

Holder’s Inequality

=1

2π

[ ∫R||R(is, A)x||2ds

]1/2[ ∫R||R(is, A∗)y||2ds

]1/2≤ 1

2π·√

(1 +Mω)2 · L · ||x||2 ·√

(1 +Mω)2 · L · ||y||2

=L(1 +Mω)2||x|| · ||y||

2π

and lastly, since D(A2) is dense in H , we have that,

||tT (t)|| = sup{|〈tT (t)x, y〉| | x, y ∈ D(A2), ||x|| = ||y|| = 1}

≤ L(1 +Mω)2

2π

which implies limt→∞ ||T (t)|| = 0, so by the uniform stability theorem, we musthave ω(A) < 0, as required.

3 Spectral Theory of Semigroups and Generators

When A generates a strongly continuous semigroup on some Hilbert space, thelocation of σ(A) is important in terms of Gearhart-Pruss. More precisely, σ(A)needs to be contained by the strict left-half plane, C−, if there is to be any hopefor uniform stability.

14

Definition. (Spectral Bound) Letting A : D(A)→ X be a closed operator, it’sspectral bound is the number s(A) = sup{<(λ) | λ ∈ σ(A)}.

In the event that A generates a strongly continuous semigroup, the relations(A) ≤ ω(A) always holds, and we are especially interested in when equalitypersists, as this will allow us to investigate the stability of semigroup solutionsin terms of the spectrum of A.

The spectrum of any closed operator can be decomposed according to exactlyhow λ−A : D(A)→ X fails to be bijective. Specifically, when equality betweenthe spectral and growth bounds fails, these decompositions help us to pinpointprecisely what has gone wrong.

Definition. (Point Spectrum) Letting A : D(A)→ X be a closed operator, thesubset of the spectrum given by,

Pσ(A) = {λ ∈ σ(A) | λ−A is not injective}

is called the point spectrum of A.

For every λ ∈ Pσ(A), we must have that N [λ−A] 6= ∅, so each such λ is, inthis sense, a true eigenvalue of the linear map, A. Sometimes, we need to extendthe notion of a true eigenvalue to the collection of complex numbers which are“almost eigenvalues.”

Definition. (Approximate Point Spectrum) Letting A : D(A)→ X be a closedoperator, the subset of the spectrum given by,

Aσ(A) = {λ ∈ σ(A) | λ−A is not injective or R[λ−A] is not closed in X}

is called the approximate point spectrum of A.

Clearly, Pσ(A) ⊆ Aσ(A) always holds, and the following lemma makes itclear why the term “approximate” should be used for the larger set.

Lemma. (Approximate PS Characterization) Letting A : D(A)→ X be a closedoperator, we have that λ ∈ Aσ(A) iff there exists a unit-norm sequence (xn)n ⊂D(A) such that limn→∞ ||(λ−A)xn|| = 0.

For any λ ∈ σ(A) which is not an approximate eigenvalue, we must havethat λ−A is injective and R[λ−A] is closed in X. This forces the range not tobe a dense subset of X (else λ would be in the resolvent set), so we may collectthe non-approximate members of the spectrum as follows.

Definition. (Residual Spectrum) Letting A : D(A)→ X be a closed operator,the subset of the spectrum given by,

Rσ(A) = {λ ∈ σ(A) | R[λ−A] is not dense in X}

is called the residual spectrum of A.

15

Note that the approximate and residual spectra need not be disjoint, as anyλ ∈ Rσ(A) may give rise to a non-injective λ− A, or a range, R[λ− A], whichis neither closed, nor dense in X. However, we do have σ(A) = Aσ(A)∪Rσ(A).

Theorem. (Resolvent Spectral Mapping) Let A : D(A)→ X be a closed opera-tor with a nonempty resolvent set, ρ(A), and fix λ0 ∈ ρ(A). Then,

σ(R(λ0, A)) \ {0} ={ 1

λ0 − µ

∣∣∣ µ ∈ σ(A)}

and moreover, this relation holds individually for the point, approximate point,and residual spectra of A and R(λ0, A).

Proof. Fixing λ0 ∈ ρ(A), we have that for each nonzero λ ∈ C, and any x ∈ X,

[λ−R(λ0, A)]x = λ[1− 1

λR(λ0, A)

]x

= λ[(λ0 −

1

λ

)−A

]R(λ0, A)x

implying that if λ ∈ Pσ(R(λ0, A)), then λ0 − 1λ ∈ Pσ(A), and hence,

λ =1

λ0 − (λ0 − 1λ )∈{ 1

λ0 − µ

∣∣∣ µ ∈ Pσ(A)}

which shows the ⊆ containment for the point spectra. Conversely, given λ0− 1λ ∈

Pσ(A), then we must have λ ∈ Pσ(R(λ0, A)), so since

1

λ0 − (λ0 − 1λ )

= λ ∈ Pσ(R(λ0, A))

which shows the ⊇ containment, so the desired relation will hold for the pointspectra. Similar arguments show that the nonzero approximate point and resid-ual spectra must satisfy similar relations, and this completes the proof.

Indeed, so-called “spectral mapping” results will be the key to showing theopposite inequality, ω(A) ≤ s(A), which will allow us to investigate the stabilityof semigroup solutions through the lens of generator spectra. There are two mainspectral mapping results.

Theorem. (Spectral Inclusion One) Let A : D(A) → X generate the stronglycontinuous semigroup, {T (t)}t≥0, on the Banach space X. Then, for each t ≥ 0,the inclusion given by

etσ(A) = {etλ | λ ∈ σ(A)} ⊆ σ(T (t))

holds, and further, the above inclusion holds when restricted to the point, ap-proximate point, or residual parts of the spectrum.

16

Proof. First, we note that for any λ ∈ C, the shifted semigroup {e−λtT (t)}t≥0

is generated by A− λ. Then, by the Toolbox Lemma, we have that,

(eλt − T (t))x = −(A− λ)

∫ t

0

eλ(t−s)T (s)xds (x ∈ X)

=

∫ t

0

eλ(t−s)T (s)(λ−A)xds (x ∈ D(A))

and we now make the following observations:

1. Clearly, λ ∈ Pσ(A) implies etλ ∈ Pσ(T (t)), so etPσ(A) ⊆ Pσ(T (t)) holds.

2. In general, if λ ∈ Aσ(A), then the sequential characterization of approx-imate eigenvalues shows etAσ(A) ⊆ Aσ(A), and the point spectrum inclu-sion holds within this larger inclusion.

3. One can show that the norm-closure of R[etλ − T (t)] is contained by thenorm-closure of R[λ−A], so if λ ∈ Rσ(A), then etλ ∈ Rσ(T (t)) as well.

and hence,

etσ(A) = etAσ(A) ∪ etRσ(A)

⊆ Aσ(T (t)) ∪Rσ(T (t)) = σ(T (t))

where we have that etPσ(A) ⊆ Pσ(T (t)) within the approximate point spectruminclusion as well, and this completes the proof.

It turns out that once zero is removed from consideration (since e raised toany power is nonzero), the spectral inclusions for the point and residual spectrawill actually hold with equality.

Theorem. (Spectral Inclusion Two) For the generator, A : D(A)→ X, of thestrongly continuous semigroup {T (t)}t≥0 on the Banach space X, the identities,

etPσ(A) = Pσ(T (t)) \ {0}etRσ(A) = Rσ(T (t)) \ {0}

will hold for each t ≥ 0.

These spectral mapping results point towards our goal of showing that forthe generator, A, of a strongly continuous semigroup {T (t)}t≥0, the relation

etσ(A) = σ(T (t)) \ {0} (SMP)

holds for every t ≥ 0. When the SMP holds, we see that for t∗ > 0,

r(T (t∗)) = max{

sup{|λ|∣∣∣ λ ∈ σ(T (t∗)) \ {0}

}, 0}

≤ sup{et∗|µ|

∣∣∣ µ ∈ σ(A)}≤ et∗s(A)

17

where the penultimate inequality follows by the SMP. Now, taking the log ofboth sides and dividing by t∗, we get,

ω(A) =log r(T (t∗))

t∗≤ s(A)

and recalling that ω(A) ≥ s(A) always holds, the SMP implies that the spectralbound and growth bound of the generator A must coincide. Lastly, one has, bySpectral Inclusion Two, that any failure in the SMP must be due to a nonzeroapproximate eigenvalue, λ ∈ Aσ(A) \ Pσ(A).

4 Additional Topics: Analytic Semigroups, FractionalPowers, and Asymptotics Revisited

So far, we have focused primarily on the properties of generic strongly continuoussemigroups, briefly touching on uniformly continuous semigroups as necessary.In terms of regularity, the so-called analytic semigroups fall between strong anduniform continuity, and arising naturally from the concept of sectorial operators,they are of independent interest with respect to asymptotics.

Definition. (Sectorial Operators) A closed and densely-defined linear operator,A : D(A) → X, is said to be sectorial (of angle δ) if there exists a δ ∈ (0, π2 ]such that

Σπ2 +δ :=

{λ ∈ C

∣∣∣ 0 ≤ | arg(λ)| < π

2+ δ}\ {0}

is a subset of the resolvent set, ρ(A), and if, for every ε ∈ (0, δ), the estimate

||R(λ,A)|| ≤ Mε

|λ|

holds for some Mε ≥ 1 and every 0 6= λ ∈ Σπ2 +ε.

Given a sectorial operator of angle δ, we may define a one-parameter familyof operators on Σδ ∪ {0} as follows:

T :=

{T (0) := I

T (z) := 12πi

∫γeλzR(λ,A)dλ z ∈ Σδ

where, for a fixed ε ∈ (| arg(z)|, δ), the piecewise smooth curve γ is as depictedon the next page,

18

and where we observe that for any z ∈ Σδ, there will always exist such an ε, andwhere among all suitable ε, we may, by analyticity, choose the radius r > 0 tobe any number we wish. Indeed, the value of T (z), if it exists, will be invariantamong suitable ε and positive radii as a consequence of Cauchy’s Theorem. Thisis now the task at hand - to show that T (z) is well-defined for each z ∈ Σδ. Weestimate the three components of γ as follows, first:∣∣∣∣∣∣ 1

2πi

∫γ1

eλzR(λ,A)dλ∣∣∣∣∣∣ ≤ 1

2π

∫ −r−∞

∣∣∣e−s|z|ei(arg(z)−(π2

+ε))∣∣∣ · Mε

|s|ds

=Mε

2π

∫ ∞r

∣∣∣eu|z| sin(arg(z)−ε)∣∣∣ · 1

|u|du u = −s

=Mε

2π

∫ ∞r

1

|u|eu|z| sin(ε−arg(z))du <∞

where the convergence happens because 0 < ε − arg(z) < π, so it follows that0 < sin(ε − arg(z)) < 1. Now, the integral along the other unbounded portionof γ will converge absolutely by a similar argument, and the integral along thecircle of radius r is just a proper Riemann integral (so it is absolutely convergentalready). To summarize, we have not only shown that T (z) is well-defined foreach z ∈ Σδ, but that T ⊂ L (X) as well.

Theorem. (Properties of T ) Let A : D(A)→ X be a sectorial operator of angleδ and let T ⊂ L (X) be the family of operators defined as above. Then, we havethat z 7→ ||T (z)|| is analytic on Σδ, uniformly bounded on Σδ′ ∪ {0} for anyδ′ ∈ (0, δ), and strongly continuous for any such δ′ in the sense that,

limz→0

z∈Σδ′∪{0}

T (z)x = x (x ∈ X)

19

and lastly, that T (z + w) = T (z)T (w) for all z, w ∈ Σδ.

Proof. Fixing δ′ ∈ (0, δ), we see that z 7→ T (z) is continuous on the compact setΣδ′∩B1(0), and by the form of the estimate for absolute convergence, uniformlybounded on Σδ′ ∩B1(0)c. In other words, z 7→ T (z) must be uniformly boundedon Σδ′ ⊃ Σδ′ ∪ {0}, which proves the second claim.

Now, fixing z0 ∈ Σδ, there exists a closed ball Bρ(z0) ⊂ Σδ so that, for anyz ∈ Bρ(z0), we may express the (analytic) integrand as a power series to get,

T (z) =1

2πi

∫γ

[ ∞∑i=0

λi

i!(z − z0)i

∞∑j=0

(z0 − z)jR(z0, A)j+1]dλ

=1

2πi

∫γ

∞∑k=0

[ k∑l=1

λl

l!R(z0, A)k−1

](z − z0)kdλ

and since the integrand converges uniformly on Bρ(z0), we may pass the integralthrough both summations to see that T (z) may be locally expanded as a powerseries about z0, and hence, must be analytic on Σδ.

For strong continuity, it suffices to show the required identity on D(A), whichis dense in X by definition. Fixing x ∈ D(A), suppose that (zn)n ⊂ Σδ′ ∪ {0}with zn → 0 as n→∞, and consider

T (zn)x− x =1

2πi

∫γ

eznλR(λ,A)xdλ− 1

2πi

∫γ

eznλ

λxdλ = (#)

where we have used the identity, 1 =∫

Γeznλ

λ dλ, for the closed curve,

Γ := {−se−i(π2 +ε) | −N ≤ s ≤ 1} ∪ {eiθ | −(π

2+ ε) ≤ θ ≤ π

2+ ε}

∪ {sei(π2 +ε) | 1 ≤ s ≤ N} ∪ {<(Nei(π2 +ε)) + is | −N ≤ s ≤ N}

and taking N → ∞, we are left with∫γeznλ

λ dλ = 1. Returning to our originalproblem, we may then write,

(#) =1

2πi

∫γ

eznλ(R(λ,A)− 1

λ

)xdλ

=1

2πi

∫γ

eznλ

λR(λ,A)Axdλ

where we are recalling (from Hille-Yosida) that λR(λ,A)x − x = AR(λ,A)x =R(λ,A)Ax for x ∈ D(A). We may then estimate,∣∣∣∣∣∣eznλ

λR(λ,A)Ax

∣∣∣∣∣∣ ≤ Mε||Ax|||λ|2

|eznλ| <∞

where, taking r = 1 along γ, this integrand will be bounded above by the sameintegrand showing T ∈ T to be well-defined. Regarding these integrals as being

20

of Bochner type, we may then apply dominated convergence to get,

limn→∞

[T (zn)x− x] =1

2πi

∫γ

1

λR(λ,A)Axdλ = 0

where the equality on the RHS follows by a similar argument to our earlier useof the residue theorem, except we are now closing sections of γ on the right withcircles of increasing radius, instead of vertical lines on the left. As the sequence(zn)n ⊂ Σδ′ ∪{0} was arbitrary, the desired strong continuity of T now follows.Lastly, the semigroup property of T follows by arguments of a similar flavor.

Having established the strongly continuous one-parameter family T ⊂ L (X)which satisfies the semigroup property, it should be noted that a “usual” stronglycontinuous semigroup is embedded within this family. Perhaps not surprisingly,the sectorial operator A : D(A) → X, from which T is defined, is actually thegenerator of this embedded semigroup. This yields two important ideas:

1. Any sectorial operator of angle δ ∈ (0, π2 ] generates a strongly continuoussemigroup which can be extended analytically to a sector containing thenon-negative real numbers.

2. Verifying the sectoriality of an operator does not require checking infinitely-many conditions, as would be required by Hille-Yosida.

The family of operators T can be called an analytic semigroup in the sense thatit satisfies the conditions of the above theorem.

Definition. (Analytic Semigroup) For a fixed δ ∈ (0, π2 ], a family of boundedlinear operators, T := {T (z)}z∈Σδ ⊂ L (X), is called an analytic semigroup (ofangle δ) if,

1. T (0) = I and for all z, w ∈ Σδ, T (z + w) = T (z)T (w).

2. The map z 7→ T (z) is analytic on Σδ

3. For any δ′ ∈ (0, δ), the strong continuity condition, given by,

limz→0

z∈Σδ′∪{0}

T (z)x = x

holds for every x ∈ X.

4. For any δ′ ∈ (0, δ), the map z 7→ T (z) is uniformly bounded on Σδ′ ∪ {0}.

To summarize: any sectorial operator will give rise to an analytic semigroupof the form T , and generate the embedded strongly continuous semigroup. Asa final note, there are a variety of conditions which are equivalent to the secto-riality of an operator, see [2, 4].

The connection between analytic semigroups and asymptotics is establishedby the notion of fractional powers of a closed operator. Indeed, if A : D(A)→ X

21

is a sectorial operator of angle δ ∈ (0, π2 ] and 0 ∈ ρ(A), then for any α > 0, wedefine the operator,

A−α :=1

2πi

∫γ

λ−αR(λ,A)dλ

where γ runs from ∞e−iδ′ to ∞eiδ′ for a given δ′ ∈ (0, δ), and avoids R+ ∪ {0}by traversing a circle of sufficiently small radius to the left of zero. Additionally,for this definition to be unambiguous, we specify,

λ−α := e−α log(λ)

where the logarithm is the principal value logarithm of λ 6= 0, that is, log(λ) =log |λ|+ i arg(λ).

This definition of A−α actually holds for a general class of closed operatorswhich satsify certain conditions on their resolvent sets and on the norms of theirresolvent operators. In our present situation (where A is sectorial), we have theequivalent representation,

B−α =1

Γ(α)

∫ ∞0

tα−1T (t)dt B := −A

where T (t) is the embedded strongly continuous semigroup within the analyticsemigroup defined by A (see [4, 2.6]).

Let us now turn to the useful asymptotics results one may find in [1]. To bespecific, the following result will be our starting point.

Theorem. Let {T (t)}t≥0 be a bounded and strongly continuous semigroup on aBanach space, X, with an invertible generator, A : D(A)→ X. If, in addition,iR ⊂ ρ(A), then we have

||T (t)A−1|| → 0 (t→∞)

In other words, for any y ∈ D(A), with y = Ax for some x ∈ D(A), the solution

u(t) := T (t)y

to the ACP converges to 0 as t→∞.

We are interested in examining the rate of this solution decay. In particular,let us define the following functions,

M(η) := maxt∈[−η,η]

||R(it, A)|| (η ≥ 0)

which is continuous, and obviously non-decreasing. Then, we write an associatedfunction,

Mlog(η) := M(η)[

log(1 +M(η)) + log(1 + η)]

(η ≥ 0)

which is clearly also continuous, and strictly increasing as well. To be specific,Mlog satisfies these properties on [0,∞), and accordingly, has an inverse, M−1

log ,defined on [Mlog(0),∞). This leads to the Batty-Duyckaerts Theorem, whichputs a specific rate on the solution decay from the above theorem.

22

Theorem. (Batty-Duyckaerts) Let {T (t)}t≥0 be a bounded and strongly con-tinuous semigroup on a Banach space, X, with an invertible generator, A :D(A)→ X. If, in addition, iR ⊂ ρ(A), and M and Mlog are defined as above,then there must exist constants B,C ≥ 0 such that

||T (t)A−1|| ≤ C

M−1log ( tC )

(t ≥ B)

and in particular, given some α > 0, with M(η) ≤ C(1 + ηα), then (via somealgebra) the above estimate becomes

||T (t)A−1|| ≤ C( log(t)

t

) 1α

(t ≥ B)

and this estimate is essentially sharp, in the sense that it cannot be improvedin arbitrary Banach spaces, but it can be made better if the underlying spacehappens to be a Hilbert space.

Let us now discuss this type of semigroup decay in more detail, for whichwe will need a few lemmas found in [1].

Lemma. (Bounded Semigroups on a Hilbert Space) Let {T (t)}t≥0 be a stronglycontinuous semigroup on a Hilbert space, H , with generator A : D(A) → H .Then, {T (t)}t≥0 is bounded (i.e. ||T (t)|| ≤M) if and only if C+ ⊂ ρ(A) and

supζ>0

ζ

∫R

(||R(ζ + iη, A)x||2 + ||R(ζ + iη, A∗x||2

)dη <∞

for every x ∈H .

Proof. Let us assume first that {T (t)}t≥0 is bounded so ||T (t)|| ≤ M for eacht ≥ 0. Then, as ω(A) ≤ 0, we must have C+ ⊂ ρ(A) automatically, and forx ∈H , we may write

R(ζ + iη, A)x =

∫ ∞0

e−(ζ+iη)tT (t)xdt

=

∫Re−iηtTζ(t)xdt

where Tζ(t) is defined as in Tω(t) in the proof of Gearhart-Pruss. Then, byPlancharel’s Theorem, it follows that∫

R||R(ζ + iη, A)x||2dη = 2π

∫ ∞0

||e−ζT (t)x||dt ≤ C1

ζ

and a similar estimate holds so that∫R||R(ζ + iη, A∗)x||2dη ≤ C2

ζ

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and hence for any ζ > 0 and x ∈H , we have

ζ

∫R

(||R(ζ + iη, A)x||2 + ||R(ζ + iη, A∗)x||2

)dη ≤ ζ

(C1 + C2

ζ

)= C1 +C2 <∞

which proves the desired result upon taking the supremum over positive ζ.Conversely, let us assume that C+ ⊂ ρ(A) and that

supζ>0

ζ

∫R

(||R(ζ + iη, A)x||2 + ||R(ζ + iη, A∗)x||2

)dη <∞

for every x ∈H , and we’ll try to show that {T (t)}t≥0 is a bounded semigroup.Indeed, for x ∈H and λ ∈ C+, let us write

λ 7→ R(λ,A)x =

∫ ∞0

e−λtT (t)xdt

Then, we may (complex) differentiate both sides with respect to λ to get

−R(λ,A)2x =d

dλ

∫ ∞0

e−λtT (t)xdt

= −∫ ∞

0

te−λtT (t)xdt = −L(tT (t)x)

where L is the Laplace Transform. Applying the inverse Laplace Transform,L−1, and setting ζ := <(λ), we get that

t 7→ tT (t)x = L−1(R(λ,A)2x)

=1

2πi

∫ ∞−∞

e(ζ+is)tR(ζ + is, A)2xds

so for a fixed t ≥ 0, let us now divide by t and for an arbitrary h ∈H , consider

〈T (t)x, h〉 =⟨ 1

2πit

∫ ∞−∞

e(ζ+is)tR(ζ + is, A)2xds, h⟩

=1

2πit

∫ ∞−∞

e(ζ+is)t〈R(ζ + is, A)2x, h〉ds

where we may move the inner product under the integral sign just as in theproof of Gearhart-Pruss. Now, by Cauchy-Schwarz and a special case of Young’sInequality (2ab ≤ a2 + b2), we get that

|〈T (t)x, h〉| ≤ eζt

2πt

∫ ∞−∞|〈R(ζ + is, A)x,R(ζ − is, A∗)h〉|ds

≤ eζt

2πt

∫ ∞−∞||R(ζ + is, A)x|| · ||R(ζ − is, A∗)h||ds

≤ eζt

4πt

∫ ∞−∞

(||R(ζ + is, A)x||2 + ||R(ζ − is, A∗)h||2

)ds

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Then, since we were free to choose ζ := <(λ) = 1t , we may write

|〈T (t)x, h〉| ≤ Cζ∫ ∞−∞

(||R(ζ + is, A)x||2 + ||R(ζ − is, A∗)h||2

)ds

which, by some algebra and our supremum assumption is finite. Thus, {T (t)}t≥0

is a bounded semigroup by the uniform boundedness principle.

Rather than assuming the invertibility of A as in the previous two theoremsof this section, we may just define its fractional powers as a closed operator,and this yields the following idea.

Lemma. (Bounded Semigroups on a Hilbert Space 2) Let {T (t)}t≥0 be a bounded,strongly continuous semigroup on a Hilbert space, H , with generator A : D(A)→H . Then, if iR ⊂ ρ(A), we have that for a fixed α > 0,

||R(λ,A)(−A)−α|| ≤ C (<(λ) > 0)

if and only if||R(is, A)|| = O(|s|α)

Proof. It has already been shown (see [1] references) that

||R(λ,A)|| ≤ C(1 + |λ|α) (0 < <(λ) < 1)

holds if and only if the condition

||R(λ,A)(−A)α|| < C1 (0 < <(λ) < 1)

holds as well. Then, let us define the function

F (λ) := R(λ,A)λ−α(

1− λ2

B2

)B � 1

where λ−α = e−α log(λ) with the complex logarithm taking its principle value,and with λ being a member of the set,

D := {λ ∈ C | <(λ) ≥ 0 and 1 ≤ |λ| ≤ B}.

We observe that ∂D consists of three individual pieces, namely,

1. D1 = {eiθ | −π2 ≤ θ ≤π2 }

2. D2 = {is | 1 ≤ s ≤ B}

3. D3 = {Beiθ | −π2 ≤ θ ≤π2 }

and then, applying the maximum principle to F , we need only worry about it’svalue on the boundary, which, by the form of F , implies the desired result.

Let us now state and prove the central result of Borichev and Tomilov (asstated in [1])

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Theorem. (Borichev and Tomilov) Let {T (t)}t≥0 be a bounded, strongly con-tinuous semigroup on a Hilbert space, H with generator A such that iR ⊂ ρ(A).Then, for a fixed α > 0, the following conditions are equivalent:

1. ||R(is, A)|| = O(|s|α) as s→∞

2. ||T (t)(−A)−α|| = O(t−1) as t→∞

3. ||T (t)(−A)−α|| = o(t−1) as t→∞

4. ||T (t)A−1|| = O(t−1/α) as t→∞

5. ||T (t)A−1x|| = o(t−1/α) as t→∞

Proof. Every statement besides 1. =⇒ 3. has been previously shown (see [1]references), so let us tackle this remaining implication. Write

H := H ⊕H

and define the operator

A :=

(A (−A)−α

O A

)on the diagonal domain D(A) := D(A)⊕D(A). Then, it is not hard to see thatσ(A) = σ(A), and for each λ ∈ ρ(A), that the resolvent operator takes the form

R(λ,A) =

(R(λ,A) R2(λ,A)(−A)−α

O R(λ,A)

)λ ∈ ρ(A) = ρ(A)

Next, let us define the one-parameter family of operators

T (t) :=

(T (t) tT (t)(−A)α

O T (t)

)which, as it is easy to see, is a strongly continuous semigroup on H. Moreinteresting is its generator. Indeed, consider x := (x1, x2) ∈ D(A), and write

T (h)x− xh

=

(T (h) hT (h)(−A)−α

O T (h)

)·(x1

x2

)−(x1

x2

)h

=

(T (h)x1 + hT (h)(−A)−αx2

T (h)x2

)−(x1

x2

)h

=

(T (h)x1−x1

h + T (h)(−A)−αx2T (h)x2−x2

h

)→(Ax1 + (−A)−αx2

Ax2

)(h→ 0+)

In other words, {T (t)}t≥0 is a strongly continuous semigroup whose generatortakes the form of A acting on the diagonal domain D(A) as defined above. By

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our assumption 1. and the fact that iR ⊂ ρ(A) = ρ(A), it follows by the secondlemma that

||R(λ,A)(−A)−α|| ≤ C (<(λ) > 0)

which then implies for x = (x1, x2) ∈ H and λ ∈ C+ that

||R(λ,A)x||2 =

∥∥∥∥(R(λ,A) R2(λ,A)(−A)−α

O R(λ,A)

)·(x1

x2

)∥∥∥∥2

≤ ||R(λ,A)x1 +R2(λ,A)(−A)−αx2||2 + ||R(λ,A)x2||2

≤ (||R(λ,A)x1||+ ||R(λ,A)(−A)αx2||)2 + ||R(λ,A)x2||2

≤ (||R(λ,A)x1||+ C||R(λ,A)x2||)2 + ||R(λ,A)x2||2 = (#)

where the C ≥ 0 follows by commutativity and the bound for ||R(λ,A)(−A)α||.Then, we have

(#) = ||R(λ,A)x1||2 + 2C||R(λ,A)x1|| · ||R(λ,A)x2||+ (C2 + 1)||R(λ,A)x2||2

≤ ||R(λ,A)x1||2 + 2C( ||R(λ,A)x1||2

2+||R(λ,A)x2||2

2

)+ (C2 + 1)||R(λ,A)x2||2

which results from Young’s Inequality in the case p = q = 2. Finally, we havethat the above may be bounded as

≤ ||R(λ,A)x1||2 + C||R(λ,A)x1||2 + C||R(λ,A)x2||2 + (C2 + 1)||R(λ,A)x2||2

= (1 + C)||R(λ,A)x1||2 + (C2 + C + 1)||R(λ,A)x2||2

≤ max{1 + C,C2 + C + 1} · (||R(λ,A)x1||2 + ||R(λ,A)x2||2)

In particular, performing the same estimate for ||R(λ,A∗)x||2, we have that foreach x = (x1, x2) ∈ H and λ ∈ C+,

||R(λ,A)x||2 ≤ C1(||R(λ,A)x1||2 + ||R(λ,A)x2||2) <∞

and||R(λ,A∗)x||2 ≤ C2(||R(λ,A∗)x1||2 + ||R(λ,A∗)x2||2) <∞

by the first of the above lemmas, since {T (t)}t≥0 is a bounded semigroup. Usingthis lemma again, we see that T must be a bounded semigroup as well. Bydefinition, this means

supt≥0||tT (t)(−A)−α|| <∞

and by the density of D(A) in H, along with the first theorem of this decaysection, it follows that

||tT (t)(−A)−α|| = o(1) (t→∞ and x ∈H )

which completes the proof.

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References

[1] Borichev, A., Tomilov, Y. (2010). Optimal Polynomial Decay of Functionsand Operator Semigroups. Mathematische Annalen. 347: 455-478. Springer-Verlag.

[2] Engel, K.J., Nagel, R. (2000). One-Parameter Semigroups for Linear Evo-lution Equations. New York, NY. Springer-Verlag.

[3] Satbir’s Notes

[4] Pazy, A. (1983). Semigroups of Linear Operators and Applications to PartialDifferential Equations. New York, NY. Springer-Verlag.

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