Some elements of semigroup theory
J´erˆome Le Rousseau January 8, 2019
Contents
1 Strongly continuous semigroups 2 1.1 Definition and basic properties ...... 2 1.2 The Hille-Yosida theorem ...... 5 1.3 The Lumer-Phillips theorem ...... 6
2 Differentiable and analytic semigroups 9
3 Mild solution of the inhomogeneous Cauchy problem 10
4 The case of a Hilbert space. 12
Semigroup theory is at the heart of the understanding of many evolution equa- tions that can be put in the form d x(t)+ Ax(t)= f(t), t> 0, x(t)= x , (0.1) dt 0 with x(t) and x0 in a proper function space, usually a Banach space, denoted by X below, if not a Hilbert space, with A an unbounded operator on X, with dense do- main, and f a function of the time variable t taking values in X. First, in Sections 1 and 2, we review the case of a homogeneous equation, that is f ≡ 0. Second, in Section 3 we consider the more general case of an inhomogeneous equation. In par- ticular, we provide the necessary form of the solution, given by the so-called mild solution based on the Duhamel formula. Third, in Section 4 we show how some results improve in the case the function space X is a Hilbert space.
1 1 Strongly continuous semigroups
Consider the homogeneous equation associated with the evolution problem (0.1), that is, d x(t)+ Ax(t)=0, t> 0, x(t)= x , (1.1) dt 0
Under proper assumptions on A we can write the solution in the form x(t)= S(t)x0, where S(t): X → X is a bounded operator. As some sort of differentiation with respect to time is expected in (1.1), a minimal assumption is then that
S(0)x = x and t 7→ S(t)x be continuous for all x ∈ X. (1.2)
With t 7→ x(t) solution to (1.1), if the evolution problem is well posed, we expect from uniqueness that solving the following problem, for some t0 ≥ 0, d y(t)+ Ay(t)=0, t>t , y(t )= x(t ), (1.3) dt 0 0 0 yield a solution that satisfies y(t) = x(t) for t ≥ t0. In particular, this implies the following property
S(t + t′)= S(t) ◦ S(t′), for t, t′ ∈ [0, +∞). (1.4)
Properties (1.2) and (1.4) are precisely the starting point of semigroup theory in Banach spaces.
1.1 Definition and basic properties Let X be a Banach space.
Definition 1. A family S(t) of bounded operators on X, indexed by t ∈ [0, +∞) is called a semigroup if:
′ ′ ′ S(0) = IdX and S(t + t )= S(t) ◦ S(t ) for t, t ∈ [0, +∞). (1.5)
The semigroup is called strongly continuous if, moreover, for all x ∈ X we have limt→0+ S(t)x = x. One says that S(t) is a C0-semigroup for short.
For each C0-semigroup there exist M ≥ 1 and ω ∈ R such that
S ωt k (t)kL (X) ≤ Me , (1.6) by the uniform boundedness principle [3, Theorem 1.2.2]. It follows that the map
(t, x) 7→ S(t)x is continuous from [0, +∞) × X into X. A C0-semigroup S(t) is said
2 S to be bounded if there exists M ≥ 1 such that k (t)kL (X) ≤ M, for t ≥ 0. In the case M = 1 one says that the C0-semigroup is of contraction. We define the unbounded linear operator A from X to X, with domain
D(A)= {x ∈ X; lim t−1(x − S(t)x) exists}, (1.7) t→0+ and given by
Ax = lim t−1(x − S(t)x), x ∈ D(A). (1.8) t→0+ For basic aspects of unbounded operators we refer to [1].
This operator (A, D(A)) is called the generator of the C0-semigroup. One can prove that the generator of a C0-semigroup has a dense domain in X and is a closed operator [3, Corollary 1.2.5] (see [1] for the notion of closed operators). The domain D(A) can be equipped with the graph norm
kxkD(A) = kxkX + kA(x)kX .
Since A is closed one finds that (D(A), k.kD(A)) is complete. Note that the map
S(t) 7→ A (1.9) is injective [3, Theorem 1.2.6]. The following proposition shows that computing S(t)x yields a solution of an evolution equation.
Proposition 2. Let T ∈ R+ ∪{+∞} and let x ∈ D(A). We have u(t) = S(t)x ∈ C 0([0, T ],D(A)) ∩ C 1([0, T ],X) and d u(t)+ Au(t)=0, 0 ≤ t ≤ T, u(0) = x. (1.10) dt Moreover, S(t)x is the unique solution to (1.10) in C 0([0, T ],D(A)) ∩ C 1([0, T ],X). In addition, we have AS(t)x = S(t)Ax. Here, [0, T ] means [0, +∞) if T =+∞.
Proof. Let x ∈ D(A). We write, for h> 0,
−1 −1 h IdX −S(h)S(t)x = S(t)h IdX −S(h)x →h→0+ S(t)Ax, as S(t) is bounded on X. Hence, we have S(t)x ∈ D(A) and AS(t)x = S(t)Ax by d+ S S (1.7)–(1.8). Moreover, we have the following right derivative dt (t)x = −A (t)x. To compute the left derivative we write, for h> 0,
h−1(S(t)x − S(t − h)x)= S(t − h)h−1(S(h)x − x).
3 Since h−1(S(h)x − x) → −Ax as h → 0+ and since (t, x) 7→ S(t)x is continuous on [0, T ] × X we obtain
h−1(S(t)x − S(t − h)x) → −S(t)Ax. h→0+ d S S S S We thus obtain dt (t)x + A (t)x = 0. From A (t)x = (t)Ax and the continuity of t 7→ S(t)y for all y ∈ X we conclude that S(t)x ∈ C 0([0, T ],D(A)) and finally using the equation we have S(t)x ∈ C 1([0, T ],X). Uniqueness. Let u ∈ C 0([0, T ],D(A)) ∩ C 1([0, T ],X) be a solution to (1.10) satisfying u(0) = x. Let 0 < s ≤ T . For t ∈ [0,s] we set v(t)= S(s − t)u(t). With the first part we have d d v(t)= S(s − t) u(t)+ S(s − t)Au(t)=0. dt dt We thus find S(s)x = S(s)u(0) = v(0) = v(s)= u(s).
We provide also the following result that shows that integration with respect to time yields a gain of regularity. T S Lemma 3. For x ∈ X and T > 0, we have ∫0 (t)x dt ∈ D(A) and
T S(T )x − x + A ∫ S(t)x dt =0. 0 Proof. For h> 0 we compute
T T −1 −1 Fh = h (IdX −S(h)) ∫ S(t)x dt = h ∫ S(t) − S(t + h)x dt 0 0 h = h−1 ∫ S(t)x − S(t)S(T )x dt. 0
With the continuity of t 7→ S(t)x and t 7→ S(t)S(T )x, the fundamental theorem of calculus yields the result, by (1.7)–(1.8).
We shall use the following version in which a smooth window function is intro- duced.
C ∞ Lemma 4. For x ∈ X and χ ∈ c (0, ∞) we have
∞ ∞ ∞ ∫ χ(t)S(t)x dt ∈ D(A) and A ∫ χ(t)S(t)x dt = ∫ χ′(t)S(t)x dt. 0 0 0 We see that the result of Lemma 3 formally coincides with that of Lemma 4 in the case χ = 1(0,T ).
4 Proof. For h> 0 we compute
∞ ∞ −1 −1 Fh = h (IdX −S(h)) ∫ χ(t)S(t)x dt = h ∫ χ(t) S(t) − S(t + h)x dt. 0 0 Observe that we have ∞ ∞ ∞ ∫ χ(t)S(t + h)x dt = ∫ χ(t − h)S(t)x dt = ∫ χ(t − h)S(t)x dt, 0 h 0 because of the support of χ. We thus obtain
∞ −1 Fh = h ∫ χ(t) − χ(t − h)S(t)x dt. 0 With the continuity of t 7→ S(t)x, the Lebesgue dominated-convergence theorem yields
∞ ′ lim Fh = ∫ χ (t)S(t)x dt. h→0+ 0 Consequently, by (1.7)–(1.8), we obtain the result.
zt Observe that if S(t) is a C0-semigroup and z ∈ C then e S(t) satisfies (1.5). The following proposition is then clear from what precedes.
zt Proposition 5. Let S(t) be a C0-semigroup and z ∈ C. Then e S(t) is also a
C0-semigroup and its generator is A − z IdX .
Note that, because of the uniqueness of the generator of a C0-semigroup [3,
Theorem 1.2.6], conversely, if A generates a C0-semigroup, then A − z IdX is the zt generator of a C0-semigroup, namely e S(t).
1.2 The Hille-Yosida theorem The next natural question is to wonder if an unbounded operator on X is the generator of a C0-semigroup. The Hille-Yosida theorem is central in the semigroup theory, providing a clear answer to this question. We refer to [3, Theorem 1.3.1] for a proof. Theorem 6. Let (A, D(A)) be a linear unbounded operator on a Banach space X.
It generates a C0-semigroup of contraction if and only if: 1. A is closed and D(A) is dense in X.
2. The resolvent set ρ(A) of A contains (−∞, 0) and we have the following esti- mate:
−1 kRλ(A)kL (X) ≤ 1/|λ|, λ< 0, Rλ(A)=(λ IdX −A) .
5 For the notions of closed operators, resolvent set, spectrum, and resolvent oper- ator Rλ(A), we refer to [1, 2, 5].
The previous result is limited to contraction C0-semigroups. The following corol- lary provides a characterization of all generators of C0-semigroups; we refer to [3, Theorem 1.5.3] for a proof.
Corollary 7. Let (A, D(A)) be a linear unbounded operator on a Banach space X. S S ωt It generates a C0-semigroup (t) such that k (t)kL (X) ≤ Me , for some M ≥ 1 and ω ∈ R, if and only if:
1. A is closed and D(A) is dense in X.
2. The resolvent set ρ(A) of A contains (−∞, −ω) and we have the following estimate:
n n N∗ −1 kRλ(A) kL (X) ≤ M/|ω + λ| , λ< −ω, n ∈ , Rλ(A)=(λ IdX −A) .
The Hille-Yosida theorem has the following simple consequence.
Corollary 8. Let (A, D(A)) be the generator of a bounded C0-semigroup S(t), that S is, k (t)kL (X) ≤ M, for t ≥ 0, for some M > 0. Then, its spectrum satisfies sp(A) ⊂{z ∈ C; Re z ≥ 0}.
ibt ibt Proof. Let b ∈ R, the C0-semigroup e S(t) is generated by A − ib IdX . As e S(t) ibtS satisfies ke (t)kL (X) ≤ M, for t ≥ 0, the conclusion follows from Corollary 7 in the case ω = 0.
1.3 The Lumer-Phillips theorem The Lumer-Phillips theorem provides another characterisation of generators of con- traction semigroups. Let X′ be the dual space of X equipped with the strong topology (see [1, 4]). For x ∈ X we set
′ 2 2 F (x)= φ ∈ X ; φ(x)= hφ, xiX′,X = kxkX = kφkX′ , (1.11) which is not empty by the Hahn-Banach theorem.
Definition 9. A linear unbounded operator (A, D(A)) on X is said to be mono- tone (or accretive) if for all x ∈ D(A), x =6 0, there exists φ ∈ F (x) such that
Rehφ, AxiX′,X ≥ 0.
6 Definition 10. A linear unbounded operator (A, D(A)) on X is said to be maximal monotone if it is monotone and if moreover there exists λ0 > 0 such that the range of λ0 IdX +A, Ran(λ0 IdX +A), is equal to X, The Lumer-Philips theorem reads as follows.
Theorem 11. Let (A, D(A)) be a linear unbounded operator. It generates a C0- semigroup of contraction if and only if
1. A has a dense domain.
2. A is maximal monotone.
A proof based on the Hille-Yosida theorem directly follows from Lemmata 13 and 14 given below.
Remark 12. Observe that there is no need to assume that the operator A is closed in the converse part of the Lumer-Philips theorem as in the Hille-Yosida theorem. In fact, as proven below, a maximal monotone operator is closed (see Lemma 14). In the case of a reflexive Banach space, the dense domain assumption may be dropped in the converse part of the Lumer-Philips theorem: a maximal monotone operator has a dense domain; see [3, Theorem 1.4.6] (see also [1, Proposition 7.1] for the Hilbert space case).
The next lemma gives a characterization of monotone operators.
Lemma 13. An unbounded operator (A, D(A)) on X is monotone if and only if
k(λ IdX +A)xkX ≥ λkxkX , x ∈ D(A) and λ> 0. (1.12) Proof. First, we assume that A is monotone. Let λ > 0 and x ∈ D(A). We may then write, for some φ ∈ F (x),
2 λkxkX ≤ λ Rehφ, xiX′,X + Rehφ, AxiX′,X = Rehφ,λx + AxiX′,X
≤kφkX′ kλx + AxkX = kxkX kλx + AxkX , yielding (1.12). Second, we assume that (1.12) holds. Let x ∈ D(A) with x =6 0. For λ> 0 we let
φλ ∈ F (λx + Ax). By (1.12), we have λx + Ax =6 0 and thus φλ =6 0. We normalize it setting ψλ = φλ/kφλkX . We then have kλx + AxkX = hψλ, λx + AxiX′,X . We may thus write, with (1.12),
λkxkX ≤k(λ IdX +A)xkX = hψλ, λx + AxiX′,X
= λhψλ, xiX′,X + hψλ, AxiX′,X .
7 As kψλkX′ = 1 the conclusion is twofold:
hψλ, AxiX′,X ≥ 0, hψλ, xiX′,X ≥kxkX −kAxkX /λ, λ> 0.
As the unit ball of X′ is compact for the weak star topology1 by the Banach-Alaoglu ′ theorem, there exists ψ ∈ X with kψkX′ ≤ 1 and an increasing sequence (λn)n that ∗ diverges to +∞ such that ψλn ⇀ ψ implying
hψ, AxiX′,X ≥ 0, hψ, xiX′,X ≥kxkX .
This yields kψkX′ = 1. We set φ = kxkX ψ and we have φ ∈ F (x) and hφ, AxiX′,X ≥ 0. As x =6 0 is arbitrary in D(A), this yields that A is monotone.
The value of λ0 > 0 in Definition 10 is not of great significance. In fact, we have the following result.
Lemma 14. Let A be a maximal monotone operator on X. Then, A is closed and for all λ > 0 the operator λ IdX +A is bijective from D(A) onto X. Moreover, if −1 λ > 0, its inverse, (λ IdX +A) , is a bounded operator and we have the following −1 −1 estimation k(λ IdX +A) kL (X) ≤ λ .
Proof. Let λ> 0. The injectivity of λ IdX +A follows from Lemma 13.
As A is maximal monotone, there exists λ0 > 0 such that λ0 IdX +A is also −1 surjective. Its inverse (λ0 IdX +A) is thus well defined on X. By Lemma 13 we −1 −1 have k(λ0 IdX +A) kL (X) ≤ λ0 . By the closed-graph theorem (see [1]), the graph −1 of (λ0 IdX +A) is closed in X × X and thus so is the graph of A. ′ ′ We now prove that if λ IdX +A is surjective then so is λ IdX +A for any λ such that 0 <λ′ < 2λ. The proof is based on the Banach contraction fixed point theorem.
By induction, starting with λ = λ0 we then reach the conclusion that λ IdX +A is onto for any λ> 0 and then the boundedness of its inverse follows from Lemma 13. ′ ′ Let λ,λ > 0 be such that λ IdX +A is onto and 0 < λ < 2λ. Let y ∈ X. We wish to find x ∈ X such that λ′x + Ax = y. This reads λx + Ax = y +(λ − λ′)x −1 ′ and thus thus we have x = (λ IdX +A) (y +(λ − λ )x), meaning that we seek a −1 ′ fixed point for the bounded affine map H : x 7→ (λ IdX +A) (y +(λ − λ )x). By −1 the computation above we have k(λ IdX +A) kL (X) ≤ 1/λ, we thus find
′ ′ ′ kH(x) − H(x )kX ≤|1 − λ /λ| kx − x kX .
As 0 < |1 − λ′/λ| < 1, the Banach contraction fixed point theorem applies. 1The weak star topology is often referred to as the σ(E′, E) topology en E′ (see e.g. [4, 1]).
8 2 Differentiable and analytic semigroups
Above, we have considered C0-semigroups. If the continuity assumption of a semi- group S(t) is reinforced, say by assuming uniform continuity instead of strong con- + tinuity: kS(t) − IdX kL (X) → 0 as t → 0 . Then, one can prove that the generator A of S(t) is a bounded operator on X and that one simply has S(t) = exp(−tA) = n Pn≥0(−tA) /n! (see e.g. [3, Section 1.1]). Other regularity assumptions with respect to t can be made, yet not making the semigoup becoming a trivial exponential of a bounded operator.
Definition 15. A C0-semigroup S(t) is called differentiable for t > t0 if for all x ∈ X the map t 7→ S(t)x is differentiable for t > t0. S d S S If x ∈ D(A), then t 7→ (t)x is differentiable for t ≥ 0 and dt (t)x = −A (t)x by Proposition 2. If the semigroup is differentiable for t > t0, this means that this property extends to all x ∈ X if t > t0. The following proposition states properties of differentiable semigroups; see [3, Lemma 2.4.2] for a proof.
Proposition 16. Let S(t) be a differentiable semigroup for t > t0. Then for t > nt0, n ∈ N, S(t) maps X in D(An), and for all x ∈ X, the map t → S(t)x is n d nS nS L S times differentiable, and dt (t)=(−A) (t) ∈ (X). Moreover, t → (t) ∈ n C (((n + 1)t0, +∞), L (X)).
k ℓ Note that if t0 = 0 then S(t) ∈ C ((0, +∞), L (X,D(A ))), for any k,ℓ ∈ N. Like many other semigroup properties, the differentiability of a semigroup can be characterized through a resolvent estimate; see [3, Theorem 2.4.8] for a proof.
S S ωt Theorem 17. Let (t) be a C0-semigroup that satisfies k (t)kL (X) ≤ Me with A as its generator. The semigroup S(t) is differentiable for t > 0 if and only if for every b> 0 there exist a> 0 and C0 > 0 such that
Σb = {z ∈ C; Re z ≤ −a + b log | Im z|} ⊂ ρ(A), and
kz IdX −AkL (X) ≤ C0| Im z| z ∈ Σb, Re z ≥ −ω.
We now consider analytic semigroups. For 0 <θ<π/2, we set
Σθ = {z; | arg(z)| ≤ θ}∪{0}.
Definition 18. A map S : Σθ → L (X) is said to be an analytic semigroup on Σθ if
9 1. z 7→ S(z) is analytic on Σθ \{0} in the topology of L (X).
2. S(0) = IdX and lim S(z)x = x for all x ∈ X; z→0 z∈Σθ\{0}
3. S(z1 + z2)= S(z1)S(z2), for all z1, z2 ∈ Σθ.
The following result yields a characterization of an analytic semigroup; we refer to [3, Theorem 2.5.2] for a proof.
Theorem 19. Let S(t) be a C0-semigroup on X and let A be its generator such that 0 ∈ ρ(A) and kS(t)kL (X) ≤ M, for some M > 0. The following statements are equivalent.
1. There exists 0 <θ<π/2 such that S(t) can be extended as an analytic semi- ′ ′ group on Σθ and is such that kS(z)kL (X) ≤ M for z ∈ Σθ for some M > 0.
2. There exists C > 0 such that
−1 σ + iτ ∈ ρ(A) and k(σ + iτ − A) kL (X) ≤ C/|τ|, σ< 0, τ =06 .
3. There exist 0
−1 z ∈ ρ(A) and k(z − A) kL (X) ≤ C/|z|,
if z =06 and π/2 − a< arg(z) < 3π/2+ a.
4. The semigroup S(t) is differentiable for t> 0 and there exists C > 0 such that
S kA (t)kL (X) ≤ C/t, t> 0.
3 Mild solution of the inhomogeneous Cauchy prob- lem
In what precedes, we have seen that semigroups can be used to solve the abstract homogeneous Cauchy problem in a Banach space X: d u(t)+ Au(t)=0 for t> 0 u(0) = u . dt 0
In fact, if A generates a semigroup on X and if u0 ∈ D(A) then the unique solution 0 in C ([0, +∞),D(A)) is given by u(t)= S(t)u0 by proposition 2. Note that u(t) ∈ C 1([0, +∞),X) and moreover the equation is even satisfied for t ≥ 0. In the case of a selfadjoint generator on a Hilbert space this can be extended to x ∈ X by
10 Corollary 27. Then, the unique solution is in C 0([0, +∞),X) ∩ C k((0, ∞),D(Aℓ)), k,ℓ ∈ N and note that the equation is only satisfied for t> 0. We are now interested into solving a nonhomogeneous abstract Cauchy problem of the form d u(t)+ Au(t)= f(t) ∈ X for t> 0 and u(0) = u . (3.1) dt 0
Here, A is assumed to generate a C0-semigroup on X. A classical solution is a function
u ∈ C ([0, ∞),X) ∩ C 1((0, ∞); X), such that u(t) ∈ D(A) for t > 0 and that satisfies (3.1), that is, both the equation for t > 0 and the initial condition. If u(t) is such a solution, we choose T > 0 and we set w(t)= S(T − t)u(t). We have, for t> 0,
d d w(t)= S(T − t) u(t)+ Au(t) = S(T − t)f(t). dt dt
1 If f|(0,T ) ∈ L (0, T ; X) we find
T T u(T )= w(T )= w(0) + ∫ S(T − t)f(t) dt = S(T )u0 + ∫ S(T − t)f(t) dt. 0 0 This is precisely the Duhamel formula in an abstract setting. We thus have the following uniqueness result.
1 Proposition 20. If f ∈ Lloc(0, ∞; X) and if u is a classical solution it is given by
t u(t)= S(t)u0 + ∫ S(t − s)f(s) ds, t ≥ 0 0 We are thus naturally led to the following definition.
1 Definition 21. Let f ∈ Lloc(0, ∞; X) and u0 ∈ X. The function
t u(t)= S(t)u0 + ∫ S(t − s)f(s) ds, t ≥ 0 0 is such that u ∈ C ([0, ∞); X) and is called the mild solution of the inhomogeneous abstract Cauchy problem (3.1).
1 C If f ∈ Lloc(0, ∞; D(A)) ∩ ((0, ∞); X) and u0 ∈ D(A) then the mild solution is a classical solution as can be readily checked.
11 4 The case of a Hilbert space.
In the case the space X is a Hilbert space. It can be identified with its dual space X′ by the Riesz theorem. Out of habit, we shall denote by H the Hilbert rather than by X. Then if x ∈ H we have, for the set defined in (1.11), F (x)= {x}. We then have the following definition in agreement with Definition 9.
Definition 22. A linear unbounded operator (A, D(A)) on H is said to be monotone
(or accretive) if for all x ∈ D(A) one has Re(Ax, x)H ≥ 0.
In the Hilbert case the Lumer-Phillips theorem reads, in agreement with Re- mark 12.
Theorem 23. Let A be a linear unbounded operator. It generates a C0-semigroup of contraction if and only if it is maximal monotone.
Proposition 24. Let (A, D(A)), be an unbounded operator on H, that generates a ∗ ∗ C0-semigroup S(t). Then, the operator (A ,D(A )) generates a semigroup Σ(t) and Σ(t)= S(t)∗.
Recall that if (A, D(A)) generates a semigroup then it is closed and densely defined. This allows one to properly define its adjoint operator (see [1]).
Proof. The operator (A∗,D(A∗)) is closed and densely defined [1]. By Corollary 7, consequence of the Hille-Yosida theorem (Theorem 6), there exists M ≥ 1 and ω ∈ R −1 such that, for λ < −ω, the operator Rλ(A)=(λ IdX −A) exists and is bounded on H and moreover n n kRλ(A) kL (H) ≤ M/|ω + λ| . ∗ n ∗ −1 n By Theorem III.5.30 in [2], the operator Rλ(A ) = (λ IdX −A ) exists and we have ∗ n n n kRλ(A ) kL (H) = kRλ(A) kL (H) ≤ M/|ω + λ| . ∗ ∗ Hence, by Corollary 7, the operator (A ,D(A )) generates a C0-semigroup. We denote by Σ(t) the semigroup generated by A∗. Let t> 0 and let x ∈ D(A) and x′ ∈ D(A∗). For s ∈ [0, t] we introduce the function
′ h(s)=(S(t − s)x, Σ(s)x )H .
′ ′ ∗ ′ We compute h (s)= −(AS(t − s)x, Σ(s)x )H +(S(t − s)x, A Σ(s)x )H = 0, as S(t − s)x ∈ D(A) and Σ(s)x′ ∈ D(A∗). We thus have h(t)= h(0), that is,
′ ′ ′ ∗ (x, Σ(t)x )H =(S(t)x, x )H , x ∈ D(A), x ∈ D(A ).
12 Since the domains of the two operators are dense in H we conclude that we have
′ ′ ′ (x, Σ(t)x )H =(S(t)x, x )H , x, x ∈ H, which yields the conclusion.
In the case of a self adjoint operator (A, D(A)), that is, D(A∗) = D(A) and A = A∗, we have the following consequence.
Corollary 25. Let (A, D(A)) be a selfadjoint operator on H. If it generates a ∗ C0-semigroup S(t) on H then S(t) = S(t).
If the operator (A, D(A)) is selfadjoint, the conclusions of the Lumer-Philips theorem are even stronger; we refer to [1, Theorem 7.7] for a proof.
Theorem 26. Let A be a linear unbounded selfadjoint operator on a Hilbert space H. It generates an analytic semigroup S(t) of contraction if and only if it is maximal monotone.
From Proposition 16 and Theorem 19 we have the following properties.
Corollary 27. Let S(t) be the analytic semigroup of contraction generated by A unbounded and selfadjoint on H a Hilbert space. Then:
1. We have t 7→ S(t) ∈ C k((0, ∞); L (H,D(Aℓ))), for any k,ℓ ∈ N. S 2. There exists C > 0 such that kA (t)kL (H) ≤ C/t. 3. For all x ∈ H, there exists a unique solution in
C 0([0, ∞); H) ∩ C 1((0, ∞); H) ∩ C 0((0, ∞); D(A))
d S to dt u(t)+ Au(t)=0, for t> 0. It is given by u(t)= (t)x. 4. Moreover t 7→ S(t)x ∈ C 0([0, ∞); H) ∩ C k((0, ∞),D(Aℓ)), k,ℓ ∈ N.
The important property, as compared to the result of Theorems 11 and 23, lays S d in the fact that here the function u(t)= (t)x solves the equation dt u(t)+Au(t)=0 for t> 0 not only for x ∈ D(A) but also for x ∈ H, as the semigroup is differentiable for t> 0. The following result is often handy to assess that an operator is selfadjoint in the framework of semigroup theory. We refer to [1, Proposition 7.6].
Proposition 28. Let A be an unbounded operator on H a Hilbert space that is symmetric, in the sense that (Ax, y)H =(x, Ay)H for x, y ∈ D(A). If A is maximal monotone then A is selfadjoint, that is, D(A∗)= D(A) and A = A∗.
13 If A is a symmetric unbounded operator with dense domain then one has D(A) ⊂ D(A∗). Hence, A∗ also has a dense domain, but it may happen that D(A) =6 D(A∗). 2 R 2 R C ∞ R A simple example is the operator A : L ( ) → L ( ) with domain D(A)= c ( ), defined by Au = u′′. We see that A is symmetric and yet we have D(A∗)= H2(R).
References
[1] Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equa- tions. Universitext. Springer, New York, 2011.
[2] Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, 1980.
[3] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differen- tial Equations. Springer-Verlag, New York, 1983.
[4] Treves, F. Topological Vector Spaces, Distributions and Kernels. Academic Press, New York, 1967.
[5] Yosida, K. Functional Analysis, sixth ed. Springer-Verlag, Berlin, 1980.
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