Lectures on Operator Semigroups

Yuri G. Kondratiev

2017 2 Contents

1 Linear Dynamical Systems5 1.1 Cauchy’s Functional Equation...... 5 1.2 Finite-Dimensional Systems: Matrix Semigroups...... 8 1.3 Uniformly Continuous Operator Semigroups...... 10 1.4 Strongly Continuous Semigroups...... 11

3 4 Contents Chapter 1 Linear Dynamical Systems

There are many good reasons why an ”autonomous deterministic system” should be described by maps T ptq, t ě 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 the system into itself. These maps completely determine the time evolution of the system in the following way: If the system is in state x0 at time t0 “ 0 then at time t it is in state T ptqx. However, in most cases a complete knowledge of the maps T ptq is hard, if not impossible, to obtain. It was one of the great discoveries of mathematical physics, based on the invention of calculus, that, as a rule, it is much easier to understand the ”infinitesimal changes” occurring at any given time. In this case, the system can be described by a differential equation. In this chapter we analyze this phenomenon in the mathematical context of linear operators on Banach spaces. For this purpose, we take two opposite views. pV 1q. We start with a solution t ÞÑ T ptq and ask which assumptions imply that it is differentiable and satisfies a differential equation. pV 2q. We start with a differential equation and ask how its solution can be related 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, without any further motivation, the following question: D’eterminer la fonction φpxq de mani’ere qu’elle reste continue entre deux limites r’eelles quelconques de la variable x, et que l’on ait pour toutes les valeurs r’eelles des variables x et y φpx ` yq “ φpxqφpyq. Using modern notation, we restate his question as follows dropping the continuity requirement for the moment.

5 6 Chapter 1. Linear Dynamical Systems

p1.1q Problem. Find all maps T p¨q : R` ÞÑ C satisfying the functional equation

T pt ` sq “ T ptqT psqfor all t, s ě 0, (1.1) T p0q “ 1. Evidently, the exponential functions

t ÞÑ eta satisfy (FE) for any a P C. With his question, Cauchy suggested that these canonical solutions should be all solutions of (FE). Before giving an answer to Problem 1.1, we take a closer look at the exponential functions and observe that they, besides solving the algebraic identity (FE), also enjoy some important analytic properties. Determine the function φpxq in such a way that it remains continuous be- tween two arbitrary real limits of the variable x, and that, for all real values of the variables x and y, one has

φpx ` yq “ φpxqφpyq.

Proposition 1.1. Let T ptq :“ eta for some a P C all t ě 0. Then the function T p¨q is differentiable and satisfies the differential equation (or, more precisely, the initial value problem) (DE)

d T ptq “ aT ptq dt for all t ě 0, T p0q “ 1. Conversely, the function T p¨q : R` Ñ C defined by T ptq “ eta for some a P C is the only differentiable function satisfying (DE). d Finally, we observe that a “ dtT ptq for t “ 0.

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

Qpsq :“ T psqSpt ´ sq for 0 ď s ď t

d for some fixed t ą 0 is differentiable with derivative dsQpsq ” 0. This shows that T ptq “ Qptq “ Qp0q “ Sptq for arbitrary t ą 0. This proposition shows that, in our scalar-valued case, pV 2q can be answered easily using the exponential function. It is now our main point that continuity is already sufficient to obtain differentiability 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 differentiable, 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

t V ptq :“ T psqds, t ě 0, ż0 is differentiable with V 1ptq “ T ptq. This implies

V ptq lim “ V 1p0q “ T p0q “ 1. tÑ0` t

Therefore, V pt0q is different from zero, hence invertible, for some small t0 ą 0. The functional equation (FE) now yields

t ´1 ´1 T ptq “ V pt0qV pt0qT ptq “ V pt0q T pt ` sqds ż0

t`t0 ´1 ´1 “ V pt0q T psqds “ V pt0qpV pt ` t0q ´ V ptqq żt for all t ě 0. Hence, T p¨q is differentiable with derivative

d T pt ` hq ´ T ptq “ lim “ dt hÑ0` h

T phq ´ T p0q lim T ptq “ T 1p0qT ptq hÑ0` h for all t ě 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’s Problem 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 ě 0.

1.2 Finite-Dimensional Systems: Matrix Semi- groups

In this section we pass to a more general setting and consider finite dimensional vector spaces X “ Cn. The space LpXq of all linear operators on X will then be identified with the space MnpCq of all complex n ˆ n matrices, and 8 Chapter 1. Linear Dynamical Systems a linear dynamical system on X will be given by a matrix-valued function T p¨q : R` Ñ MnpCq satisfying the functional equation (FE)

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

As before, the variable t will be interpreted as ”time”. The ”time evolution” of a state x0 P X is then given by the function ξ : R` Ñ X defined as ξptq “ T ptqx0. We also call tT ptqx0 | t ě 0u the orbit of x0 under T p¨q. From the functional equation (FE) it follows that an initial state x0 arrives after an elapsed time t ` 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 restate Cauchy’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 dynamical systems. As in Section 1 the candidates for solutions of (FE) are the ”expo- nential functions”. tA Definition 1.4. For any A P MnpCq and t P R the matrix exponential e is defined by 8 tkAk etA :“ . (1.2) k! k`0 ÿ n Taking any norm on C and the corresponding matrix-norm on MnpCq one shows that the partial sums of the series above form a Cauchy sequence; hence the series converges and satisfies

}etA} ď et}A}. for all t ě 0. Moreover, the map t ÞÑ etA has the following properties.

Proposition 1.5. For any A P MnpCq the map

tA R` Q t Ñ e P MnpCq is continuous and satisfies (FE)

ept`sqA “ etAesA for t, ě 0, e0A “ I.

Proof. Since the series 8 tkAk k! k`0 ÿ 1.2. Finite-Dimensional Systems: Matrix Semigroups 9 converges, one can show, as for the Cauchy product of scalar series, that

8 tkAk 8 skAk k! k! k`0 k`0 ÿ ÿ 8 n tn´kAn´k skAk “ “ n k ! k! n“0 k“0 p ´ q ÿ ÿ 8 pt ` sqnAn . n! n“0 ÿ This proves (FE). In order to show that t Ñ etA is continuous, we first observe that by (FE) one has et`h ´ etA “ etApehA ´ Iq for all t, h P R. Therefore, it hA suffices to show that limhÑ0e “ I. This follows from the estimate 8 hkAk }ehA ´ I}“} } ď k! k“1 ÿ |h|k}A}k k “ 18 “ e|h|}A} ´ 1. k! ÿ tA Obviously, the range of the function t Ñ T ptq :“ e 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 semigroup pR`, `q into the multiplicative semigroup MnpCq. Keeping this in mind, we start to use the following terminology. Definition 1.6. We call etA, t ě 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 and even complex t. Then the map

T : t Ñ etA extends to a continuous (even analytic) homomorphism from the additive group pR, `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 generated by A. Example 1.7. (i) The (semi) group generated by a diagonal matrix A “ diagpa1, . . . anq is given by

etA “ diagpeta1 , . . . , etan q. 10 Chapter 1. Linear Dynamical Systems 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 can restate 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 ě 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 much more complex than what we encountered up to now. For every function T : R` Ñ LpXq satisfying (FE) the set tT ptq | t ě 0u is a commutative sub-semigroup of LpXq. This justifies calling the functional equation (FE) the semigroup law and using the following terminology. Definition 1.8. A family T ptq, t ě 0 of bounded linear operators on a Banach space X is called a (one-parameter) semigroup (or linear dynamical system) on X if it satisfies the functional equation (FE). If (FE) holds even for all t, s P R we call T ptq, t P R a (one-parameter) group on X. We now introduce the ”typical” examples of one-parameter semigroups of operators on a Banach space X. Take any operator A P LpXq. As in the matrix case, we can define an operator-valued exponential function by

8 tkAk etA “ , k! k“0 ÿ where the convergence of this series takes place in the Banach algebra LpXq. Using the same arguments as above, one shows that etA, t ě 0 satisfies the functional equation (FE) and the differential equation (DE), and hence Theorem 1.10 below follows as in Section 2. Definition 1.9. A one-parameter semigroup T ptq, t ě 0 on a Banach space X is called uniformly continuous (or norm continuous) if R` Q t Ñ T ptq P LpXq is continuous with respect to the uniform operator topology on LpXq. Theorem 1.10. Every uniformly continuous semigroup T ptq, t ě 0 on a Banach space 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 is sufficient. For a uniformly continuous semigroup T ptq the operators

t V ptq “ T psqds ż0 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 ą 0 sufficiently small, the operator V ptq becomes invertible. Repeat now the computations from the proof of Theorem 1.4 in order to obtain that t Ñ T ptq is differentiable and satisfies (DE). (i) The operator A in Theorem 1.10 is determined uniquely as the derivative of 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 uniformly continuous group etA, t P R, or to etA, t P C, respectively. (iii) From the differentiability of T ptq it follows that for each x P X the orbit map R` Q t Ñ T ptqx P X is differentiable as well. Therefore, the map xptq :“ T ptqx is the unique solution of the X-valued initial value problem (or abstract Cauchy problem) (ACP)

x1ptq “ Axptq, t ě 0, xp0q “ x.

1.4 Strongly Continuous Semigroups

In many essential examples uniform continuity is too strong a requirement for many natural semigroups defined on concrete function spaces. Definition 1.11. A family T ptq, t ě 0 of bounded linear operators on a Banach space 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 ě 0 is a strongly continuous semigroup if the functional equation (FE) T pt ` sq “ T ptqT psq for all t, s ě 0,T p0q “ I holds and the orbit maps (SC)

ξx : t Ñ ξxptq :“ 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 the following frequently used equivalence. Lemma 1.12. Let X be a Banach space and let F be a function from a compact set K Ă R into LpXq. Then the following assertions are equivalent. (a) F is continuous for the strong operator topology, i.e., the mappings K 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 X are continuous for all x in some dense subset D Ă X. 12 Chapter 1. Linear Dynamical Systems

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

Proof. The implication pcq Ñ paq is trivial, while paq Ñ pbq follows from the uniform 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} ď M for all t P K and fix some  ą 0 and a compact set C Ă X. Then there exist finitely many x1, . . . , xm P D such that  C ĂYm px ` Uq, i“1 i M where U denotes the unit ball of X. Now choose δ ą 0 s.t.

}F ptqxi ´ F psqxi} ď  for all i “ 1, . . . , m, and for all t, s P K, such that |t ´ s| ď δ. For arbitrary x, y P C and t, s P K with }x ´ y} ď M ď δ this yields

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

`}F psqpxj ´ xq} ` }F psqpx ´ yq} ď 4,  where we choose j P t1, . . . , mu such that }x ´ xj} ď 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

ξx : t Ñ T ptqx at each t ą 0 and for each x P X is already implied by much weaker properties. Proposition 1.13. For a semigroup T ptq, t ě 0 on a Banach space X, the following assertions are equivalent. (a) T ptq is strongly continuous. (b)limtÑ0` T ptqx “ x for all x P X. (c) There exist δ ą 0,M ě 1 and a dense subset D Ă X such that (i) }T ptq} ď M for all t P r0, δs, 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 pδnqnPN Ă R` converg- ing to zero such that }T pδnq} Ñ 8 as n Ñ 8. Then, by the uniform bound- edness principle, there exists x P Xsuch that T pδnqx}, n P N is unbounded, contradicting the fact that T p¨q is continuous at t “ 0. 1.4. Strongly Continuous Semigroups 13

In order to verify that pcq Ñ pbq, we put K :“ ttn | n P Nu Y t0u for an arbitrary sequence ptnqnPN Ă 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 can apply Lemma (b) to obtain lim T pt qx “ x nÑ8 n for all x P X. Since ptnqnPN was chosen arbitrarily, this proves pbq. To show that pbq Ñ paq, let t0 ą 0 and let x P X. Then

lim }T pt0 ` hqx ´ T pt0qx} ď }T pt0q} lim }T phqx ´ x}“ 0, tÑ0`0 tÑ0`0 which proves right continuity. If h ă 0, the estimate

}T pt0 ` hqx ´ T pt0qx} ď }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, δs by the uniform boundedness principle and then on each compact interval using (FE). Since in many cases the uniform boundedness of the operators T ptq for t P r0, δs is obvious, one obtains strong continuity by checking (right) continuity of the orbit maps ξx 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 ě 0 the finite orbits tT ptqx | t P r0, t0su are continuous images of a compact interval, hence compact and therefore bounded for each x P X. So by the uniform boundedness principle, each strongly continuous semigroup is uniformly bounded on each compact interval, a fact that implies exponential boundedness on R`. Proposition 1.14. For every strongly continuous semigroup T ptq, t ě 0 there exist constants ω P R and M ě 1 such that }T ptq} ď Meωt (1.3) for all t ě 0. Proof. Choose M ě 1 such that }T psq} ď M for all 0 ď s ď 1 and write t ě 1 as t “ s ` n for n P N and 0 ď s ă 1. Then }T ptq} ď }T psq}}T p1q}n ď M n`1 “ Men log M ď Meωt holds for ω :“ log M ω :“ logM and each t ě 0.

Definition 1.15. For a strongly continuous semigroup T “ pT ptqqtě0 we call ωt ω0 :“ ω0pT q :“ inftω P R | DMω ě 1 }T ptq} ď Mωe @t ě 0u. its growth bound (or type). Moreover, a semigroup is called bounded if we can take ω “ 0, and contractive if ω “ 0 and M “ 1 is possible. Finally, the semigroup is called isometric if }T ptqx}“}x} for all t ě 0, x P X. 14 Chapter 1. Linear Dynamical Systems

We would like to show that using the weak operator topology instead of the strong operator topology will not change our class of semigroups. This is a surprising result, and its proof needs more sophisticated tools from functional analysis. Theorem 1.16. A semigroup T ptq, t ě 0 on a Banach space X is strongly continuous if and only if it is weakly continuous, i.e., if the mappings R` Q t ÞÑă T ptqx, x1 ąP C are continuous for each x P X, x1 P X1. Proof. We have only to show that weak implies strong continuity. As a first step, we use the principle of uniform boundedness twice to conclude that on compact intervals, T ptq is pointwise and then uniformly bounded. Therefore (use Proposition 1.13(c)), it suffices to show that

E “ tx P X | lim pT ptqx ´ xq “ 0u tÑ0`0 is a dense subspace of X. To this end, we define for x P X and r ą 0 a linear 1 form xr on X by 1 r ă x , x1 ą:“ ă T psqx, x1 ą ds r r ż0 1 1 for x P X . 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 space X endowed with the weak topology, hence is weakly compact in X. Mark Krein’s theorem implies that its closed convex hull cchtT psqx | s P r0, rsu 2 1 is still weakly compact in X. Since xr is a σpX ,X q-limit of Riemann sums, it follows 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 ą 0, x P Xu is weakly dense in X. On the other hand, for xr P D we obtain

}T ptqxr ´ xr}“ 1 t`r r sup } ă T psqx, x1 ą ds ´ ă T psqx, x1 ą ds} 1 r }x }ď1 żt ż0 1 t 1 t`r ď } ă T psqx, x1 ą ds} ` } ă T psqx, x1 ą ds} r r ż0 żr 1.4. Strongly Continuous Semigroups 15

2t ď }x} sup }T psq} Ñ 0 r 0ďsďr`t as t Ñ 0 ` 0, i.e., D Ă E. We conclude that E is weakly, hence strongly, dense in X.