]> 2.3 Semigroups and Riesz bases

2.3 Semigroups and Riesz bases

Definition 2.3.1. Let X be a Banach space. A family {S(t)}t0 of operators in L(X) is said to be a C0semigroup on X if the following conditions are satisfied:

  1. S(0) = I, S(t + s) = S(t)S(s) for all t,s 0,
  2. lim t0+S(t)x = x for all x X.

If, additionally,

  1. For each fixed x X, the map tS(t)x is real analytic function on (0,),

then {S(t)}t0 is an analytic semigroup on X. The linear operator

𝒜x = lim t0+S(t)x x t ,D(𝒜) = x X : lim t0+S(t)x x t

is called the infinitesimal generator of a linear C0–semigroup {S(t)}t0 on X.

The following two theorems characterize the generators of analytic and C0–semigroups.

Theorem 2.3.1 (Hille). [48], [69]. A linear operator 𝒜 : (D(𝒜) X)X acting on a Banach space X is the infinitesimal generator of a linear analytic semigroup on X iff 𝒜 is closed and densely defined and there exist ω , ϕ (π 2 ,π), and M 1 such that the sector Sω,ϕ = {λ : ϕ arg(λ ω) ,λω} is contained in the resolvent set ρ(𝒜) and the resolvent of 𝒜 satisfies the estimate

(λI 𝒜)1 L(X) M λ ωλ Sω,ϕ (2.10)

Theorem 2.3.2 (Hille–Phillips–Yosida). A linear operator 𝒜 : (D(𝒜) X)X acting on a Banach space X is the infinitesimal generator of a linear C0–semigroup on X iff 𝒜 is closed and densely defined, and there exist ω and M 1 such that the halfplane Πω = {λ : Re λ > ω} is contained in the resolvent set ρ(𝒜) and the resolvent of 𝒜 satisfies the estimate

(λI 𝒜)n L(X) M (Re λ ω)nλ Πωn (2.11)

If the above conditions are satisfied then for any x0 D(𝒜), T > 0 there exists a unique classical solution x C1([0,T],X) C([0,T],D 𝒜), x(t) = S(t)x0 of problem (2.3), where D𝒜 denotes the Banach space D(𝒜) equipped with the norm x𝒜 = xX + 𝒜xX. The function x(t) = S(t)x0, where x0 X, is called a weak solution of (2.3).

In what follows H will denote a Hilbert space with a scalar product ,H.

Definition 2.3.2. The system {φi}iJ H is called a Riesz basis of H if there exist T L(H) with T1 L(H) and an orthonormal basis {ei}iJ such that φi = Tei.

For other characterizations of Riesz bases, see [3], [28], [90] and [67].

Definition 2.3.3. Assume that H1 and H2 are Hilbert spaces with scalar products ,H 1, ,H 2, respectively. Let 𝒯 : (D(𝒯 ) H1)H2 be a closed, densely defined linear operator.

D(𝒯) := {v H 2 : (!)hv H1 : 𝒯 x,vH2 = x,hvH1x D(𝒯 )}

is the domain of the adjoint operator 𝒯 : (D(𝒯) H 2)H1 with respect to 𝒯, defined as 𝒯v := h v, v D(𝒯).

Recall that the operator 𝒜 is called discrete if its resolvent (λI 𝒜)1 is a compact operator for some, or equivalently, for all λ ρ(𝒜) [69, p. 49]. The name discrete is motivated by the fact that operators with a compact resolvent have the spectrum consisting entirely of eigenvalues, however it is not ensured that the set of eigenvalues (point spectrum) is nonvoid. There exist also discrete operators with a finite number of eigenvalues. The special situation holds for discrete normal operators. The operator 𝒜 : (D(𝒜) H)H is called normal if D(𝒜) = D(𝒜) and 𝒜xH = 𝒜xH for all x D(𝒜).

Theorem 2.3.3 (Spectral theorem for discrete normal operators). An operator with a compact resolvent is normal iff it possesses a system of eigenvectors forming an orthonormal basis of H.

Proof. For the proof of necessity see [53, pp. 260 - 263 and pp. 276 - 277], [76, pp. 250 - 255] or, less explicitly [86, Theorem 7.2, p. 167]. Sufficiency can be deduced from [86, Theorem 7.2, p. 167]. □

Definition 2.3.4. 𝒜 : (D(𝒜) H)H is similar to a normal operator if there exists a similarity transformation T L(H) with T1 L(H) such that 𝒩 = T1𝒜T is normal.

The following conclusion can be drawn from Theorem 2.3.3 and Definition 2.3.4.

Corollary 2.3.1. An operator 𝒜 : (D(𝒜) H)H with a compact resolvent is similar to a normal operator iff 𝒜 possesses a system of eigenvectors forming a Riesz basis of H. To be more precise, {ei}iJ is an orthonormal basis formed by the system of eigenvectors of a discrete normal operator 𝒩, corresponding to its eigenvalues {λi}iJ iff the system {φi}iJ, φi = Tei, where T stands for the similarity transformation, is the Riesz basis of eigenvectors of 𝒜, corresponding to its eigenvalues {λi}iJ, or equivalently, the system {ψi}iJ, ψi = (T)1e i is the Riesz basis of eigenvectors of 𝒜, corresponding to its eigenvalues {λi¯}iJ.

Definition 2.3.5. The semigroup {S(t)}t0 is asymptotically stable (AS) if

lim tS(t)x = 0x H (2.12)

and weakly asymptotically stable, if

lim tS(t)x1,x2H = 0x1,x2 H .

It is exponentially stable (EXS), if there exist M 1, α > 0 such that

S(t) L(H) Meαtt 0 .

Clearly EXS implies AS, and for compact semigroups both concepts are equivalent. In turn, AS implies weak asymptotic stability, and for semigroups having discrete infinitesimal generator both concepts are equivalent [79], [4, p. 25]. The semigroup {S(t)}t0, generated by 𝒜, is EXS if the adjoint semigroup {S(t)} t0, generated by 𝒜, is EXS. AS of {S(t)}t0 implies weak asymptotic stability of {S(t)} t0.

Exercise 2.3.1. Consider the Hilbert space H = L2(0,) with the standard scalar product

x1,x2H =0x 1(θ)x2(θ)dθ .

A family of operators {S(t)}t0,

(S(t)g)(θ) := g(θ t), θ > t 0, t > θ > 0 (2.13)

defines a C0–semigroup of right–shifts on H. The most difficult fact to be verified is the axiom (ii) of the semigroup. By the Parseval theorem

S(h)g gL2(0,)2 = 1 2πS(h)ĝ (jω) ĝ(jω) 2dω .

But

S(h)ĝ (s) =hesθg(θ h)dθ =0es(ξ+h)g(ξ)dξ = eshĝ(s)

and hence, by the Lebesgue dominated convergence theorem, we have

lim h0+ S(h)g gL2(0,)2 = lim h0+ 1 2πejωh 1 2 ĝ(jω) 2dω = 0 .

The infinitesimal generator of this semigroup is

𝒜x = x D(𝒜) ={x L2(0,) : x L2(0,),x(0) = 0} := W 01,2[0,) .

Indeed, if x D(𝒜) then x is continuous on [0,) and lim θx(θ) = 0.

2x,𝒜xL2(0,) = 20x(θ)x(θ)dθ = 2 lim t0tx(θ)x(θ)dθ =

= lim tx2(t) lim tx(t) .

The limit cannot be nonzero as otherwise the integral 0x2(t)dt would be divergent. Now, applying the Parseval theorem and the Lebesgue dominated convergence theorem we obtain

lim h0+ 1 h[S(h)g g] 𝒜gL2(0,)2 =

= lim h0+ 1 2π1 h ejωh 1 + jω2 ĝ(jω) 2dω = 0g D(𝒜) .

For comparison observe that 𝒜 is densely defined and closed, and satisfies (2.11). Indeed, x(θ) =0θeλ(θτ)g(τ)dτ solves the system

x(θ) = λx(θ) + g(θ) x(0) =0

uniquely and thus we have

(λI 𝒜)1g L2(0,) = eλ() g L2(0,)

eλ() L1(0,) gL2(0,) 1 Re λ gL2(0,)

provided that g H, Reλ > 0. By iterations one gets (2.11) with ω = 0. Notice that (sI 𝒜)1g is the Laplace transform of the function tS(t)g as we have

((sI 𝒜)1g)(θ) =0θes(θξ)g(ξ)dξ =0θestg(θ t)dt =

=0est g(θ t),0 < t < θ 0, t > θ dt =0est(S(t)g)(θ)dt .

This confirms (2.13).

By Definition 2.3.3 the adjoint operator with respect to 𝒜 is

𝒜v =v D(𝒜) ={v L2(0,) : v L2(0,)} := W1,2[0,) .

Since D(𝒜)D(𝒜) the operator 𝒜 is not normal. The operator 𝒜 generates the adjoint semigroup {S(t)} t0 with respect to {S(t)}t0. By Definition 2.3.3,

S(t)x,vL2(0,) =0x(θ t), θ > t 0, t > θ > 0 v(θ)dθ = =tx(θ t)v(θ)dθ =0x(ξ)v(ξ + t)dξ = x,S(t)v L2(0,) (2.14)

for all x,v H. This means that the adjoint semigroup

(S(t)v)(θ) = v(t + θ),θ,t 0 (2.15)

is the semigroup of left–shifts. Prove by direct solving of the equation (λI 𝒜)v = g

((λI 𝒜)1g)(θ) =θeλ(θξ)g(ξ)dξ

and verify the result applying the Laplace transform to (2.15). Here g H and Reλ > 0.

The resolvent of 𝒜 is not compact. Indeed, if the resolvent of 𝒜 were compact then the number of eigenvectors of 𝒜 would be equal to the number of eigenvectors of 𝒜. However, this is not the case as 𝒜 has no eigenvectors while 𝒜 has them. By the same arguments 𝒜 is not similar to a normal operator.

The semigroup {S(t)} t0 is AS as we have

lim tS(t)g L2(0,)2 = lim t0g2(t + θ)dθ = lim ttg2(θ)dθ = 0g H .

This semigroup does not decay exponentially (consider g(θ) = 1 θ + 1 as a counterexample). The semigroup {S(t)}t0 is not AS. Indeed, by (2.13) we have

S(t)gL2(0,)2 =tg2(θ t)dθ =0g2(ξ)dξ = g L2(0,)2 ,

so any nonzero trajectory does not tend to 0 as t . However, this semigroup tends weakly to 0 which follows from (2.14) and AS of the semigroup {S(t)} t0.

The next theorem is a particular case of known results [13, Theorem 2], [72, pp. 25 - 28], however, the proof given here is elementary.

Theorem 2.3.4. Let H be a Hilbert space with an inner product ,H and let 𝒜 : (D(𝒜) H)H be a linear discrete operator which is similar to a normal one.

A necessary and sufficient conditon for 𝒜 to be the infinitesimal generator of a linear analytic semigroup on H is the existence of ω and ϕ0 (π 2 ,π) such that ρ(𝒜) contains the sector Sω,ϕ0 = {λ : ϕ0 arg(λ ω) ,λω}.

A necessary and sufficient condition for 𝒜 to be the infinitesimal generator of a linear C0–semigroup on H is the existence of ω such that ρ(𝒜) contains the halfplane Πω = {λ : Re λ > ω}. Moreover, this semigroup is EXS iff

sup λiσ(𝒜) Re λi < 0 ,

where σ(𝒜) denotes the spectrum of 𝒜.

Proof. The necessary conditions easily follow from Theorems 2.3.1, 2.3.2, and Definition 2.3.5. We now prove the sufficient condition for the generation of analytic semigroup. 𝒜 is closed because (λI 𝒜)1 L(H) [86, p. 89]. Also, span {φi}iJ D(𝒜), and {φi}iJ is complete, which implies the density of D(𝒜) in H. Now consider the sector Sω,ϕ, defined as Sω,ϕ0 but with the angle of obtuseness ϕ < ϕ0, with ϕ (π 2 ,π). Then Sω,ϕ ω,ϕ0 ρ(𝒜). So, by Hille’s theorem, it suffices to show the existence of M 1 for which (2.10) holds. Let us inspect Figure 2.2.


PICT
Figure 2.2: The auxiliary diagram for the proof of analytic–semigroup generation

Since 0 ϕ0 π 2 π 2 < ϕ we have Sω,ϕ0π 2 Sω,ϕ. For λ belonging to the cone Sω,ϕ0π 2 , the inequalities

λ ω λ λi ,i J

hold. If λ belongs to the remaining part of Sω,ϕ, then

λ ω sin(ϕ0 ϕ) λ λi i J .

Theses estimates together yield

1 λ λi 2 csc 2(ϕ 0 ϕ) λ ω2 i Jλ Sω,ϕ .

By virtue of Corollary 2.3.1,

(λI 𝒜)1fH2 = T(λI 𝒩)1T1fH2 TL(H)2 T1 L(H)2 (λI 𝒩)1fH2 = = TL(H)2 T1 L(H)2 iJ (λI 𝒩)1f,e iH 2 = = TL(H)2 T1 L(H)2 iJ f, (λ¯I 𝒩)1e iH 2 = = TL(H)2 T1 L(H)2 iJ 1 λ λi 2 f,eiH 2 TL(H)2 T1 L(H)2csc 2(ϕ 0 ϕ) λ ω2 iJ f,eiH 2 = = TL(H)2 T1 L(H)2csc 2(ϕ 0 ϕ) λ ω2 fH2f Hλ S ω,ϕ (2.16)

Hence (2.10) holds with M = TL(H) T1 L(H) csc(ϕ0 ϕ) 1.

To prove the sufficient condition for generation of a semigroup we consider the sequence of identities

(λI 𝒜)n = (λI T𝒩T1)n = (λI T𝒩T1)n+1(λI T𝒩T1)1 =

= (λI T𝒩T1)n+1(T(λI 𝒩)T1)1 = (λI T𝒩T1)n+1T(λI 𝒩)1T1 =

= (λI T𝒩T1)n+2(λI T𝒩T1)1T(λI 𝒩)1T1 =

= (λI T𝒩T1)n+2T(λI 𝒩)1T1T(λI 𝒩)1T1 =

= (λI T𝒩T1)n+2T(λI 𝒩)2T1 = = T(λI 𝒩)nT1 .

Hence

(λI 𝒜)n L(H) = T(λI 𝒩)nT1 L(H) TL(H) T1 L(H) (λI 𝒩)n L(H) (2.17)

Now,

(λI 𝒩)1xH2 = iJ (λI 𝒩)1x,e iH 2 = iJ x, (λ¯I 𝒩)1e iH 2 = = iJ 1 λ λi 2 x,eiH 2x H .

Let us inspect Figure 2.3.


PICT
Figure 2.3: The auxiliary diagram for the proof of semigroup generation

If λ Πω then

λ λi Re λ ω > 0,ω := sup iJ Re λi

and therefore

(λI 𝒩)1x H 1 Re λ ω xHx H .

By iterations we get

(λI 𝒩)nx H 1 (Re λ ω)n xHx H (2.18)

Finally, (2.11) with M = TL(H) T1 L(H) 1 follows from (2.17) and (2.18). □ □