OPERATOR SEMIGROUPS

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OPERATOR SEMIGROUPS

St´ephane ATTAL

Abstract This lecture is an introduction to the theory of Operator Semi- groups and its main ingredients: different types of continuity, associated generator, dual and predual semigroups, Stone’s Theorem. The lecture also starts with a complete introduction to the Bochner integral.

3.1 Bochner Integral . . . . 2

3.1.1 Dual Topologies on Banach Spaces . . . . 2

3.1.2 Bochner Measurability . . . . 4

3.1.3 Construction of the Bochner Integral . . . . 6

3.1.4 Actions of Operators . . . . 10

3.2 Operator Semigroups. . . . 11

3.2.1 Definitions . . . . 11

3.2.2 Uniform Continuity . . . . 12

3.2.3 Strong and Weak Continuity . . . . 15

3.3 Generators. . . . 18

3.3.1 Definition . . . . 18

3.3.2 Spectrum . . . . 20

3.3.3 Dual Semigroup . . . . 21

3.3.4 Predual Semigroups . . . . 25

3.3.5 Stone’s Theorem . . . . 26

References . . . . 28

St´ephane ATTAL

Institut Camille Jordan, Universit´e Lyon 1, France e-mail: attal@math.univ-lyon1.fr

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This lecture makes use of basic elements of bounded and unbounded operator theory. The elements we use are rather elementary in most of the lecture:

domains, closed operators, spectrum, etc; only when dealing with Stone’s Theorem we make use of the functional calculus for general self-adjoint operators. All these ingredients may be found in Lecture 1.

3.1 Bochner Integral

Before entering into the theory of operator semigroups we shall make clear basic constructions and properties of the Bochner integrals, that is, the integration theory for functions taking values in Banach spaces. These notions and results are not only useful when considering operator semigroups but they are also be used over and ever in many subject of interest for us, in particular when dealing with integration on Fock spaces or with Quantum Stochastic Calculus.

3.1.1 Dual Topologies on Banach Spaces

We shall make use of many well-known properties associated to the different topologies associated to Banach spaces. We recall without proof the main results that we use. They are all very classical theorems, proofs may be easily found in the usual literature (see the notes at the end of the chapter).

Definition 3.1.LetBbe a Banach space with a normk·k. The dual ofB is the spaceB^∗ of continuous linear forms onB.

Recall that we denote the action onBof a linear form φbyB 7→ hφ , Bi instead ofB7→φ(B).

Theorem 3.2.When equipped with the norm

kφk= sup{|hφ , Bi|; B∈ B, kBk= 1}

the space B^∗ is a Banach space.

Definition 3.3.The dual spaceB^∗ofB allows to define aweak topology on B. It is the topology on B induced by the seminorms B 7→ |hφ , Bi|, for all φ∈ B^∗.

Definition 3.4.Note that every elementB ∈ B defines a continuous linear formφ7→ hφ , BionB^∗, henceBis a subset ofB^∗∗, the dual ofB^∗. In general B^∗∗ is not equal toB. When this is the case, the spaceB is calledreflexive.

Every Hilbert space is a reflexive Banach space, but this is not the only case.

For example, the spacesL^p(X,F, µ) are reflexive if 1< p <∞. The space

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L¹(X,F, µ) is a well-known example of Banach space which is not reflexive in general.

Definition 3.5.On the Banach spaceB^∗ one can define also a weak topology: the∗-weak topology, also often calledσ-weak topology. It is the smallest topology on B^∗ which makes the linear formsφ7→ hφ , Bibeing continuous, for allB∈ B. It is generated by the seminormsφ7→ |hφ , Bi|for allB∈ B.

Concerning these two weak topologies, here are the main theorems which will be of much use for us. First of all, here are two very basic results concerning the weak and the norm topologies onB.

Proposition 3.6.

1) The weak topology is weaker than the norm topology.

2) The weak and the norm closure of convex sets coincide.

The first basic results concerning the∗-weak topology onB^∗ are the following.

Theorem 3.7.

1) OnB^∗ the weak topology is stronger than the∗-weak topology.

2) The space B^∗ is closed for the ∗-weak topology.

3) A linear form λ on B^∗ is ∗-weakly continuous if and only if there exists an element B∈ B such that

λ(φ) =hφ , Bi

for allφ∈ B^∗.

The main interest of the∗-weak topology is the following result: this topology makes the bounded balls ofB^∗ being compact, which is not the case of the other topologies in infinite dimension.

Theorem 3.8 (Banach-Alaoglu Theorem). The unit ball ofB^∗ is compact for the ∗-weak topology.

Finally, a theorem which is very useful in order to prove that a linear form is∗-weakly continuous.

Theorem 3.9 (Krein-Smulian Theorem). A convex subsetC inB^∗ is∗- weakly closed if and only if its intersection with the balls{φ∈ B^∗; kφk ≤r}

of B^∗ are ∗-weakly closed.

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3.1.2 Bochner Measurability

Before entering into the construction of the Bochner integral we shall establish a few technical lemmas concerning the notion of measurability we shall use for defining these integrals.

The general setup in which we shall be interested is the following. We consider a measured space (E,E, µ), with the measureµ being σ-finite. We consider a Banach spaceBand a functionf from E toB.

Definition 3.10.We say that the functionf isweakly measurable if, for all φ∈ B^∗, the mapping x7→ hφ , f(x)iis measurable from (E,E) toC.

We say that the function f has a separable range if its range f(E) is a separable subset ofB.

There are two cases for which this second property is satisfied and which will be very useful for us.

Proposition 3.11.

1) If B is a separable Banach space, then any function f : E 7→ B has a separable range.

2) Whatever is the Banach space B, ifE is a separable topological space and if the functionf :E7→ Bis continuous, then f has a separable range.

Proof. The proof of 1) is immediate for any subset of a separable metric space is separable (exercise). The proof of 2) is also immediate for the continuous image of a separable space is always separable (exercise). ut

Remark:Actually all the discussion that we shall have here does not exactly needsf to have a separable range, but analmost surely separable range, that is, there existsN ∈ Esuch thatµ(N) = 0 and such that the setf(E\N)⊂ B is separable. If the function f from E to B has an almost surely separable range, then we can changef on the null setN into a function fewhich has full range being separable. But it is easy to check that the constructions and the theorems which follow concerning the Bochner integral off with respect toµare not affected by this change on a null set.

Definition 3.12.From now on, a function f from E to B which is weakly measurable and has a separable range is called a Bochner measurable function.

We now prove a kind of separability property ofB^∗, whenB is separable.

Put

C₁={φ∈ B^∗; kφk= 1}

Lemma 3.13.IfBis a separable Banach space, then there exists a countable setY included in C1such that, for all φ∈ C1, there exists a sequence (φn)in Y satisfying

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n→+∞lim hφn, yi=hφ , yi for ally∈ B.

Proof. Let X ={yn; n ∈ N} be a countable dense subset of B. For every n∈N^∗ consider the mappingΦ_n fromC₁to Rⁿ defined by

Φ_n(φ) = (hφ , y₁i, . . . ,hφ , y_ni).

As Rⁿ is a separable metric space then Φn(C1) ⊂ Rⁿ is also separable.

Hence, for all n ∈ N^∗, there exists {φn,k ∈ C1; k ∈ N} such that the set {Φn(φn,k) ; k∈N} is dense in Φn(C1). In particular, for every φ∈ C1 there exists a sequence{φn,k_n; n∈N}such that

|hφ_n,k_n, y_ii − hφ , y_ii| ≤ 1 n

for alli= 1, . . . , n. In particular, lim_nhφ_n,k_n, y_ii=hφ , y_iifor alli∈N. This implies easily that limnhφn,k_n, yi=hφ , yifor ally∈ B. ut

This lemma has the following useful consequence.

Lemma 3.14.If f is a Bochner measurable function fromE toB then the function x7→ kf(x)k is measurable from E toR.

Proof. Fora >0 andφ∈ B^∗consider the sets A={x∈E; kf(x)k ≤a}

and

A_φ ={x∈E; |hφ , f(x)i| ≤a}. We obviously have

A⊂ \

φ∈C1

A_φ.

By the Hahn-Banach Theorem, for every x∈ E there exists φ0 ∈ C1 such that hφ0, f(x)i=kf(x)k; this implies that

\

φ∈C1

Aφ⊂A

and hence the two sets are equal.

Consider the closureXof the linear span of the range off. It is a separable Banach space by hypothesis. Let Y be the countable set associated to X by Lemma 3.13. We claim that we have

A= \

φ∈C1

Aφ= \

φ∈Y

Aφ.

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Indeed, one inclusion is obvious and, for the converse inclusion, consider x ∈ E such that |hψ , f(x)i| ≤ a for all ψ ∈ Y. By Lemma 3.13 every hφ , f(x)i, with φ∈ C₁, is a limit of a sequence hψ_n, f(x)i), with (ψ_n)⊂ Y.

Hence, we have |hφ , f(x)i| ≤ a for all φ ∈ C₁. This proves the announced equality of sets.

Finally, asf is weakly measurable, the sets Aφ are measurable, hence so isA. This proves the measurability of x7→ kf(x)k. ut

3.1.3 Construction of the Bochner Integral

We can now pass to the construction of the Bochner integral. Let f be Bochner measurable function from E to B. By Lemma 3.14 the function x7→ kf(x)k is measurable fromE to R.

Definition 3.15.We say thatf isBochner integrable if furthermore Z

E

kf(x)kdµ(x)<∞.

We wish to show thatf being Bochner integrable is a sufficient condition for the integral

Z

E

f(x) dµ(x) to be a well-defined and to be an element ofB.

The construction actually follows the usual construction of real-valued integrals. First of all, assume thatf is asimple function, that is, f is of the form

f(x) =X

n∈N

fn1lA_n(x)

for some measurable partition (An)⊂ E ofEand somefn∈ B. Clearly such a function f is Bochner measurable. Iff is furthermore Bochner integrable, that is, if

X

n∈N

kfnk µ(An)<∞, we put

Z

E

f(x) dµ(x) =X

n∈N

µ(A_n)f_n.

This way we have defined an element ofB and clearly we have

Z

E

f(x) dµ(x)

≤ Z

E

kf(x)kdµ(x).

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The difference with the usual integration theory for real-valued functions is that, in the general Banach context, approximating measurable functions by simple functions is not obvious.

Lemma 3.16.Let f be a Bochner measurable function from E to B, then there exists a sequence(fn)of simple functions from E toB such that

n→+∞lim sup

x∈E

kf(x)−f_n(x)k= 0.

Proof. LetXbe the closure of the range off. AsXis separable by hypothesis there exists a countable subsetY ={yn; n∈N} ⊂ X which is dense in X.

For alln∈Ndefine the functiongn from E toRby gn(x) =kf(x)−ynk .

Asf is weakly measurable all the functions x7→f(x)−yn are weakly measurable. They also obviously have separable range. Hence, by Lemma 3.14, the functionsgn are measurable.

For alln∈Nand allk∈N^∗define the setsAⁿ_k ={x∈E; gn(x)≤1/k}. They are measurable subsets ofE. AsY is dense inX, for everyx∈E, every k∈N^∗ there exists at least ay_n ∈ Y such thatkf(x)−y_nk ≤1/k. Hence we have E=∪nAⁿ_k, for allk∈N^∗.

For all fixedk∈N^∗ define for alln≥1

B_k⁰=A⁰_k and B_kⁿ=Aⁿ_k\ [

j<n

A^j_k.

They are measurable subsets of E and, for each fixed k ∈ N^∗, the family (B_kⁿ)_n∈N forms a partition ofE. Define the functions

fk(x) =X

n∈N

yn1lBⁿ_k(x). We clearly have

kf(x)−fk(x)k ≤ 1 k,

for allx∈E. We have proved the uniform approximation property. ut Corollary 3.17.Iff is a Bochner integrable function fromEtoBthen there exists a sequence(fn) of simple and Bochner integrable functions fromE to B such that

n→+∞lim Z

E

kf(x)−f_n(x)kdµ(x) = 0.

Proof. Asµisσ-finite andkf(·)kis integrable, for everyε >0 there exists a measurable setK∈ E such thatµ(K)<∞and

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Z

E\K

kf(x)k dµ(x)≤ε .

Consider a sequence (fk) such as given by Lemma 3.16. We have Z

E

kf(x)−f_k1l_K(x)kdµ(x)≤ Z

E

kf(x)−f(x)1l_K(x)kdµ(x)+

+ Z

E

kf(x)−f_k(x)k1l_K(x) dµ(x)

≤ε+1 kµ(K). It is easy to conclude now. ut

In particular, the sequence (R

Efn(x) dµ(x))_n∈Nis Cauchy inBfor

Z

E

f_n(x)−f_m(x) dµ(x)

≤ Z

E

kfn(x)−f_m(x)k dµ(x)

which tends to 0 whennandmtend to +∞. Hence this sequence converges in B.

Definition 3.18.We denote by Z

E

f(x) dµ(x)

its limit inB. This integral is called theBochner integral off with respect to the measure µ.

Note that this limit does not depend on the choice of the sequence (fn) (exercise). This limit obviously satisfies the inequality

Z

E

f(x) dµ(x)

≤ Z

E

kf(x)kdµ(x). (3.1)

We now give a useful characterization of the Bochner integral in terms of the action of elements ofB^∗

Theorem 3.19.Letf be a Bochner measurable and Bochner integrable function from(E,E, µ)toB. Then its Bochner integralR

Ef(x) dµ(x)is the unique element ofB which satisfies

φ ,

Z

E

f(x) dµ(x)

= Z

E

hφ , f(x)idµ(x), (3.2) for allφ∈ B^∗.

Proof. Let us first check that the integrals

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Z

E

hφ , f(x)idµ(x)

are well-defined. As f is Bochner measurable, the functionx7→ hφ , f(x)iis measurable fromE toC. It is also absolutely integrable for

Z

E

|hφ , f(x)i|dµ(x)≤ Z

E

kφk kf(x)k dµ(x). Hence the integral above is well-defined.

We claim that the mapping λ : φ 7→ R

Ehφ , f(x)idµ(x) is *-weakly- continuous. Indeed, if (φn) converges *-weakly to φ ∈ B^∗ then hφn, f(x)i converges to hφ , f(x)i for all x (by hypothesis) and the sequence (φn) is bounded (by the Uniform Boundedness Principle). By Lebesgue’s Theorem this shows that λ(φn) converges to λ(φ). This proves that λ is *-weakly- continuous.

By Theorem 3.7 this proves that there exists a unique elementg∈ Bsuch that

λ(φ) =hφ , gi. The fact the the Bochner integral g = R

Ef(x)dx satisfies the relation (3.2) is obtained easily for simple functions. One then passes to the limit for general Bochner integrable functions. ut

In the case of separable Hilbert spaces the definition of the Bochner integral becomes much simpler. This result is an easy consequence of the results established previously, we leave the proof to the reader.

Theorem 3.20.Let H be a separable Hilbert space and let f be a function fromE toH.

1) The function f is Bochner measurable if and only if, for all φ∈ H, the function x7→ hφ , f(x)iis measurable from E toC.

2) If f is Bochner integrable then, for allφ∈ H, we have Z

E

|hf(x), φi|dµ(x)<∞ and the Bochner integral R

Ef(x) dµ(x) is the unique element of H which satisfies

Z

E

f(x) dµ(x), φ

= Z

E

hf(x), φidµ(x) (3.3) for allφ∈ H.

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3.1.4 Actions of Operators

The usual properties of Integration Theory, such as additivity, linearity, Change of Variable Formula (whenE=Rⁿ), Fundamental Theorem of Cal- culus (whenE=R), Lebesgue’s Dominated Convergence Theorem, ... remain valid in the context of Banach-valued functions. We shall not develop these results here. In this subsection we concentrate only on results which are spe- cific to Bochner integrals and which admit no interesting equivalent result for usual integrals in finite dimension: the behavior of Bochner integrals with respect to the action of linear operators.

Proposition 3.21.If f is a Bochner integrable function from E to B and if T is a bounded operator on B, then x7→Tf(x) is Bochner integrable too.

Furthermore we have T

Z

E

f(x) dµ(x) = Z

E

Tf(x) dµ(x). (3.4)

Proof. Ifφ is a continuous linear form onB then φ◦T is also a continuous linear form on B. This shows that Tf is weakly measurable. The range of Tf is the continuous image by T of the range of f, hence it is separable too. The norm-integrability ofTf is obvious. This proves thatTf is Bochner integrable.

The identity (3.4) is immediate whenf is simple. We then conclude easily in the general case, using an approximation by simple functions (Corollary 3.17) and using the continuity ofT. ut

In the case of Hilbert spaces we have an extension of Proposition 3.21, but first of all we need a little lemma on measurability.

Lemma 3.22.Let Tbe a closable operator fromDomT⊂ HtoHand letf be a Bochner measurable function from E toHsuch that f(x)∈DomT for all x∈E. Then the function x7→Tf(x)is Bochner measurable.

Proof. As T is closable it admits a densely defined adjoint T^∗. For all ψ ∈ DomT^∗ and allx∈Ewe have

hψ ,Tf(x)i=hT^∗ψ , f(x)i.

In particular, for allψ∈DomT^∗, the functionx7→ hψ ,Tf(x)iis measurable (by Proposition ??). As DomT^∗ is dense in H, it is easy to see that the mapping x7→ hψ ,Tf(x)i is measurable from E to C, for all ψ ∈ H. This gives the Bochner measurability ofTf, by Proposition??again. ut

Proposition 3.23.Let f be a Bochner integrable function fromE to a separable Hilbert space H. Let T be a closable operator on H, with closure T, such that f(x)belongs toDomT for allx∈E and such that

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Z

E

kTf(x)kdµ(x)<∞. ThenR

Ef(x) dµ(x)belongs toDomT and T

Z

E

f(x) dµ(x) = Z

E

Tf(x) dµ(x). (3.5)

Proof. By Lemma 3.22 the functionTfis Bochner measurable. By hypothesis it is also norm-integrable, hence the Bochner integralR

ETf(x) dµ(x) is well- defined.

AsTis closable it admits a densely adjointT^∗. Letψ∈DomT^∗, then

T^∗ψ , Z

E

f(x) dµ(x)

= Z

E

hT^∗ψ , f(x)idµ(x) (by Theorem 3.20)

= Z

E

hψ ,Tf(x)idµ(x)

=

ψ , Z

E

Tf(x) dµ(x)

(by Theorem 3.20). This proves thatR

Ef(x) dµ(x) belongs to the domain ofT^∗∗ =T and that T

Z

E

f(x) dµ(x) = Z

E

Tf(x) dµ(x). This proves the announced relation (3.5). ut

3.2 Operator Semigroups

We now discuss the basic properties of operator semigroups and in particular their different notions of continuity.

3.2.1 Definitions

Definition 3.24.LetBbe a Banach space. Anoperator semigroup, or simply asemigroup, onBis a family (Tt) of bounded linear operators onB, indexed byt∈R⁺, such that

i)T0= I,

ii)TsTt=Ts+tfor alls, t∈R⁺.

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In general these two conditions are far too general and a continuity assumption has to be added. There are several types of continuity conditions that are usually considered.

Definition 3.25.A semigroup (Tt) onBis

– uniformly continuous ift7→Tt is continuous for the operator-norm, – strongly continuous if, for all f ∈ B, the mapping t 7→Ttf is continuous fromR⁺toB,

– weakly continuous if, for allf ∈ B, allφ∈ B^∗, the mappingt7→ hφ ,Ttfi is continuous fromR⁺toC,

– continuous if the mapping (t, f)7→T_tf is continuous fromR⁺× BtoB.

Clearly the uniform continuity implies the strong continuity, which itself implies the weak continuity. The continuity implies the strong continuity. We shall see later on more relations between these different continuity notions for semigroups.

3.2.2 Uniform Continuity

The case of uniformly continuous semigroups is the simplest one and is easy to characterize. But, first of all, let us make clear some properties of the Bochner integral in this context.

Proposition 3.26.Let (T_t)be a uniformly continuous semigroup of operators onB. Then the mappingt7→T_t, fromR⁺to the space of bounded operators on B, is Bochner integrable on any compact interval. For all f ∈ B, the mappingt7→Ttf, from R⁺toB, is also Bochner integrable on any compact interval. The Bochner integralRt

0Tsdsdefines a bounded operator onBwhich satisfies

Z t

0

T_sds

f = Z t

0

T_sfds (3.6)

for allf ∈ B.

Furthermore, if H is any bounded operator on B then (H T_t)_t∈R+ and (T_tH)_t∈R+are Bochner integrable on any compact interval and we have

H Z t

0

Tsds

= Z t

0

H Tsds , (3.7)

Z t

0

T_sds

H= Z t

0

T_sHds . (3.8)

Proof. The mappingt7→Ttis continuous by hypothesis, hence it is weakly- measurable and by Proposition 3.11 it has separable range. This means that

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it is Bochner measurable. By the continuity oft7→ kTtk, this mapping is also Bochner integrable on all compact interval.

Exactly the same arguments show thatt7→T_tf is Bochner integrable on any compact interval.

The identity (3.6) is obvious when t7→T_tis a simple function, it is then rather easy to see that it remains true in the general case, by an approximation argument.

The fact that (H Tt)_t∈_R+ and (TtH)_t∈_R+ are Bochner integrable on any compact interval is immediate. Now, we have, by (3.6)

Z t

0

T_sds

Hf = Z t

0

T_sHfds= Z t

0

T_sHds

f , for allf ∈ B. This gives (3.8).

On the other hand, by Proposition 3.21 and Equation (3.6), we have H

Z t

0

T_sds

f =H Z t

0

T_sfds

= Z t

0

H T_sfds

= Z t

0

H T_sds

f ,

for allf ∈ B. This proves (3.7). tu

We can now pass to the main characterization theorem concerning uniformly continuous semigroups.

Theorem 3.27.Let(Tt)be a semigroup onB. Then the following assertions are equivalent.

i) The semigroup (Tt)is uniformly continuous.

ii) There exists a bounded operatorZ onB such that T_t= e^tZ

for allt∈R⁺.

In this holds, then the operator Zis given by

Z= lim

h→0

1

h(Th−I), on all B.

Proof. IfZis a bounded operator onBthen the series e^t^Z=X

n∈N

tⁿZⁿ n!

is convergent in operator norm. This defines a bounded operator e^t^Z onB and obviously we have

e^t^Z

≤e^tkZk.

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It is also easy to check thatT_t= e^t^Z, t∈R⁺, defines a uniformly continuous semigroup (proof left to the reader). We have proved that ii) implies i).

Assume that i) is satisfied. The mapping t7→Ttis Bochner integrable on any compact interval and by the Fundamental Theorem of Calculus, one can choose hsmall enough such that

h⁻¹ Z h

0

T_sds−I

<1.

This means that the operatorX=Rh

0 Tsdsis invertible, for the inverse series 1

h X

n∈N

I−1

hX ⁿ

converges in operator norm. In particularXhas a bounded inverseX⁻¹. Put Z= (Th−I)X⁻¹.

We have, using Proposition 3.21 and the Change of Variable Formula, (Tt−I)X=

Z t+h

t

Tsds− Z h

0

Tsds

= Z t+h

h

T_sds− Z t

0

T_sds

= (Th−I) Z t

0

Tsds .

As all the operators above commute, we get by Proposition 3.26 Tt−I =

Z t

0

TsZ ds.

In particular _dt^d T_texists and is equal toT_tZ. This means that d

dt T_te^−t^Z

= 0 and henceTt= e^tZ. ut

Definition 3.28.The operator Z given by the theorem above is called the generator of (Tt).

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3.2.3 Strong and Weak Continuity

We have seen that the uniform continuity for the semigroup is the signature of a bounded generator. Outside of this case, the strong continuity assumption is a very reasonable assumption and it has many consequences.

Before hand, we need to recall (without proof) a famous theorem of Func- tional Analysis (proofs may be found in any textbook in Functional Analysis, such as [RS80]).

Theorem 3.29 (Uniform Boundedness Principle). Let (T_i)_i∈I be any family of bounded operators from a Banach spaceB to a normed linear space A. If for each x ∈ B the set {kT_ixk ; i ∈ I} is bounded, then the set {kTik ; i∈I} is bounded.

We can now state our theorem on strong continuity.

i) The semigroup (T_t)is strongly continuous.

ii) The semigroup(Tt) is strongly continuous at0⁺. iii) The semigroup (Tt)is continuous.

In that case there exist positive constants M andβ such that

kTtk ≤Me^βt (3.9)

for allt∈R⁺.

Proof. It is obvious that i) implies ii) and that iii) implies i). We have only have to prove that ii) implies iii).

Assume that (Tt) is strongly continuous at 0⁺, that is, lim_t→0Ttf = f, for allf ∈ H. We claim that there exists an∈N^∗ such that

Cn= sup{kTtk; 0≤t≤1/n}

is finite. Indeed, if this were not true one could construct a sequence (tn) tending to 0 such that kTt_nk tends to +∞. By the Uniform Boundedness Principle (Theorem 3.29), this would mean the existence ofx∈ Bsuch that (T_t_nx)_n∈_Nis unbounded. This would contradict the strong continuity in 0⁺. Put C =C_nⁿ, then, by the semigroup property of (T_t) we have kTtk ≤C for allt∈[0,1]. As a consequence, for allt∈R⁺, we have

kT_tk ≤C^[t]+1≤C^t+1.

This gives (3.9).

In particular, if limntn =t inR⁺and limnfn=f in Bthen

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kTtnf_n−T_tfk ≤ kTtn(f_n−f)k+kTtnf −T_tfk

≤Me^βtⁿ kfn−fk+Me^β^min(tⁿ^,t)

T_|t−t_n_|f −f . In particular, limnkTt_nfn−Ttfk = 0. This proves that (Tt) is continuous.

u t

Note that St = e^−βtTt, t ∈ R⁺, is also a semigroup, from which (Tt) is easily deduced. This semigroup is uniformly bounded, that is,kStk ≤M for allt. Hence, multiplying byM⁻¹, one can easily reduce the study of strongly continuous semigroups to the case of contraction semigroups, that is, such that kTtk ≤1 for allt.

We end up this discussion with an important result: another improvement of the strong continuity condition.

i)(T_t) is strongly continuous.

ii) (Tt)is weakly continuous.

iii)(Tt)is weakly continuous at0⁺.

Proof. Obviously i) implies ii) and ii) implies iii). The only non trivial di- rection to prove is that iii) implies i). Even better, the semigroup property shows easily that iii) implies ii). There just remains to prove that ii) implies i).

The main argument for proving (3.9) in Theorem 3.30 was the fact that Cn= sup{kTtk; 0≤t≤1/n}

is finite, by an application of the Uniform Boundedness Principle (Theo- rem 3.29). The rest was just applying the semigroup property. If (T_t) is only weakly continuous at 0⁺then, for allf ∈ Bandφ∈ B^∗, the quantityhφ ,T_tfi tend tohφ , fiwhenttends to 0, hencet7→ hφ ,T_tfiis bounded in a neighborhood of 0. By the Uniform Boundedness Principle applied to the linear formφ7→ hφ ,Ttfi, the mappingt7→Ttf is then bounded in a neighborhood of 0. By the Uniform Boundedness Principle again we have that t 7→ Tt is bounded near 0. Hence, developing the same argument as in Theorem 3.30, we get that (3.9) holds true if (Tt) is weakly continuous at 0⁺.

If (T_t) is weakly continuous, then the closed linear span of the set{T_tf; t∈ R⁺} (which we denote by R) is invariant under T_t and separable, for it is the weak (and hence the strong) closure of the linear space generated by {Ttf; t∈Q⁺}(see Proposition 3.6).

Letf ∈ Bandφ∈ B^∗, by the above arguments and by the weak continuity we see that t 7→ hφ ,Ttfi is locally bounded and measurable, hence the integral

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1 ε

Z ε

0

hφ ,Ttfidt converges and the linear form

λ : φ7→ 1 ε

Z ε

0

hφ ,Ttfidt is clearly a continuous linear form onB^∗.

The mapping t 7→ Ttf is weakly measurable and has a separable range R (as proved above), it is then Bochner measurable and hence t 7→ kTtfk is measurable by Lemma 3.14. We have also seen that this map is locally bounded near 0. If (φn) is a sequence in the unit ball of B^∗ converging ∗- weakly toφ∈ B^∗, then by the Banach-Alaoglu Theorem (Theorem 3.8) the mappingφbelongs to the unit ball ofB^∗too. Furthermore, by the Dominated Convergence Theorem we have that

1 ε

Z ε

0

hφn,Ttfidt converges to

1 ε

Z ε

0

hφ ,Ttfidt .

Hence the linear form λis∗-weakly continuous on the unit ball ofB^∗. This means that the intersection of the kernel of λ with the unit ball of B^∗ is

∗-weakly closed. By the Krein-Smulian Theorem (Theorem 3.9) this implies that the kernel ofλis∗-weakly closed, henceλis∗-weakly continuous. This means (Theorem 3.7) that there exists fε∈ B such that

hφ , fεi= 1 ε

Z ε

0

hφ ,T_tfidt , for allφ∈ B^∗.

As a consequence we have

|hφ ,Thfε−fεi|=|hT_h^∗φ , fεi − hφ , fεi|

=1 ε

Z ε

0

hT_h^∗φ ,Ttfidt− Z ε

0

hφ ,Ttfidt

=1 ε

Z ε+h

h

hφ ,Ttfidt− Z ε

0

hφ ,Ttfidt

=1 ε

Z ε+h

ε

hφ ,Ttfidt− Z h

0

hφ ,Ttfidt

≤kfk kφk ε

Z ε+h

ε

Me^βtdt− Z h

0

Me^βtdt

! .

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Sinceφis arbitrary, this means that

h→0limkThfε−fεk ≤ lim

h→0

kfk ε

Z ε+h

ε

Me^βtdt− Z h

0

Me^βtdt

!

= 0. LetLbe the set of allg∈ Bsuch that lim_h→0T_hg=g. It is a subspace and it is norm-closed by (3.9). In particular it is weakly closed. Now, note that the weak limit of f_ε is f, whenεtends to 0. Hence L=B and the proof is complete. ut

3.3 Generators

We have seen in Theorem 3.27 that uniformly continuous semigroups admit bounded generators. In the case of strongly continuous semigroups there still exists a good notion of associated generator.

3.3.1 Definition

Definition 3.32.Let (T_t) be a strongly continuous semigroup onB. We define thegenerator of (T_t) to be the operatorZ onB such that

DomZ=

f ∈ B; lim

t→0

1

t(Tt−I)f exists

and

Zf = lim

t→0

1

t(T_t−I)f for allf ∈DomZ.

Definition 3.33.Recall that, in the same way as for operators on Hilbert space, an operator Ton a Banach space is a closed operator if it is densely defined operator and if, for all sequence (f_n) ⊂DomT which converges to some f inB and such that (Tf_n) converges inB thenf ∈DomTand Tf = limTf_n.

The following result shows that the operatorZ defined this way is always an “interesting” operator.

Proposition 3.34.The space DomZ is dense in B and the operator Z is closed.

Proof. SetZhf = ¹_h(Thf−f) andYsf =¹_sRs

0Tufdu. The operatorsZhand Ysare bounded and

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Z_hY_s=Y_sZ_h=Z_sY_h=Y_hZ_s,

as can be checked easily (making use of Proposition 3.26). In particular, for every s >0 andf ∈ B we have

h→0limZhYsf = lim

h→0Zs(Yhf) =Zsf .

ThereforeY_sf belongs to DomZand since lim_s→0Y_sf =f we get that DomZ is dense.

Now, if (fn) is a sequence in DomZ converging to f in B and such that (Zf_n)_n∈

N converges tog∈ B, then Ysg= lim

n YsZfn = lim

n Ys

lim

h Zhfn

= lim

n lim

h Zs(Yhfn) = lim

n Zsfn=Zsf .

It follows that lims→0Zsf exists, hencef ∈DomZandZf = lims→0Zsf = g. We have proved thatZis closed. ut

The following is a set of important relations relating the generator and its semigroup.

Proposition 3.35.If f ∈DomZthen the following holds.

1) For all t∈R⁺we haveT_tf ∈DomZ.

2) The function f 7−→Ttf is strongly differentiable in Band d

dtT_tf =Z T_tf =T_tZf . (3.10) 3) We have

Ttf −f = Z t

0

TsZfds= Z t

0

Z Tsfds . (3.11) Proof. We have, forf ∈DomZ,

s→0lim 1

s[Ts(Ttf)−Ttf] = lim

s→0Tt

1

s(Tsf−f)

=TtZf.

This proves 1) and Z Ttf =TtZf. This also proves thatt7→Ttf has right- hand derivative equal toTtZf, in each pointt∈R. This right-hand derivative is continuous in t, hence it is a classical theorem of real analysis (see for example [Rud87], Theorem 8.21, to be adapted here to the case of Banach- valued functions) that this implies that t 7→ T_tf is differentiable at every point.

ThusT_tf =f+Rt

0T_sZfds. This proves 2) and 3). ut

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In Theorem 3.31 we have seen that the weak continuity of the semigroup is sufficient for proving its strong continuity. In the same spirit, the following result shows that weak and strong generators coincide.

Proposition 3.36.Let (Tt) be a weakly continuous semigroup on B. Con- sider the operator LonB defined by

DomL=

f ∈ B; weak - lim

t→0

1

t(Tt−I)f exists

and

Lf = weak - lim

t→0

1

t(Tt−I)f

for allf ∈DomL. ThenLcoincides with the generatorZ of(T_t).

Proof. As the norm convergence implies the weak convergence, we obviously have DomZ⊂DomLandZ=Lon DomZ. Now we just need to prove that DomL⊂DomZ.

For allf ∈DomL, allφ∈ B^∗ put F(t) =

φ ,(Tt−I)f − Z t

0

TsLf ds

.

We have lim

h→0

1

h(F(t+h)−F(t)) = lim

h→0

T_t^∗φ , 1

h(T_hf−f)

−lim

h→0

* φ , 1

h Z t+h

t

TsLf ds +

=hT_t^∗φ ,Lfi − hφ ,TtLfi= 0.

This proves that F(t) = 0 for all t. As this is true for allφ∈ B^∗, this shows that

(T_t−I)f = Z t

0

T_sLf ds so that

t→0lim 1

t(Tt−I)f =Lf . In other wordsf belongs to DomZ andZf =Lf. ut

3.3.2 Spectrum

Consider the generatorZof a strongly continuous semigroup (T_t) satisfying

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kTtk ≤Me^βt

for all t (Theorem 3.30). The following theorem gives information on the spectrumσ(Z) ofZand the resolvent operatorsR_λ(Z) = (λI−Z)⁻¹.

Theorem 3.37.If (Tt) is a strongly continuous semigroup on B, with generator Z and satisfying (3.9), then the spectrum of Z is included in the set {λ∈C; Re(λ)≤β}. Moreover, for allλsuch that Re(λ)> β we have

Rλ(Z) = Z ∞

0

e^−λtTt dt . Proof. If Re(λ)> β, the integral

Y^λ= Z ∞

0

e^−λtT_t dt

is convergent in operator norm, by (3.9). Putg=Y^λf, we have, by Propo- sition 3.26,

1

h(Thg−g) = 1 h

Z ∞

0

e^−λtTt+hf dt− Z ∞

0

e^−λtTtf dt

= 1 h

Z ∞

h

e^−λ(t−h)Ttf dt− Z ∞

0

e^−λtTtf dt

= 1

h (e^λh−1) Z ∞

0

e^−λtT_tf dt−e^λh Z h

0

e^−λtT_tf dt

! .

Passing to the limith→0, we getλg−f on the right hand side.

This proves thatgbelongs to DomZand (λI−Z)g=f, henceg=Rλ(Z)f.

u t

3.3.3 Dual Semigroup

It is very useful, in particular for semigroups of operators acting on Hilbert spaces, to consider the dual of operator semigroups. We study here their main properties.

First recall the definition of adjoint operators on Banach spaces.

Definition 3.38.LetBbe a Banach space and letB^∗be its dual. LetZbe a densely defined operator onB, with domain DomZ. One defines the operator Z^∗ onB^∗ as follows. Consider the set

DomZ^∗={φ∈ B^∗; f 7→ hφ ,Zfi is continuous on DomZ}.

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For everyφ∈DomZ^∗, the mappingλ:f 7→ hφ ,Zfiis continuous on DomZ which is dense. Henceλextends to a unique continuous linear functional on B, that is, λbelongs toB^∗. We denote byZ^∗φthis unique element of B^∗. To be clear, the elementZ^∗φis characterized by the relation

hφ ,Zfi=hZ^∗φ , fi for allf ∈DomZ,φ∈DomZ^∗.

Definition 3.39.If (T_t) is a strongly continuous semigroup of operators on a Banach spaceB, one can consider thedual semigroup(T_t^∗) onB^∗. Clearly it is a semigroup of operators onB^∗. The point is that, in general, the semigroup (T_t^∗) may not be strongly continuous. Let us see that with a counter-example.

Consider the operatorsT_t onL¹(R) defined by (T_tf)(s) =f(s+t) for all s, t∈R. Clearly theT_t’s are norm 1 operators and they form a semigroup. Let us show that they constitute a strongly continuous semigroup. The fact that lim_h→0kT_tf −fk₁= 0 is clear on the dense subspace of continuous functions with compact support, by Lebesgue’s Theorem. As the norm ofT_tis uniformly bounded in t, it is easy to show that the property lim_h→0kTtf−fk₁ = 0 extends to allf ∈L¹(R).

The dual semigroup of (Tt) is the semigroup (T_t^∗f)(s) =f(s−t) defined onL^∞(R). It is not strongly continuous onL^∞(R), as can be seen easily by taking forf any indicator function.

Definition 3.40.The dual semigroup (T_t^∗) is not strongly continuous in general, but it is always *-weakly continuous, for

hThφ−φ , fi=hφ ,Thf−fi

for allφ∈ B^∗, allf ∈ B, and hence the limit is 0 whenhtends to 0.

One defines the *-weak generatorZ⁰ of (T_t^∗) as follows DomZ⁰ =

φ∈ B^∗; ∗-weak - lim

h→0

1

h(T_h^∗φ−φ) exists

and

Z⁰φ=∗-weak - lim

h→0

1

h(T_h^∗φ−φ) for allφ∈DomZ⁰.

We now wish to prove the following theorem characterizingZ⁰.

Theorem 3.41.Let(T_t)be a strongly continuous semigroup of operators on B with generatorZ. The *-weak generatorZ⁰ of(T_t^∗)and its domainDomZ⁰ coincide with the operatorZ^∗ and its domainDomZ^∗.

The proof of this theorem will be achieved in several steps. The first step is the definition of a*-weak integral onB^∗.

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Definition 3.42.Let φ be a function from (E,E) to B^∗. We say that φ is

*-weakly-measurable ifx7→ hφ(x), fiis measurable for allf ∈ B.

Ifφis *-weakly measurable and if we have Z

E

|hφ(x), fi|dµ(x)<∞ for allf ∈ B, we shall say that φis*-weakly integrable.

Proposition 3.43.If φ is a *-weakly integrable function from (E,E, µ) to B^∗ then the mapping

f 7→

Z

E

hφ(x), fidµ(x) is a continuous linear form onB.

Proof. LetTbe the operator fromBtoL¹(E,E, µ) defined by (Tf)(x) =hφ(x), fi.

We claim thatTis a closed operator. Indeed, if (f_n) converges tof inBand if (Tf_n) converges toyinL¹(E), then there exists a subsequence (f_n_k)_k∈_Nsuch that Tf_n_k tends toy almost surely. ButTf_n_k(x) =hφ(x), f_n_kiconverges to hφ(x), fifor allx∈E. Hencey=Tf andTis closed.

By the Closed Graph Theorem this means that T is bounded, for it is everywhere defined and closed.

The linear form of the proposition is the composition of T and of the mapping f 7→R

EfdµfromL¹(E) toC. One concludes easily. ut

The linear form above is thus an element ofB^∗. This allows the following definition.

Definition 3.44.One defines the *-weak-integral

*-w- Z

E

φ(x) dµ(x)

as being the unique element ofB^∗which satisfies

*-w- Z

E

φ(x) dµ(x), f

= Z

E

hφ(x), fidµ(x) for allf ∈ B.

As a second step of the proof of Theorem 3.41 we derive useful identities for the semigroup (T_t^∗), using the *-weak integral. They are the one for (T_t^∗) corresponding to the (3.11) for (Tt).

Proposition 3.45.Let (T_t^∗)be the dual semigroup of a strongly continuous semigroup (Tt)on B. Then for allφ∈ B^∗ we have

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*-w- Z t

0

T_s^∗φ ds∈DomZ^∗ and

Z^∗

*-w- Z t

0

T_s^∗φds

=T_t^∗φ−φ . (3.12) Furthermore, if φbelongs toDomZ^∗ then

Z^∗

*-w- Z t

0

T_s^∗φds

= *-w- Z t

0

T_s^∗Z^∗φ ds . (3.13) Proof. For allf ∈DomZwe have

*-w- Z t

0

T_s^∗φds ,Zf

= Z t

0

hT_s^∗φ ,Zfids

= Z t

0

hφ ,TsZfids

=

φ , Z t

0

T_sZfds

=hφ ,Ttf−fi

=hT_t^∗φ−φ , fi. This proves that *-w-Rt

0T_s^∗φdsbelongs to DomZ^∗ and this proves (3.12).

If furthermoreφbelongs to DomZ^∗then the same computation gives

Z^∗

*-w- Z t

0

T_s^∗φds

, f

= Z t

0

hφ ,TsZfids

= Z t

0

hT_s^∗Z^∗φ , fids

=

*-w- Z t

0

T_s^∗Z^∗φds , f

.

This proves (3.13). ut

We can now conclude for the proof of Theorem 3.41; this is our third step.

Proof (of Theorem 3.41). Let φbelong to DomZ^∗ andf ∈ B. We have, by Proposition 3.45

1

hhT_h^∗φ−φ , fi= 1 h

*

Z^∗ *-w- Z h

0

T^∗_sφds

! , f

+

= 1 h

Z h

0

hT_s^∗Z^∗φ , fids .

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Taking the limit when htends to 0 we obtain hZ^∗φ , fi. This proves thatφ belongs to DomZ⁰ and that Z⁰φ=Z^∗φ. We have proved thatZ^∗⊂Z⁰.

Conversely, ifφbelongs to DomZ⁰ and iff ∈DomZ, then hZ⁰φ , fi= lim

h→0

1

hhT_h^∗φ−φ , fi

= lim

h→0

1

hhφ ,T_hf−fi

=hφ ,Zfi.

This proves thatφ belongs to DomZ^∗ and thatZ^∗φ=Z⁰φ. The theorem is proved. ut

The case of reflexive Banach space is much more easy. The following theorem is now easy to prove and left to the reader.

Theorem 3.46.If (Tt) is a strongly continuous semigroup on a reflexive Banach space B, with generator Z, then (T_t^∗)_t∈

R⁺ is a strongly continuous semigroup onB^∗, with generatorZ^∗.

3.3.4 Predual Semigroups

Another situation which is very useful is the case where the BanachBadmits a predual space.

Definition 3.47.Thepredual of a Banach spaceB, when it exists, is a Ba- nach spaceB_∗ such thatBis the dual ofB_∗.

This is typical of the situation whereBisB(H) andB_∗=L1(H) or more generally whenBis a, so-called, von Neumann algebra.

Lemma 3.48.If L is a bounded operator on B and if L is also ∗-weakly continuous, then there exists a unique bounded operatorL_∗ on B_∗ such that

hLf , φi=hf ,L_∗φi

for allφ∈ B_∗, all f ∈ B. In particular Lis the adjoint ofL_∗.

Proof. The linear form λ : f 7→ hLf , φi is ∗-weakly continuous on B, by the ∗-weak continuity of L. Hence by Theorem 3.7, there exists an element φ_∗ ∈ B_∗ such that λ(f) = hf , φ_∗i for all f ∈ B. The element φ_∗ depends linearly onφ, we denote it byL_∗φ. The mappingL_∗ is bounded onB_∗for

|hf ,L_∗φi|=|hLf , φi| ≤ kφk kLk kfk.

Uniqueness is obvious and the relationL= (L∗)^∗ is also immediate now. ut

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Definition 3.49.The mapL_∗ is call thepredual map ofL.

Theorem 3.50.Let B be a Banach space which admits a predual B_∗. Let (Tt) be a strongly continuous semigroup of operators on B such that all the mappings Tt are ∗-weakly continuous. Consider the semigroup (T_∗t) on B_∗, defined by

hf ,T_∗tφi=hTtf , φi,

for all φ∈ B∗ and all f ∈ B. Then(T∗t)is a strongly continuous semigroup of bounded operators onB∗. The semigroup(Tt)is the dual of the semigroup (T∗t).

Proof. By Lemma 3.48 each of the maps T_t admit a predual map T_∗t. It is easy to check that (T_∗t) is a semigroup on B_∗. Furthermore, for all f ∈ B and allφ∈ B_∗ we have

hTtf−f , φi=hf ,T_∗tφ−φi.

Hence the semigroup (T_∗t) is weakly continuous. By Theorem 3.31 it is strongly continuous. The rest of the properties have been already proved in Lemma 3.48. ut

3.3.5 Stone’s Theorem

We end this section and this lecture with the very important theorem of Stone, characterizing unitary groups of operators.

Definition 3.51.On a Hilbert spaceH, consider astrongly continuous unitary semigroup, that is, a strongly continuous semigroup of operators (Ut)_t∈_R+

onHsuch that each of the operatorsUtis unitary. For everyt∈R⁺, put U−t=U⁻¹_t =U^∗_t. (3.14) Then it is very easy to check that the family (U_t)_t∈_Rnow satisfies

U_tU_s=U_s+t

for alls, t∈R. That is, the unitary semigroup can be extended into aunitary group, indexed byRnow.

Conversely, if (Ut)_t∈Ris a unitary group then the relation U_−tUt=U0= I

leads to

U−t=U⁻¹_t =U^∗_t (3.15)

for allt∈R.

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Theorem 3.52 (Stone’s Theorem). A family (U_t)_t∈_R of operators on H is a strongly continuous unitary (semi)group if and only its generator is of the formiHwhereHis self-adjoint.

In that case we have

U_t= e^itH,

for allt∈R, in the sense of the functional calculus for self-adjoint operators.

Proof. Let H be a self-adjoint operator on H. If, for each t ∈ R we put U_t = e^it^H for some self-adjoint operator H, in the sense of the functional calculus of self-adjoint operators, then (U_t)_t≥0 is clearly a unitary group by the functional calculus. It is strongly continuous for, using the spectral measureµ_ϕ associated toH, we have

kU_tϕ−ϕk²= Z

R

|e^itλ−1|²dµ_ϕ(λ) which converges to 0 asttends to 0. Finally, if ϕis such that

t→0lim

U_tϕ−ϕ t

exists then define the operatorBby Bϕ=1

i limUtϕ−ϕ

t .

We then easily check that

hϕ,Bψi=hBϕ, ψi and thusBis symmetric.

Now, ifϕbelongs to DomHwe have by the Spectral Theorem again

Uhϕ−ϕ h −iHϕ

2

= Z

R

e^ihλ−1 h −iλ

dµϕ(x).

The right-hand side converges to 0 by Lebesgue’s Theorem (use |e^x−1| ≤

|x|). This proves thatH⊂B. AsBis symmetric, this gives finallyB=H.

Conversely, let (Ut)_t≥0be a strongly continuous unitary semigroup. It has a generatorKwhich is a closed operator (Proposition 3.34).

Put Vt = U^∗_t for all t ∈ R⁺. This defines another semigroup, the dual semigroup of (Ut), with generator equal toK^∗by Theorem 3.46. Iff belongs to DomK^∗then, for allt > h >0, we have

1

h(Uh−I)Vtf = 1

h(Vt−h−Vt)f .

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Hence, whenhtends to 0, the limit of the above exists and is equal to−K^∗V_tf.

This proves thatV_tDomK^∗⊂DomK and thatK=−K^∗ onV_tDomK^∗. Every g ∈ DomK^∗ can be written as g = U_tV_tg. Hence V_tg belongs to DomK and finally g = U_tV_tg belongs to DomK by Proposition 3.35. We have proved that DomK^∗⊂DomK.

Inverting the roles of (Ut) and (Vt), that is, considering the relationUt= V_t^∗ we would obtain in the same way: DomK⊂DomK^∗.

We have proved that K = −K^∗, that is, K = iH for some self-adjoint operatorH. The theorem is proved. ut

Notes

In order to write this chapter we have used the following famous reference books on Functional Analysis and Operator Semigroups: the book by Davies [Dav80] on semigroups, Dunford-Schwarz’s famous bible on linear operators [DS88], the book by Yosida [Yos80]. Complements, in particular on weak topologies for Banach spaces, can be found in Rudin’s book [Rud91] or Con- way’s book [Con85].

There are other references that may be consulted on the subject of Opera- tor Semigroups. The first volume of Bratelli-Robinson’s book [BR87] contains a nice chapter on semigroups. The book by Clement et al [CHA⁺87] is quite advanced in the subject. The second volume of Reed-Simon’s series [RS75]

contains a summary of the main results.

References

[BR87] Ola Bratteli and Derek W. Robinson.Operator algebras and quantum sta- tistical mechanics. 1. Texts and Monographs in Physics. Springer-Verlag, New York, second edition, 1987.C^∗- andW^∗-algebras, symmetry groups, decomposition of states.

[CHA⁺87] Ph. Cl´ement, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter. One-parameter semigroups, volume 5 ofCWI Monographs.

North-Holland Publishing Co., Amsterdam, 1987.

[Con85] John B. Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1985.

[Dav80] Edward Brian Davies. One-parameter semigroups, volume 15 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1980.

[DS88] Nelson Dunford and Jacob T. Schwartz. Linear operators. Part I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1988. General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.

[RS75] Michael Reed and Barry Simon.Methods of modern mathematical physics.

II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.

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[RS80] Michael Reed and Barry Simon.Methods of modern mathematical physics.

I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, 1980. Functional analysis.

[Rud87] Walter Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, third edition, 1987.

[Rud91] Walter Rudin. Functional analysis. International Series in Pure and Ap- plied Mathematics. McGraw-Hill Inc., New York, second edition, 1991.

[Yos80] Kˆosaku Yosida.Functional analysis, volume 123 ofGrundlehren der Math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences]. Springer-Verlag, Berlin, sixth edition, 1980.