Operators: Theory and Applicationskondrat/course/ota1.pdfOperators: Theory and Applications Yuri G....

Operators: Theory and Applications

Yuri G. Kondratiev

2018/2019

Part II

Contents

1 Linear Dynamical Systems 5

1.1 Cauchy’s Functional Equation . . . . . . . . . . . . . . . . . . . 51.2 Finite-Dimensional Systems: Matrix Semigroups . . . . . . . . . 81.3 Uniformly Continuous Operator Semigroups . . . . . . . . . . . 101.4 Strongly Continuous Semigroups . . . . . . . . . . . . . . . . . . 11

3

2 Semigroups, Generators,and Resolvents 162.1 Spectral Theory for Closed Operators . . . . . . . . . . . . . . . 162.2 Generators of Semigroups and Their Resolvents . . . . . . . . . 182.3 Hille-Yosida Generation Theorems . . . . . . . . . . . . . . . . . 25

4 Contents

Chapter 1

Linear Dynamical Systems

There are many good reasons why an ”autonomous deterministic system” shouldbe described by maps T ptq, t • 0, satisfying the functional equation (FE)

T pt ` sq “ T ptqT psq.

Here, t is the time parameter, and each T ptq maps the ”state space” of thesystem into itself. These maps completely determine the time evolution of thesystem in the following way: If the system is in state x0 at time t0 “ 0 thenat time t it is in state T ptqx. However, in most cases a complete knowledgeof the maps T ptq is hard, if not impossible, to obtain. It was one of the greatdiscoveries of mathematical physics, based on the invention of calculus, that,as a rule, it is much easier to understand the ”infinitesimal changes” occurringat any given time. In this case, the system can be described by a di↵erentialequation. In this chapter we analyze this phenomenon in the mathematicalcontext of linear operators on Banach spaces. For this purpose, we take twoopposite views.

pV 1q. We start with a solution t fiÑ T ptq and ask which assumptions implythat it is di↵erentiable and satisfies a di↵erential equation.

pV 2q. We start with a di↵erential equation and ask how its solution can berelated to a family of mappings.

1.1 Cauchy’s Functional Equation

As a warm-up, this program will be performed in the scalar-valued case first.In fact, it was A. Cauchy who in 1821 asked in his Cours d’Analyse, withoutany further motivation, the following question: D’eterminer la fonction �pxqde mani’ere qu’elle reste continue entre deux limites r’eelles quelconques de lavariable x, et que l’on ait pour toutes les valeurs r’eelles des variables x et y�px`yq “ �pxq�pyq. Using modern notation, we restate his question as followsdropping the continuity requirement for the moment.

5

6 Chapter 1. Linear Dynamical Systems

p1.1q Problem. Find all maps T p¨q : R` fiÑ C satisfying the functionalequation

T pt ` sq “ T ptqT psqfor all t, s • 0, (1.1)

T p0q “ 1.Evidently, the exponential functions

t fiÑ eta

satisfy (FE) for any a P C. With his question, Cauchy suggested that thesecanonical solutions should be all solutions of (FE). Before giving an answerto Problem 1.1, we take a closer look at the exponential functions and observethat they, besides solving the algebraic identity (FE), also enjoy some importantanalytic properties.

Determine the function �pxq in such a way that it remains continuous be-tween two arbitrary real limits of the variable x, and that, for all real values ofthe variables x and y, one has

�px ` yq “ �pxq�pyq.

Proposition 1.1. Let T ptq :“ eta for some a P C all t • 0. Then the functionT p¨q is di↵erentiable and satisfies the di↵erential equation (or, more precisely,the initial value problem) (DE)

d

dtT ptq “ aT ptq

for all t • 0, T p0q “ 1. Conversely, the function T p¨q : R` Ñ C defined byT ptq “ eta for some a P C is the only di↵erentiable function satisfying (DE).Finally, we observe that a “ d

dtT ptq for t “ 0.

Proof. We show only the assertion concerning uniqueness. Let Sp¨q : R` Ñ Cbe another di↵erentiable function satisfying (DE). Then the new function Qp¨q :r0, ts Ñ C defined by

Qpsq :“ T psqSpt ´ sq for 0 § s § t

for some fixed t ° 0 is di↵erentiable with derivative ddsQpsq ” 0. This shows

thatT ptq “ Qptq “ Qp0q “ Sptq

for arbitrary t ° 0. This proposition shows that, in our scalar-valued case,pV 2q can be answered easily using the exponential function. It is now our mainpoint that continuity is already su�cient to obtain di↵erentiability in pV 1q.

1.2. Finite-Dimensional Systems: Matrix Semigroups 7

Proposition 1.2. Let T p¨q : R` Ñ C be a continuous function satisfying (FE).Then T p¨) is di↵erentiable, and there exists a unique a P C such that (DE)holds.

Proof. Since T p¨q is continuous on R`, the function V p¨q defined by

V ptq :“ª t

0T psqds, t • 0,

is di↵erentiable with V 1ptq “ T ptq. This implies

limtÑ0`

V ptqt

“ V 1p0q “ T p0q “ 1.

Therefore, V pt0q is di↵erent from zero, hence invertible, for some small t0 ° 0.The functional equation (FE) now yields

T ptq “ V ´1pt0qV pt0qT ptq “ V ´1pt0qª t

0T pt ` sqds

“ V ´1pt0qª t`t0

tT psqds “ V ´1pt0qpV pt ` t0q ´ V ptqq

for all t • 0. Hence, T p¨q is di↵erentiable with derivative

d

dt“ lim

hÑ0`T pt ` hq ´ T ptq

h“

limhÑ0`

T phq ´ T p0qh

T ptq “ T 1p0qT ptq

for all t • 0. This shows that T p¨q satisfies (DE) with a :“ T 1p0q.

The combination of both results leads to a satisfactory answer to Cauchy’sProblem 1.1.

Theorem 1.3. Let T p¨q : R` Ñ C be a continuous function satisfying (FE).Then there exists a unique a P C such that T ptq “ eta for all t • 0.

1.2 Finite-Dimensional Systems: Matrix Semi-

groups

In this section we pass to a more general setting and consider finite dimensionalvector spaces X “ Cn. The space LpXq of all linear operators on X willthen be identified with the space MnpCq of all complex n ˆ n matrices, and


a linear dynamical system on X will be given by a matrix-valued functionT p¨q : R` Ñ MnpCq satisfying the functional equation (FE)

T pt ` sq “ T ptqT psq for all t, s • 0, T p0q “ I.

As before, the variable t will be interpreted as ”time”. The ”time evolution” of astate x0 P X is then given by the function ⇠ : R` Ñ X defined as ⇠ptq “ T ptqx0.We also call tT ptqx0 | t • 0u the orbit of x0 under T p¨q. From the functionalequation (FE) it follows that an initial state x0 arrives after an elapsed timet ` s at the same state as the initial state y0 :“ T psqx0 after time t.

In this new context we study (V1) and (V2) from Section 1 and restateCauchy’s Problem 1.1.

Problem. Find all maps T p¨q : R` Ñ MnpCq satisfying the functional equa-tion (FE).

Imitating the arguments from Section 1 we first look for ”canonical” solu-tions of (FE) and then hope that these exhaust all (natural) linear dynamicalsystems. As in Section 1 the candidates for solutions of (FE) are the ”expo-nential functions”.

Definition 1.4. For any A P MnpCq and t P R the matrix exponential etAisdefined by

etA :“8ÿ

k`0

tkAk

k!. (1.2)

Taking any norm on Cn and the corresponding matrix-norm on MnpCq oneshows that the partial sums of the series above form a Cauchy sequence; hencethe series converges and satisfies

}etA} § et}A}.

for all t • 0. Moreover, the map t fiÑ etA has the following properties.

Proposition 1.5. For any A P MnpCq the map

R` Q t Ñ etA P MnpCq

is continuous and satisfies (FE)

ept`sqA “ etAesA

for t,• 0, e0A “ I.

Proof. Since the series8ÿ

k`0

tkAk

k!

1.2. Finite-Dimensional Systems: Matrix Semigroups 9

converges, one can show, as for the Cauchy product of scalar series, that

8ÿ

k`0

tkAk

k!

8ÿ

k`0

skAk

k!

“8ÿ

n“0

nÿ

k“0

tn´kAn´k

pn ´ kq!skAk

k!“

8ÿ

n“0

pt ` sqnAn

n!.

This proves (FE). In order to show that t Ñ etA is continuous, we first observethat by (FE) one has et`h ´ etA “ etApehA ´ Iq for all t, h P R. Therefore, itsu�ces to show that limhÑ0ehA “ I. This follows from the estimate

}ehA ´ I} “ }8ÿ

k“1

hkAk

k!} §

ÿk “ 18 |h|k}A}k

k!“ e|h|}A} ´ 1.

Obviously, the range of the function t Ñ T ptq :“ etA in MnpCq is a commu-tative semigroup of matrices depending continuously on the parameter t P R`.In fact, this is a straightforward consequence of the following decisive property:

The mapping t Ñ T ptq is a homomorphism from the additive semigrouppR`,`q into the multiplicative semigroup MnpCq. Keeping this in mind, westart to use the following terminology.

Definition 1.6. We calletA, t • 0

the (one-parameter) semigroup generated by the matrix A P MnpCq. The def-inition, the continuity, and the functional equation (FE) hold for any real andeven complex t. Then the map

T : t Ñ etA

extends to a continuous (even analytic) homomorphism from the additive grouppR,`q (or pC,`q) into the multiplicative group GLpn,Cq of all invertible, com-plex n ˆ n matrices. We call etA, t P R the (one-parameter) group generatedby A.

Example 1.7. (i) The (semi) group generated by a diagonal matrix A “diagpa1, . . . anq is given by

etA “ diagpeta1, . . . , etanq.


1.3 Uniformly Continuous Operator Semigroups

From now on, we take X to be a complex Banach space with norm } ¨ }.We denote by LpXq the Banach algebra of all bounded linear operators on

X endowed with the operator norm. In analogy to Sections 1 and 2, we canrestate Cauchy’s question in this new context.

Problem. Find all maps T p¨q : R` Ñ LpXq satisfying functional equation(FE)

T pt ` sq “ T ptqT psq for all t, s • 0, T p0q “ I.

The search for answers to this question will be the main theme of our lec-tures, and due to the infinite-dimensional framework, the answers will be muchmore complex than what we encountered up to now.

For every function T : R` Ñ LpXq satisfying (FE) the set tT ptq | t • 0uis a commutative sub-semigroup of LpXq. This justifies calling the functionalequation (FE) the semigroup law and using the following terminology.

Definition 1.8. A family T ptq, t • 0 of bounded linear operators on a Banachspace X is called a (one-parameter) semigroup (or linear dynamical system) onX if it satisfies the functional equation (FE). If (FE) holds even for all t, s P Rwe call T ptq, t P R a (one-parameter) group on X.

We now introduce the ”typical” examples of one-parameter semigroups ofoperators on a Banach space X. Take any operator A P LpXq. As in the matrixcase, we can define an operator-valued exponential function by

etA “8ÿ

k“0

tkAk

k!,

where the convergence of this series takes place in the Banach algebra LpXq.Using the same arguments as above, one shows that etA, t • 0 satisfies thefunctional equation (FE) and the di↵erential equation (DE), and hence Theorem1.10 below follows as in Section 2.

Definition 1.9. A one-parameter semigroup T ptq, t • 0 on a Banach space Xis called uniformly continuous (or norm continuous) if R` Q t Ñ T ptq P LpXqis continuous with respect to the uniform operator topology on LpXq.Theorem 1.10. Every uniformly continuous semigroup T ptq, t • 0 on a Banachspace X is of the form T ptq “ etA for some bounded operator A P LpXq.Proof. Since the following arguments were already used in the scalar and matrix-valued cases (see Sections 1 and 2), we think that a brief outline of the proof issu�cient. For a uniformly continuous semigroup T ptq the operators

V ptq “ª t

0T psqds

1.4. Strongly Continuous Semigroups 11

are well-defined, and t´1V ptq) converges (in norm!) to T p0q “ I as t Ñ 0`.Hence, for t ° 0 su�ciently small, the operator V ptq becomes invertible. Repeatnow the computations from the proof of Theorem 1.4 in order to obtain thatt Ñ T ptq is di↵erentiable and satisfies (DE).

(i) The operator A in Theorem 1.10 is determined uniquely as the derivativeof T ptq , i.e., A “ T 1p0q. We call it the generator of T ptq.

(ii) Since the definition for etA works also for t P R and even for t P C, it fol-lows that each uniformly continuous semigroup can be extended to a uniformlycontinuous group etA, t P R, or to etA, t P C, respectively.

(iii) From the di↵erentiability of T ptq it follows that for each x P X theorbit map R` Q t Ñ T ptqx P X is di↵erentiable as well. Therefore, the mapxptq :“ T ptqx is the unique solution of the X-valued initial value problem (orabstract Cauchy problem) (ACP)

x1ptq “ Axptq, t • 0, xp0q “ x.

1.4 Strongly Continuous Semigroups

In many essential examples uniform continuity is too strong a requirement formany natural semigroups defined on concrete function spaces.

Definition 1.11. A family T ptq, t • 0 of bounded linear operators on a Banachspace X is called a strongly continuous (one-parameter) semigroup (or C0-semigroup if it satisfies the functional equation (FE) and is strongly continuous.Hence, T ptq, t • 0 is a strongly continuous semigroup if the functional equation(FE)

T pt ` sq “ T ptqT psq for all t, s • 0, T p0q “ I

holds and the orbit maps (SC)

⇠x : t Ñ ⇠xptq :“ T ptqx

are continuous from R` into X for every x P X.Our first goal is to facilitate the verification of the strong continuity (SC).

This is possible thanks to the uniform boundedness principle, which implies thefollowing frequently used equivalence.

Lemma 1.12. Let X be a Banach space and let F be a function from a compactset K Ä R into LpXq. Then the following assertions are equivalent.

(a) F is continuous for the strong operator topology, i.e., the mappingsK Q t Ñ F ptqx P X are continuous for every x P X.

(b) F is uniformly bounded on K, and the mappings K Q t Ñ F ptqx P Xare continuous for all x in some dense subset D Ä X.


(c) F is continuous for the topology of uniform convergence on compactsubsets of X, i.e., the map K ˆC Q pt, xq Ñ F ptqx P X is uniformly continuousfor every compact set C Ä X.

Proof. The implication pcq Ñ paq is trivial, while paq Ñ pbq follows from theuniform boundedness principle, since the mappings t Ñ F ptqx are continuous,hence bounded, on the compact set K. To show pbq Ñ pcq, we assume

}F ptq} § M for all t P K and fix some ✏ ° 0 and a compact set C Ä X.Then there exist finitely many x1, . . . , xm P D such that

C Ä Ymi“1pxi ` ✏

MUq,

where U denotes the unit ball of X. Now choose � ° 0 s.t.

}F ptqxi ´ F psqxi} § ✏

for all i “ 1, . . . ,m, and for all t, s P K, such that |t ´ s| § �. For arbitraryx, y P C and t, s P K with }x ´ y} § ✏M § � this yields

}F ptqx ´ F psqy} § }F ptqpx ´ xjq} ` }pF ptq ´ F psqqxj}

`}F psqpxj ´ xq} ` }F psqpx ´ yq} § 4✏,

where we choose j P t1, . . . ,mu such that }x ´ xj} § ✏M . This estimate proves

that pt, xq Ñ F ptqx is uniformly continuous with respect to t P K and x P C.As an easy consequence of this lemma, in combination with the functional

equation (FE), we obtain that the continuity of the orbit maps

⇠x : t Ñ T ptqx

at each t ° 0 and for each x P X is already implied by much weaker properties.

Proposition 1.13. For a semigroup T ptq, t • 0 on a Banach space X, thefollowing assertions are equivalent.

(a) T ptq is strongly continuous.(b)limtÑ0` T ptqx “ x for all x P X.(c) There exist � ° 0,M • 1 and a dense subset D Ä X such that(i) }T ptq} § M for all t P r0, �s,ii) limtÑ0`0 T ptqx “ x for all x P D.

Proof. The implication paq Ñ pc.iiq is trivial. In order to prove that paq Ñ pc.iq,we assume, by contradiction, that there exists a sequence p�nqnPN Ä R` converg-ing to zero such that }T p�nq} Ñ 8 as n Ñ 8. Then, by the uniform bound-edness principle, there exists x P Xsuch that T p�nqx}, n P N is unbounded,contradicting the fact that T p¨q is continuous at t “ 0.


In order to verify that pcq Ñ pbq, we put K :“ ttn | n P Nu Y t0u foran arbitrary sequence ptnqnPN Ä R` converging to t “ 0. Then K is compact,T p¨q|K is bounded, and T p¨q|Kx is continuous for all x P D. Hence, we canapply Lemma (b) to obtain

limnÑ8T ptnqx “ x

for all x P X. Since ptnqnPN was chosen arbitrarily, this proves pbq.To show that pbq Ñ paq, let t0 ° 0 and let x P X. Then

limtÑ0`0

}T pt0 ` hqx ´ T pt0qx} § }T pt0q} limtÑ0`0

}T phqx ´ x} “ 0,

which proves right continuity. If h † 0, the estimate

}T pt0 ` hqx ´ T pt0qx} § }T pt0 ` hq}}x ´ T p´hqx}implies left continuity whenever T ptq remains uniformly bounded for t P r0, t0s.This, however, follows as above first for some small interval r0, �s by the uniformboundedness principle and then on each compact interval using (FE).

Since in many cases the uniform boundedness of the operators T ptq fort P r0, �s is obvious, one obtains strong continuity by checking (right) continuityof the orbit maps ⇠x at t “ 0 for a dense set of ”nice” elements x P X only.

We repeat that for a strongly continuous semigroup T ptq, t • 0 the finiteorbits tT ptqx | t P r0, t0su are continuous images of a compact interval, hencecompact and therefore bounded for each x P X. So by the uniform boundednessprinciple, each strongly continuous semigroup is uniformly bounded on eachcompact interval, a fact that implies exponential boundedness on R`.

Proposition 1.14. For every strongly continuous semigroup T ptq, t • 0 thereexist constants ! P R and M • 1 such that

}T ptq} § Me!t (1.3)

for all t • 0.

Proof. Choose M • 1 such that }T psq} § M for all 0 § s § 1 and write t • 1as t “ s ` n for n P N and 0 § s † 1. Then

}T ptq} § }T psq}}T p1q}n § Mn`1 “ Men logM § Me!t

holds for ! :“ logM ! :“ logM and each t • 0.

Definition 1.15. For a strongly continuous semigroup T “ pT ptqqt•0 we call

!0 :“ !0pT q :“ inft! P R | DM! • 1 }T ptq} § M!e!t@t • 0u.

its growth bound (or type). Moreover, a semigroup is called bounded if wecan take ! “ 0, and contractive if ! “ 0 and M “ 1 is possible. Finally, thesemigroup is called isometric if }T ptqx} “ }x} for all t • 0, x P X.


We would like to show that using the weak operator topology instead of thestrong operator topology will not change our class of semigroups. This is asurprising result, and its proof needs more sophisticated tools from functionalanalysis.

Theorem 1.16. A semigroup T ptq, t • 0 on a Banach space X is stronglycontinuous if and only if it is weakly continuous, i.e., if the mappings R` Qt fiÑ† T ptqx, x1 °P C are continuous for each x P X, x1 P X 1.

Proof. We have only to show that weak implies strong continuity. As a firststep, we use the principle of uniform boundedness twice to conclude that oncompact intervals, T ptq is pointwise and then uniformly bounded. Therefore(use Proposition 1.13(c)), it su�ces to show that

E “ tx P X | limtÑ0`0

pT ptqx ´ xq “ 0u

is a dense subspace of X. To this end, we define for x P X and r ° 0 a linearform xr on X 1 by

† xr, x1 °:“ 1

r

ª r

0† T psqx, x1 ° ds

for x1 P X 1. Then xr is bounded and hence xr P X. On the other hand, the set

tT psqx | s P r0, rsu

is the continuous image of r0, rs in the spaceX endowed with the weak topology,hence is weakly compact in X. Mark Krein’s theorem implies that its closedconvex hull

cchtT psqx | s P r0, rsuis still weakly compact in X. Since xr is a �pX2, X 1q-limit of Riemann sums, itfollows that

xr P cchtT psqx | s P r0, rsu,whence xr P X. It is clear from the definition that the set

D :“ txr | r ° 0, x P Xu

is weakly dense in X. On the other hand, for xr P D we obtain

}T ptqxr ´ xr} “

sup}x1}§1

}1r

ª t`r

t† T psqx, x1 ° ds ´

ª r

0† T psqx, x1 ° ds}

§ 1

r}

ª t

0† T psqx, x1 ° ds} ` }1

r

ª t`r

r† T psqx, x1 ° ds}


§ 2t

r}x} sup

0§s§r`t}T psq} Ñ 0

as t Ñ 0 ` 0, i.e., D Ä E. We conclude that E is weakly, hence strongly, densein X.

Chapter 2

Semigroups, Generators,and Resolvents

2.1 Spectral Theory for Closed Operators

Let pA,DpAqq be a closed operator in a Banach space X.

Definition 2.1. We call

⇢pAq :“ t� P C | � ´ A : DpAq Ñ X is bijectiveu

the resolvent set and its complement �pAq :“ Cz⇢pAq the spectrum of A.

For � P ⇢pAq, the inverse

Rp�, Aq “ p� ´ Aq´1

is, by the closed graph theorem, a bounded operator on X and will be calledthe resolvent. It follows from the definition

ARp�, Aq “ �Rp�, Aq ´ I

holds for every � P ⇢pAq.

Proposition 2.2 (Resolvent Equation=RE). For �, µ P ⇢pAq, one has

Rp�, Aq ´ Rpµ,Aq “ pµ ´ �qRp�, AqRpµ,Aq. (2.1)

Proof. The definition of the resolvent implies

r�Rp�, Aq ´ ARp�, AqsRpµ,Aq “ Rpµ,Aq

Rp�, AqrµRpµ,Aq ´ ARpµ,Aqs “ Rpµ,Aq.

If we subtract these equations and use the fact that Rp�, Aq and Rpµ,Aq

commute, we obtain (2.1).

16

17 Chapter 2. Semigroups, Generators, and Resolvents

The basic properties of the resolvent set and the resolvent map are nowcollected in the following proposition.

Proposition 2.3. For a closed operator A : DpAq Ñ X, the following propertieshold.

(i) The resolvent set ⇢pAq is open in C, and for µ P ⇢pAq one has

Rp�, Aq “

8ÿ

n“0

pµ ´ �qnRn`1

pµ,Aq

for all � P C satisfying |µ ´ �| † 1{}Rpµ,Aq}.The resolvent map is locally analytic with

dn

d�nRp�, Aq “ p´1q

nn!Rn`1p�, Aq, n P N.

(iii) Let �n P ⇢pAq with limnÑ8 �n “ �0. Then �0 P �pAq i↵

limnÑ8 }Rp�n, Aq} “ 8.

Proof. (i) For � P C write

� ´ A “ µ ´ A ` p� ´ µq “ rpI ´ pµ ´ �qRpµ,Aqspµ ´ Aq.

This operator is bijective if rI ´ pµ ´ �qRpµ,Aqs is invertible, which is the casefor |µ´�|}Rpµ,Aq} † 1. The inverse is then obtained as (von Neumann series!)

Rp�, Aq “ Rpµ,AqrI ´ pµ ´ �qRpµ,Aqs´1

“

8ÿ

n“0

pµ ´ �qnRn`1

pµ,Aq.

Assertion piiq follows immediately from the series representation for the resol-vent.

To show piiiq we use piq, which implies

}Rpµ,Aq} •1

dist pµ, �pAq• 1

for all µ P ⇢pAq. This already proves one implication.For the converse, assume that �0 P ⇢pAq. Then the continuous resolvent

map remains bounded on the compact set t�n | n • 0u. This contradicts theassumption; hence �0 P �pAq.

As an immediate consequence, we have that the spectrum �pAq is a closedsubset of C. Nothing more can be said in general. However, if A is bounded, itfollows that

�pAq Ä t� P C| | |�| § }A}u,

2.2. Generators of Semigroups and Their Resolvents 18

since

Rp�, Aq “1

�p1 ´

1

�Aq

´1“

8ÿ

n“0

An

�n`1

exists for all |�| ° }A}. In addition, an application of Liouville’s theorem to theresolvent map implies �pAq ‰ H.

Corollary 2.4. For a bounded operator A on a Banach space X, the spectrum�pAq is always compact and nonempty; hence its spectral radius

rpAq :“ supt|�| | � P �pAqu

is finite and satisfies rpAq § }A}.

2.2 Generators of Semigroups and Their Re-solvents

We recall that for a one-parameter semigroup a Banach space X uniform conti-nuity implies di↵erentiability of the map t Ñ T ptq P LpXq. The right derivativeof T at t “ 0 then yields a bounded operator A with T ptq “ etA, t P R`.We may hope that strong continuity of a semigroup might still imply somedi↵erentiability of the orbit maps

⇠x : t Ñ T ptqx P X.

We first show that di↵erentiability of ⇠x is already implied by right di↵erentia-bility at t “ 0.

Lemma 2.5. Take a strongly continuous semigroup T ptq, t • 0 and an elementx P X. For the orbit map ⇠x : t Ñ T ptqx, the following properties are equivalent.

paq ⇠xp¨q is di↵erentiable on R`.pbq ⇠xp¨q is right di↵erentiable at t “ 0.

Proof. We have only to show that pbq implies paq. For h ° 0, one has

limhÑ0`0

1

hpT pt ` hqxT ptqxq “ T ptq lim

hÑ0`0

1

hpT phqx ´ xq “

T ptq⇠1xp0q

and hence ⇠x is right di↵erentiable on R`.On the other hand, for ´t § h † 0, we write

1

hpT pt ` hqx ´ T ptqxqT ptq⇠

1xp0q “ T pt ` hqp

1

hpx ´ T p´hqxq ´ ⇠

1xp0qq


`T pt ` hq⇠1xp0q ´ T ptq⇠

1xp0q.

As h Ñ 0 ` 0, the first term on the right-hand side converges to zero, since}T pt`hq} remains bounded. The remaining part converges to zero by the strongcontinuity of T ptq. Hence, ⇠x is also left di↵erentiable, and its derivative is

⇠1xptq “ T ptq⇠

1xp0q, t • 0. (2.2)

On the subspace of X consisting of all those x P X for which the orbit maps⇠x are di↵erentiable, the right derivative at t “ 0 then yields an operator A fromwhich, in a sense to be specified later, we can hope to obtain the operators T ptqas the ”exponentials etA”.

Definition 2.6. The generator A : DpAq Ä X Ñ X of a strongly continuoussemigroup T ptq on a Banach space X is the operator

Ax :“ ⇠1xp0q “ lim

hÑ0`0

1

hpT phqx ´ xq

defined for every x in its domain

DpAq :“ tx P X | ⇠x is di↵erentiableu.

We observe that

DpAq “ tx P X | D limhÑ0`0

1

hpT phqx ´ xq.

Lemma 2.7. For the generator pA,DpAqq of a strongly continuous semigroupT ptq, t • 0, the following properties hold.

piq A : DpAq Ä X Ñ X is a linear operator.piiq If x P DpAq, then T ptqx P DpAq and

d

dtT ptqx “ T ptqAx “ AT ptqx

for all t • 0.piiiq For every t • 0 and x P X, one has

ª t

0T psqxds P DpAq.

pivq For every t • 0, one has

T ptqx ´ x “ A

ª t

0T psqxds, if x P X

“

ª t

0T psqAxds, if x P DpAq.


Proof. Assertion piq is trivial. To prove piiq take x P DpAq. Then it followsfrom (2.2) that

1

hpT pt ` hqx ´ T ptqxq

converges to T ptqAx as h Ñ 0 ` 0. Therefore,

limhÑ0`0

1

hpT phqT ptqx ´ T ptqxq

exists, and hence T ptqx P DpAq with AT ptqx “ T ptqAx.The proof of assertion piiiq is included in the following proof of pivq. For

x P X and t • 0, one has

1

hpT phq

ª t

0T psqxds ´

ª t

0T psqxdsq “

“1

h

ª t

0T ps ` hqxds ´

1

h

ª s

0T psqxds

“1

h

ª t`h

hT psqxds ´

1

h

ª s

0T psqxds

“1

h

ª t`h

tT psqxds ´

1

h

ª h

0T psqxds

which converges to T ptqx ´ x as h Ñ 0 ` 0. Hence

T ptqx ´ x “ A

ª t

0T psqxds, if x P X

holds.If x P DpAq), then the functions s Ñ T psq

T phqx´xh converge uniformly on

h P r0.ts to the function s Ñ T psqAx as h Ñ 0 ` 0. Therefore,

limhÑ0`0

1

hpT ptq ´ Iq

ª t

0T psqxds

“ limhÑ0`0

ª t

0T psq

T phq ´ I

hxds

“

ª t

0T psqAxds.

Theorem 2.8. The generator of a strongly continuous semigroup is a closedand densely defined linear operator that determines the semigroup uniquely.


Proof. Let T ptq, t • 0 be a strongly continuous semigroup on a Banach spaceX. As already noted, its generator pA,DpAqq is a linear operator. To show thatA is closed, consider a sequence pxnqnPN Ä DpAq for which limnÑinfty xn “ xand limnÑ8 Axn “ y exists. By the previous lemma, we have

T ptqxn ´ xn “

ª t

0T psqAxnds

for t ° 0.The uniform convergence of T p¨qAxn on r0, ts for n Ñ 8 implies that

T ptqx ´ x “

ª t

0T psqyds.

Multiplying both sides by 1{t and taking the limit as t Ñ 0 ` 0, we see thatx P DpAq and Ax “ y, i.e., A is closed.

By piiiq the elements1

t

ª t

0T psqds

always belong to DpAq. Since the strong continuity of T ptq implies

limtÑ0`0

1

t

ª t

0T psqxds “ x

for every x P X, we conclude that DpAq is dense in X.Finally, let Sptq be another strongly continuous semigroup having the same

generator pA.DpAqq. For x P DpAq and t ° 0, we consider the map

s Ñ ⌘xpsq :“ T pt ´ sqSpsqx

for s P r0, ts. Since for fixed s the set

tSps ` hqx ´ Spsqx

h| h P p0, 1su Y tASpsqxu

is compact, the di↵erence quotients

1

hp⌘xps ` hq ´ ⌘xpsqq

“ T pt ´ s ´ hq1

hpSps ` hqx ´ Spsqxq `

1

hpT pt ´ s ´ hq ´ T pt ´ sqqSpsqx

converge by piiq to

d

ds⌘xpsq “ T pt ´ sqASpsqx ´ AT pt ´ sqSpsqx “ 0.

From ⌘xp0q “ T ptqx and ⌘xptq “ Sptqxwe obtain T ptqx “ Sptqx for all x in thedense domain DpAq. Hence, T ptq “ Sptq for each t • 0.


Definition 2.9. A subspace D of the domain DpAq) of a linear operator A :DpAq Ñ X is called a core for A if D is dense in DpAq for the graph norm

}x}A :“ }x} ` }Ax}.

We now state a useful criterion for subspaces to be a core for the generator.

Proposition 2.10. Let pA,DpAqq be the generator of a strongly continuoussemigroup T ptq on a Banach space X. A subspace D Ä DpAq that is } ¨ }-densein X and invariant under the semigroup pT ptqqt•0 is always a core for A.

Proof. For every x P DpAq we can find a sequence pxnqnPN Ä D such thatlimnÑ8 “ x. Since for each n the map s Ñ T psqxn P D is continuous for thegraph norm } ¨ }A (use

T ptqAx “ AT ptqxq,

it follows that ª t

0T psqxnds,

being a Riemann integral, belongs to the } ¨ }A-closure of D. Similarly, the} ¨ }A-continuity of s Ñ T psqx for x P DpAq implies that

}1

t

ª t

0T psqxds ´ x}A Ñ 0

as t Ñ 0 ` 0 and

}1

t

ª t

0T psqxnds ´

1

t

ª t

0T psqxds}A Ñ 0

as n Ñ 8 and for each t ° 0. This proves that for every ✏ ° 0 we can findt ° 0 and n P N such that

}1

t

ª t

0T psqxnds ´ x}A † ✏.

Hence, x P D̄}¨}A.

Lemma 2.11. Let pA,DpAqq be the generator of a strongly continuous semi-group T ptq. Then, for every � P C and t ° 0, the following identities hold:

e´�tT ptqx ´ x “ pA ´ �q

ª t

0e�tT psqxds if x P X,

“

ª t

0e�tT psqpA ´ �qxds if x P DpAq.


Proof. It su�ces to apply pivq to the rescaled semigroup Sptq :“ e´�tT ptq, t • 0,whose generator is B “ A ´ � with domain DpBq “ DpAq.

Next, we give an important formula relating the semigroup to the resolventof its generator.

Theorem 2.12. Let T ptq be a strongly continuous semigroup on the Banachspace X and take constants ! P R,M • 1 such that

}T ptq} § Me!t, t • 0.

For the generator pA,DpAq of T ptq the following properties hold.piq If � P C

Rp�qx :“

ª 8

0e´sT psqxds

exists for all x P X, then � P ⇢pAq and Rp�q “ Rp�, Aq.piiq If <� ° !, then � P ⇢pAq, and the resolvent is given by the integral

expression in piq.piiiq

}Rp�, Aq} § M<p� ´ !q

for all <� ° !.The formula for Rp�, Aq in piq is called the integral representation of the

resolvent. Of course, the integral has to be understood as an improper Riemannintegral, i.e.,

Rp�, Aqx “ limtÑ8

ª t

0e´�sT psqxds

for all x P X.Having in mind this interpretation, we will frequently write

Rp�, Aq “

ª 8

0e�sT psqds.

Proof. piq By a simple rescaling argument we may assume that � “ 0. Then,for arbitrary x P X and h ° 0 we have

T phq ´ I

hRp0qx “

T phq ´ I

h

ª 8

0T psqxds

“1

h

ª 8

0T ps ` hqxds ´

1

h

ª 8

0T psqxds

“1

h

ª 8

hT psqxds ´

1

h

ª 8

0T psqxds

“ ´1

h

ª h

0T psqxds.


By taking the limit as h Ñ 0 ` 0, we conclude that RanpRp0qq Ä DpAq andARp0q “ ´I.

On the other hand, for x P DpAq we have

limtÑ8

ª t

0T psqxds “ Rp0qx,

limtÑ8A

ª t

0T psqxds “ lim

tÑ8

ª t

0T psqAxds “ Rp0qAx,

where we have used lemma (iv) for the second equality. Since A, this impliesRp0qAx “ ARp0qx “ ´x and therefore Rp0q “ p´Aq

´1 as claimed.Parts piiq and piiiq then follow easily from piq and the estimate

}

ª t

0e´�sT psqds} § M

ª t

0ep!´<�qsds,

since for <� ° ! the right-hand side converges to

M

<� ´ !

as t Ñ 8.

Corollary 2.13. For the generator pA,DpAqq of a strongly continuous semi-group T ptq satisfying

}T ptq} § Me!t, t • 0,

one has, for <� ° ! and n P N, that

Rnp�, Aqx “

p´1qn´1

pn ´ 1q!

dn´1

d�n´1Rp�, Aqx (2.3)

“1

pn ´ 1q!

ª 8

0sn´1e´�sT psqxds (2.4)

for all x P X. In particular,

}Rnp�, Aq} §

M

p<� ´ !qn(2.5)

for all n P N and <� ° !.

Proof. Equation (2.3) is actually valid for every operator with nonempty resol-vent set. On the other hand, by Theorem 2.12 (i), one has

d

�.Rp�, Aqx “

d

d�

ª 8

0e´�sT psqxds


“ ´

ª 8

0se´�sT psqxds

for <� ° ! and x P X. Proceeding by induction, we deduce (2.4). Finally, theestimate (2.5) follows from

}Rnp�, Aqx} § }

1

pn ´ 1q!

ª 8

0sn´1e´�sT psqxds}

§1

pn ´ 1q!}x}M

ª 8

0sn´1ep!´<�qsds

“M

p<� ´ !qn}x}.

Property piiq in Theorem 2.12 says that the spectrum of a semigroup gen-erator is always contained in a left half-plane. The number determining thesmallest such half-plane is an important characteristic of any linear operatorand is defined as follows.

Definition 2.14. To any linear operator A we associate its spectral bounddefined by

spAq :“ supt<� | � P �pAqu.

As an immediate consequence of Theorem 2.12 piiq the following relation holdsbetween the growth bound of a strongly continuous semigroup and the spectralbound of its generator.

Corollary 2.15. For a strongly continuous semigroup T ptq with generator A,one has

´8 § spAq § !0 † `8.

2.3 Hille-Yosida Generation Theorems

We now turn to the fundamental problem of semigroup theory, which is tofind ways leading from the generator (or its resolvent) to the semigroup. Moreprecisely, this means that we will discuss the following problem.

Problem. Characterize those linear operators that are the generator of somestrongly continuous semigroup.

We already saw that generators

• are necessarily closed operators,

• have dense domain, and

2.3. Hille-Yosida Generation Theorems 26

• have their spectrum contained in some proper left half-plane.

These conditions, however, are not su�cient.

Lemma 2.16. Let pA,DpAqq be a closed, densely defined operator. Supposethere exist ! P R and M ° 0 such that r!,8q Ä ⇢pAq and }�Rp�, Aq} § M forall � • !. Then the following convergence statements hold for � Ñ 8:

(i) �Rp�, Aqx Ñ x for all x P X.(ii) �ARp�, Aqx “ �Rp�, AqAx Ñ Ax for all x P DpAq.

Proof. If y P DpAq, then �Rp�, Aqy “ Rp�, AqAy`y. This expression convergesto y as � Ñ 8, since

}Rp�, AqAy} §M

�}Ay}.

Since }�Rp�, Aq} is uniformly bounded for all � • !, statement (i) follows. Thesecond statement is then an immediate consequence of the first one.

Since for contraction semigroups the technical details of the subsequentproof become much easier (and since the general case can then be deducedfrom this one), we first give the characterization theorem for generators in thisspecial case.

Theorem 2.17 (Contracting Case, Hille, Yosida, 1948). For a linear operatorpA,DpAqq on a Banach space X, the following properties are all equivalent.

(a) pA,DpAqq generates a strongly continuous contraction semigroup.(b) pA,DpAqq is closed, densely defined and for every � ° 0 one has � P ⇢pAq

and}�Rp�, Rq} § 1. (2.6)

(c) pA,DpAqq is closed, densely defined, and for every � P C with <� ° 0one has � P ⇢pAq and

}Rp�, Aq} §1

<� . (2.7)

Proof. In view of Theorem 2.8 and Theorem 2.12, it su�ces to show pbq Ñ paq.To that purpose, we define the so-called Yosida approximants

An :“ nARpn,Aq “ n2Rpn,Aq ´ nI

which are bounded operators for each n P N and commute with one another.Consider then the uniformly continuous semigroups given by

Tnptq :“ etAn, t • 0.

Since An converges to A pointwisely on DpAq (by Lemma 2.16 (ii)), we antici-pate that the following properties hold.

(i) T ptqx :“ limnÑ8 Tnptqx exists for each x P X.


(ii) T ptq is a strongly continuous semigroup on X.(iii) This semigroup has generator pA,DpAqq.By establishing these statements we will complete the proof.(i) Each Tnptq is a contraction semigroup, since

}Tnptq} § e´nten2}Rpn,Aq}t

§ e´ntent “ 1

for t • 0.So, it su�ces to prove convergence just on DpAq. By (the vector-valued

version of) the fundamental theorem of calculus, applied to the functions s Ñ

Tmpt ´ sqTnpsqx for 0 § s § t, x P DpAq and n,m P N, and using the mutualcommutativity of the semigroups Tnptq for all n P N, one has

Tnptqx ´ Tmptqx “

ª t

0

d

dspTmpt ´ sqTnpsqxqds

“

ª t

0Tmpt ´ sqTnpsqpAnx ´ Amxqds.

Accordingly,}Tnptqx ´ Tmptqx} § t}Anx ´ Amx}. (2.8)

By Lemma 2.16 (ii), pAnxqnPN is a Cauchy sequence for each x P DpAq.Therefore, Tnptqx converges uniformly on each interval r0, t0s.

(ii) The pointwise convergence of Tnptqx, n P N implies that the limit familyT ptq satisfies the functional equation (FE), hence is a semigroup, and consistsof contractions. Moreover, for each x P DpAq, the corresponding orbit map

⇠ : t Ñ T ptqx, t P r0, t0s,

is the uniform limit of continuous functions and so is continuous itself. Thissu�ces to obtain strong continuity.

(iii) Denote by pB,DpBqq the generator of T ptq and fix x P DpAq. On eachcompact interval r0, t0s, the functions

⇠n : tnptqx

converge uniformly to ⇠p¨q by (2.8), while the di↵erentiated functions

⇠1n : t Ñ TnptqAnx

converge uniformly to⌘ : t Ñ T ptqAx.

This implies di↵erentiability of ⇠ with ⇠1p0q “ ⌘p0q, i.e., DpAq Ä DpBq and

Ax “ Bx for x P DpAq.


Now choose � ° 0. Then � ´ A is a bijection from DpAq onto X, since � P

⇢pAq by assumption. On the other hand, B generates a contraction semigroup,and so � P ⇢pBq by Theorem 2.12. Hence, � ´ B is also a bijection from DpBq

onto X. But we have seen that � ´ B coincides with � ´ A on DpAq. This ispossible only if DpAq “ DpBq and A “ B.

If a strongly continuous semigroup T ptq with generator A satisfies, for some! P R, an estimate

}T pt} § e!t, (2.9)

then we can apply the above characterization to the rescaled contraction semi-group given by

Sptq :“ e´!tT ptq, t • 0.

Since the generator of Sptq is B “ A´! Theorem 2.6 takes the following form.Let ! P R. For a linear operator pA,DpAq on a Banach space X the followingconditions are equivalent.

paq pA,DpAqq generates a strongly continuous semigroup T ptqpbq pA,DpAqq is closed, densely defined, and for each � ° ! one has � P ⇢pAq

and}p� ´ !qRp�, Aq} § 1. (2.10)

pcq pA,DpAqq is closed, densely defined, and for each � P C with <� ° !one has � P ⇢pAq and

}Rp�, Aq} §1

<� ´ !. (2.11)

Semigroups satisfying ( 2.9 ) are called quasi-contractive.It is now a pleasant surprise that the characterization of generators of arbi-

trary strongly continuous semigroups can be deduced from the above result forcontraction semigroups. However, norm estimates for all powers of the resolventare needed.

Theorem 2.18 (General Case, Feller, Miyadera, Phillips, 1952). .Let pA,DpAqq be a linear operator on a Banach space X and let ! P R, M •

1 be constants. Then the following properties are equivalent.paq pA,DpAqq generates a strongly continuous semigroup T ptq satisfying

}T ptq} § Me!t. (2.12)

pbq pA,DpAq is closed, densely defined, and for every � ° ! one has � P ⇢pAq

and}rp� ´ !qRp�, Aqs

n} § M, n P N. (2.13)

pcq pA,DpAqq is closed, densely defined, and for every � P C with <� ° !one has � P ⇢pAq and

}Rnp�, Aq} §

M

p<� ´ !qn, n P N. (2.14)


Proof. The implication paq Ñ pcq has been proved in the corollary, while pcq Ñ

pbq is trivial. To prove pbq Ñ paq we use, as for Corollary 2.7, the rescalingtechnique. So, without loss of generality, we assume that ! “ 0, i.e.,

}�nRnp�, Aq § M

for all ° 0, n P N. For every µ ° 0, define a new norm on X by

}x}µ “ supn•0

}µnRnpµ,Aqx}.

These norms have the following properties.piq }x} § }x}µ § M}x}, i.e., they are all equivalent to } ¨ }.piiq

}µRpµ,Aq}µ § 1.

piiiq}�Rp�, Aq}µ § 1, @� P p0, µs.

pivq

}�nRnp�, Aq} § }�nRn

p�, Aq}µ § }x}µ

for all � P p0, µs, n P N.pvq

}x}� § }x}µ

for � P p0, µs.We give the proof only of piiiq. Due to the Resolvent Equation, we have

that

y :“ Rp�, Aqx “ Rpµ,Aqx ´ pµ ´ �qRp�, AqRpµ,Aqx “ Rpµ,Aqpx ´ pµ ´ �qyq.

This implies, by using piiq, that

}y}µ §1

µ}x}µ `

� ´ µ

µ}y}µ,

whence �}y}µ § }x}µ.On the basis of these properties one can define still another norm by

|||x||| :“ supµ ° 0}x}µ, (2.15)

which evidently satisfies

pviq }x} § |||x||| § M}x}

andpviiq |||�Rp�, Aq||| § 1


for all � ° 0.Thus, the operator pA,DpAqq satisfies condition (2.6) for the equivalent

norm |||¨||| and so, by the Theorem 2.6, generates a |||¨|||-contraction semigroupT ptq, t • 0. Using pviq again, we obtain }T ptq} § M .

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