D(T) = (
z∈X
X
k∈N
(1 +|λk|2)|φ˜k, z
X|2<∞ )
, T z=X
k∈N
λk
φ˜k, z
Xφk, z∈ D(T).
If T : D(T) ⊂X → X is self-adjoint and diagonalizable, then there exists in X an orthonormal basis (ϕk)k∈I of eigenvectors ofT (here,I ⊆Z). Denoting the eigenvalue corresponding toφk byλk, we have forλk ∈R,
D(T) = (
z∈X
X
k∈I
(1 +|λk|2)| hφk, ziX|2<∞ )
, T z=X
k∈I
λkhφk, ziXφk.
The following result gives a sufficient condition for a self-adjoint operator to be diag-onalizable.
Proposition 1.5.1. Let X be an infinite-dimensional Hilbert space and letT :D(T)⊂ X →X be a self-adjoint operator with compact resolvents. Then T is diagonalizable with an orthonormal basis(φk)k∈I of eigenvectors, whereI ⊆Z, and the corresponding family of real eigenvalues(λk)k∈I satisfieslimk→∞|λk|=∞.
Proof. This is [115, Proposition 3.2.12].
1.6 Strongly continuous semigroups
In this section we will follow the lines of [115]. However, the literature regarding strongly continuous semigroups is vast, and one can find appropriate references in [30, 40, 53, 94, 127].
Even though the construction can be done in Banach spaces, we will consider only the Hilbert space scenario, so thatX will be an infinite-dimensional Hilbert space. A familyT= (Tt)t≥0of operators inL(X) is astrongly continuous semigroupor in short C0-semigrouponX if
(i) T0=I;
(ii) Tt+s=TtTsfor every t, s≥0;
(iii) limt→0,t>0Ttz=z for allz∈X.
Thegrowth bound of a strongly continuous semigroup Tis the number ω0(T) defined by
ω0(T) = inf
t∈(0,∞)
1
t logkTtk.
Clearlyω0(T)∈[−∞,∞). The name is justified by the fact that (i) ω0(T) = limt→∞t−1logkTtk;
(ii) for anyω > ω0(T) there exists anMω∈[1,∞) such that kTtk ≤Mωeωt, ∀t∈[0,∞);
(iii) the functionϕ:R≥0×X →X defined by ϕ(t, z) =Ttzis continuous;
as it can be checked in [115, Proposition 2.1.2]. We call aC0-semigroup (Tt)t≥0 expo-nentially stableifω0(T)<0.
The linear operatorA:D(A)⊂X →X defined by D(A) =
z∈X
t→0,t>0lim
Ttz−z t exists
, Az= lim
t→0,t>0
Ttz−z
t , ∀z∈ D(A),
is called the infinitesimal generator or simply generator of the semigroup T. It is well-known thatD(A) is dense inX —see [115, Corollary 2.1.8].
We begin now to hatch what we have presented about closed operators with the semig-roup theory. For a strongly continuous semigsemig-roupT on X with generator A and for every s∈Cwith Res > ω0(T) we have s∈ρ(A), and henceAis closed. Moreover
(sI−A)−1z= Z ∞
0
e−stTtzdt , ∀z∈X
andD(A∞) is dense inX—check [115, Proposition 2.3.1 and Proposition 2.3.6]. In fact, the resolvent of the semigroup generator is the Laplace transform of the semigroup.
Further, for z0 ∈ D(A), the function z : [0,∞) → D(A) defined by z(t) := Ttz0 is continuous if we consider in D(A) the graph norm andC1([0,∞);X). Actually, z is the unique solution of theabstract Cauchy problem (ACP)
˙
z=Az, z(0) =z0.
From [115, Proposition 2.6.5] it follows that a diagonalizable operatorAis the generator of a C0-semigroupTonX if, and only if,
sup
k∈NReλk <∞.
1.6. STRONGLY CONTINUOUS SEMIGROUPS 25 If this is the case, then
sup
k∈NReλk=ω0(T) and for everyt≥0,
Ttz=X
k∈N
eλktφ˜k, z
Xφk, ∀z∈X.
Such a semigroup is calleddiagonalizable.
By having a closer look at a diagonal semigroup, it is clear that if eigenvalues ofAare, for instance, purely imaginary, one could extend the semigroup fort <0, so that we end up having a group. In the following we define such a concept.
LetTbe aC0-semigroup onX. Tis calledleft-invertible(respectively,right-invertible) if for somet >0,Ttis left-invertible (respectively, right-invertible). The semigroup is calledinvertible if it is both left-invertible and right-invertible.
A family T = (Tt)t∈R of operators in L(X) is a strongly continuous group in short C0-grouponX if
(i) T0=I;
(ii) Tt+s=TtTsfor every t, s∈R; (iii) limt→0Ttz=z for allz∈X.
The generator of such a group is defined in the same way as for semigroups. Note that given aC0-semigroupT, if for someτ >0 the operatorTτ is invertible, then Tt
is invertible for all t > 0 and T can be extended to a group by setting T−t = T−1t
—see [115, Proposition 2.7.4]. Moreover, from [115, Proposition 2.7.8], ifAgenerates a C0-semigroupT and −A a C0-semigroup S, then we can extend the family of T to all ofRby puttingT−t=StandTis aC0-group.
From [115, Proposition 2.8.5] we have that for aC0-semigroupTthe family of operators T∗ = (T∗t)t≥0 is also aC0-semigroup and its generator isA∗. It is an immediate con-sequence that ifA:D(A)⊂X→X is a diagonalizable operator, (φk) is a Riesz basis consisting of eigenvectors ofA, ( ˜φk) the biorthogonal sequence to (φk) and we denote the eigenvalue corresponding to the eigenvectorφk byλk. ThenA∗ is diagonalizable operator with eigenvectors ˜φk and eigenvaluesλk, see [115, Proposition 2.8.6].
Assuming thatAgenerates aC0-semigroupTonX, consider the spacesX1andX−1. Then the restriction of Tt to X1 is the image of Tt ∈ L(X) through the unitary operator (βI−A)−1∈L(X, X1). Therefore, these operators form aC0-semigroup on X1, whose generator is the restriction ofAtoD(A2). The operator ˜Tt∈L(X−1) is the image ofTt∈L(X) though the unitary operator (βI−A)˜ ∈L(X, X−1). Therefore, these operators form aC0-semigroup on ˜T = (˜Tt)t≥0 on X−1, whose generator is ˜A, see [115, Proposition 2.10.4]. We will often refer to this extensions as ((T|−1)t)t≥0.
Before we conclude this section, we define an important class of semigroups. A C0 -semigroupTonXis calledcontractive,semigroup of contractionsorcontraction semig-roup if for allt > 0 it holds kTtk ≤1. Further, these semigroups have a well-known characterisation, namely, for any A : D(A) ⊂ X → X the following statements are equivalent:
(i) A is the generator of a contraction semigroup onX. (ii) A is m-dissipative.
This is in fact [115, Theorem 3.8.4]. The particular case in which the norm of the semigroup is exactly one, goes via the following definition. An operator U ∈L(X) is called unitary ifU U∗=U∗U =I. A strongly continuous semigroup TonX is called unitary if Tt is unitary for every t > 0. It is clear that a unitary semigroup can be extended to a group, which is then called aunitary group. From [115, Theorem 3.8.6], for anyA:D(A)⊂X→X the following statements are equivalent:
(i) A is the generator of a unitary group onX. (ii) A is skew-adjoint.
1.6.1 Analytic semigroups and fractional powers
Here we briefly present the concept of analyticC0-semigroup, which plays a role when solving PDEs of parabolic type, whose solutions enjoy of some extra smoothness prop-erties. For this part we refer to [110, Section 3.10] and present only the case whereX is a Hilbert space, even though it can be done in the case of a Banach space.
Let 0< δ ≤π/2, and let ∆δ be the open sector
∆δ:={t∈C|t6= 0, |argt|< δ}.
The family of operators Tt ∈L(X), t ∈ ∆δ, is ananalytic C0-semigroup (with uni-formly bounded growth boundω) in ∆δ if the following conditions hold:
(i) t7→Ttis analytic in ∆δ;
(ii) T0=I andTsTt=Ts+tfor alls, t∈∆δ;
(iii) there exist constantsM ≥1 andω∈Rsuch that kTtk ≤Meωt, t∈∆δ; (iv) for allx∈X, limt→0
t∈∆δ
Ttx=x.
1.6. STRONGLY CONTINUOUS SEMIGROUPS 27 This leads us to the concept ofsectorial operator. For eachγ ∈Rand π/2 < θ < π, let Σθ,γ be the open sector
Σθ,γ ={λ∈C|λ6=γ,|arg(λ−γ)|< θ}.
A closed, densely defined linear operatorA:D(A)⊂X →X issectorialon Σθ,γ if the resolvent set ofAcontains Σθ,γ and if
k(λI−A)−1k ≤ C
|λ−γ| λ∈Σθ,γ,
for someC≥1. The operatorAissectorial if it is sectorial on some sector Σθ,γ. LetA: D(A)⊂X →X be a closed, linear operator and γ ∈R. Then the following are equivalent:
(i) A is the generator of an analytic semigroup (Tt)t≥0 with uniformly bounded growth boundγ on a sector ∆δ,δ >0;
(ii) everyλ∈Cγ belongs to the resolvent set ofAand there exists a constantCsuch that
k(λI−A)−1k ≤ C
|λ−γ| Reλ > γ;
(iii) Ais sectorial on some sector Σθ,γ withπ/2< θ < π;
(iv) Ais the generator of a semigroup (Tt)t≥0 which is differentiable on (0,∞), and there exist non-negative constantsM0, M1 such that
kTtk ≤M0eγt, k(γI−A)Ttk ≤M1t−1eγt, t >0, see [110, Theorem 3.10.6].
This last part of the section is devoted to introduce an recall some basic facts about the fractional powers Xα, for α∈ R induced by the semigroup generator A and the space X. This construction can be done in the general Banach setting, but we will consider only the Hilbert space scenario. For a more detailed discussion on the topic we refer to [110, Section 3.9].
LetA be the generator of aC0-semigroupT:= (Tt)t≥0 on the Hilbert space X with growth boundω0∈R. For eachγ∈Cω0 andα≥0 we define (γI−A)αas follows
(γI−A)0=I, (γI−A)−αx= 1
Γ(α) Z ∞
0
tα−1e−γtTtxdt , α >0, x∈X,
where Γ is the Euler Gamma function. These operators are bounded and linear onX, injective andα→ (γI−A)−α defines a semigroup, see [110, Lemma 3.9.5]. Now we
define (γI−A)αforα >0 to be the inverse of (γI−A)−α, with domainD((γI−A)α) = R((γI−A)−α).
Having the powers of γI−A at our disposal, we can construct a scale of spacesXα, forα∈R, as in the same fashion we constructed the spacesXn, n∈Z. Forα >0, we set
Xα:=R((γI−A)−α) = (γI−A)−αX.
with norm
kxkXα :=k(γI−A)αxkX.
Forα <0, we let Xαbe the completion of X with the weaker norm kxkX−α :=k(γI−A)−αxkX, α >0.
Different choices of γ∈Cω0 yield identical spaces with equivalent norms. The spaces Xα are interpreted asinterpolation spacesofXn forn∈Z, see [84, Chapters 1 & 2].
We conclude this section with an auxiliary result that will be used later on. This delivers an estimate for the norm ofTt inL(X, Xα) when the semigroup is analytic.
Lemma 1.6.1. Let A be the generator of an exponentially stable analytic semigroup (Tt)t≥0. LetXαbe the interpolation spaces defined as above associated to A. Then for all k∈Nandα∈[0,1)there exist constants M :=M(k, α)andω >0such that
kTtkL(X,Xk+α)≤M(1 +t−k−α)e−ωt, t >0.
Hence, there existsK:=K(k, α) such that sup
t∈[0,∞)
tk+αkTtkL(X,Xk+α)≤K.
Proof. For the cases in which α = 0 and k ∈ N, this is [110, Corollary 3.10.8]. For k= 0 andα∈(0,1), this follows from [110, Lemma 3.10.9] and using the exponential stability of (Tt)t≥0. Fork >1 andα∈(0,1) the result follows by induction using the former an interpolating betweenk, k+ 1 with [110, Lemma 3.9.8].
1.6.2 The abstract Cauchy problem
Throughout this section T is a C0-semigroup on X with generator A and growth boundω0(T). As we have already seen, the semigroup fully characterizes the solutions to the abstract Cauchy problem. This part of the section is dedicated to present a nonhomogeneous abstract Cauchy problem, in order to motivate the theory of abstract linear systems, where one also has inputs and outputs.
Consider the differential equation
˙
z(t) =Az(t) +f(t),
1.7. WELL-POSED LINEAR SYSTEMS 29