6.1 LetE = p with 1 ≤ p ≤ ∞(see Section 11.3). Let(λn)be a bounded sequence inRand consider the operatorT ∈L(E)defined by
T x=(λ1x1, λ2x2, . . . , λnxn, . . . ), where
6.4 Exercises for Chapter 6 171 x =(x1, x2, . . . , xn, . . . ).
Prove thatT is a compact operator fromEintoEiffλn→0.
6.2 LetEandF be two Banach spaces, and letT ∈L(E, F ).
1. Assume thatEis reflexive. Prove thatT (BE)is closed (strongly).
2. Assume thatEis reflexive and thatT ∈K(E, F ). Prove thatT (BE)is compact.
3. LetE =F =C([0,1])andT u(t )=t
0u(s)ds. Check thatT ∈ K(E). Prove thatT (BE)is not closed.
6.3 LetEandFbe two Banach spaces, and letT ∈K(E, F ). Assume dimE= ∞.
Prove that there exists a sequence(un)inEsuch thatunE =1 andT unF →0.
[Hint:Argue by contradiction.]
6.4 Let 1 ≤ p < ∞. Check thatp ⊂ c0 with continuous injection (for the definition ofpandc0, see Section 11.3).
Is this injection compact?
[Hint: Use the canonical basis(en)ofp.]
6.5 Let(λn)be a sequence of positive numbers such that limn→∞λn= +∞. Let V be the space of sequences(un)n≥1such that
∞ n=1
λn|un|2<∞.
The spaceV is equipped with the scalar product ((u, v))=∞
n=1
λnunvn.
Prove thatV is a Hilbert space and thatV ⊂2with compact injection.
6.6 Let 1 ≤ q ≤ p ≤ ∞. Prove that the canonical injection fromLp(0,1)into Lq(0,1)is continuous butnotcompact.
[Hint: Use Rademacher’s functions; see Exercise 4.18.]
6.7 LetE andF be two Banach spaces, and let T ∈ L(E, F ). Consider the following properties:
(P)
For every weakly convergent sequence(un)inE, un u,thenT un→T ustrongly inF.
(Q)
T is continuous fromEequipped with the weak topology σ (E, E)intoF equipped with the strong topology.
1. Prove that
(Q)⇔T is a finite-rank operator.
2. Prove thatT ∈K(E, F )⇒(P).
3. Assume that eitherE =1orF =1. Prove thateveryoperatorT ∈ L(E, F ) satisfies (P).
[Hint: Use a result of Problem 8.]
In what follows we assume thatEisreflexive.
4. Prove thatT ∈K(E, F )⇐⇒(P).
5. Deduce thateveryoperatorT ∈L(E, 1)is compact.
6. Prove thateveryoperatorT ∈L(c0, E)is compact.
[Hint: Consider the adjoint operatorT.]
6.8 LetEandF be two Banach spaces, and letT ∈K(E, F ). Assume thatR(T ) is closed.
1. Prove thatT is a finite-rank operator.
[Hint: Use the open mapping theorem, i.e., Theorem 2.6.]
2. Assume, in addition, that dimN (T ) <∞. Prove that dimE <∞.
6.9 LetEandF be two Banach spaces, and letT ∈L(E, F ).
1. Prove that the following three properties are equivalent:4 (A) dimN (T ) <∞andR(T )is closed.
(B)
⎧⎪
⎨
⎪⎩
There are a finite-rank projection operatorP ∈L(E) and a constantCsuch that
uE ≤C(T uF + P uE) ∀u∈E.
(C)
⎧⎪
⎨
⎪⎩
There exist a Banach spaceG, an operator Q∈K(E, G),and a constantCsuch that uE ≤C(T uF + QuG) ∀u∈E.
[Hint: When dimN (T ) <∞consider a complement ofN (T ); see Section 2.4.]
Compare with Exercise 2.12.
2. Assume thatT satisfies (A). Prove that(T +S)also satisfies (A) for everyS ∈ K(E, F ).
3. Prove that the set of all operatorsT ∈L(E, F )satisfying (A) is open inL(E, F ).
4. LetF0be a closed linear subspace ofF, and letS∈K(F0, F ).
Prove that(I+S)(F0)is a closed subspace ofF.
4Aprojection operatoris an operatorPsuch thatP2=P.
6.4 Exercises for Chapter 6 173 6.10 LetQ(t ) = p
k=1aktk be a polynomial such thatQ(1) = 0. Let E be a Banach space, and letT ∈L(E). Assume thatQ(T )∈K(E).
1. Prove that dimN (I −T ) < ∞, and thatR(I−T )is closed. More generally, prove that(I−T )(E0)is closed for every closed subspaceE0⊂E.
[Hint: WriteQ(1)−Q(t ) = Q(t )(1 −t ) for some polynomialQand apply Exercise 6.9.]
2. Prove thatN (I −T )= {0} ⇔R(I−T )=E.
3. Prove that dimN (I −T )=dimN (I −T).
[Hint for questions 2 and 3: Use the same method as in the proof of Theorem 6.6.]
6.11 LetK be a compact metric space, and letE =C(K;R)equipped with the usual normu =maxx∈K|u(x)|.
LetF ⊂Ebe aclosedsubspace. Assume that every functionu ∈F is Hölder continuous, i.e.,
∀u∈F ∃α∈(0,1] and∃L such that
|u(x)−u(y)| ≤L d(x, y)α ∀x, y∈K.
The purpose of this exercise is to show thatF is finite-dimensional.
1. Prove that there exist constantsγ ∈ (0,1]andC ≥ 0 (both independent ofu) such that
|u(x)−u(y)| ≤Cud(x, y)γ ∀u∈F, ∀x, y ∈K.
[Hint: Apply the Baire category theorem (Theorem 2.1) with Fn= {u∈F; |u(x)−u(y)| ≤nd(x, y)1/n ∀x, y∈K}.] 2. Prove thatBF is compact and conclude.
6.12 A lemma of J.-L. Lions.
LetX,Y, andZbe three Banach spaces with norms X, Y, and Z. Assume that X ⊂ Y withcompactinjection and thatY ⊂ Z withcontinuousinjection.
Prove that
∀ε >0∃Cε ≥0 satisfyinguY ≤εuX+CεuZ ∀u∈X.
[Hint: Argue by contradiction.]
Application.Prove that∀ε >0∃Cε ≥0 satisfying max[0,1]|u| ≤εmax
[0,1]|u| +CεuL1 ∀u∈C1([0,1]).
6.13 LetEandF be two Banach spaces with norms Eand F. Assume that Eis reflexive. LetT ∈K(E, F ). Consider another norm| |onE, which is weaker than the norm E, i.e.,|u| ≤CuE ∀u∈E. Prove that
∀ε >0∃Cε ≥0 satisfying T uF ≤εuE+Cε|u| ∀u∈E.
Show that the conclusion may fail whenEisnotreflexive.
[Hint: TakeE=C([0,1]),F =R,u = uL∞ and|u| = uL1.]
6.14 LetEbe a Banach space, and letT ∈L(E)withT<1.
1. Prove that(I−T )is bijective and that (I−T )−1 ≤13
(1− T).
2. SetSn =I +T + · · · +Tn−1. Prove that Sn−(I−T )−1 ≤ Tn3
(1− T).
6.15 LetEbe a Banach space and letT ∈L(E).
1. Letλ∈Rbe such that|λ|>T. Prove that I+λ(T −λI )−1 ≤ T3
(|λ| − T).
2. Letλ∈ρ(T ). Check that
(T −λI )−1T =T (T−λI )−1, and prove that
dist(λ, σ (T ))≥13
(T −λI )−1. 3. Assume that 0∈ρ(T ). Prove that
σ (T−1)=13 σ (T ).
In what follows assume that 1∈ρ(T ); set
U=(T +I )(T −I )−1=(T −I )−1(T +I ).
4. Check that 1∈ρ(U )and give a simple expression for(U−I )−1in terms ofT. 5. Prove thatT =(U+I )(U−I )−1.
6. Consider the functionf (t )=(t+1)3
(t−1), t∈R. Prove that σ (U )=f (σ (T )).
6.16 LetEbe a Banach space and letT ∈L(E).
6.4 Exercises for Chapter 6 175 1. Assume thatT2 =I. Prove thatσ (T )⊂ {−1,+1}and determine(T −λI )−1
forλ = ±1.
2. More generally, assume that there is an integern ≥2 such thatTn =I. Prove thatσ (T )⊂ {−1,+1}and determine(T −λI )−1forλ = ±1.
3. Assume that there is an integern≥2 such thatTn =0. Prove thatσ (T )= {0}
and determine(T −λI )−1forλ =0.
4. Assume that there is an integern ≥2 such thatTn<1. Prove thatI −T is bijective and give an expression for(I−T )−1in terms of(I −Tn)−1and the iterates ofT.
6.17 LetE = p with 1 ≤ p ≤ ∞and let(λn)be a bounded sequence inR.
Consider the multiplication operatorM∈L(E)defined by
Mx=(λ1x1, λ2x2, . . . , λnxn, . . . ), wherex=(x1, x2, . . . , xn, . . . ).
DetermineEV (M)andσ (M).
6.18 Spectral properties of the shifts.
An elementx ∈E=2is denoted byx =(x1, x2, . . . , xn, . . . ).
Consider the operators
Srx=(0, x1, x2, . . . , xn−1, . . . ), and
Sx=(x2, x3, x4, . . . , xn+1, . . . ), respectively called theright shiftandleft shift.
1. DetermineSrandS. DoesSr orSbelong toK(E)?
2. Prove thatEV (Sr)= ∅. 3. Prove thatσ (Sr)= [−1,+1].
4. Prove thatEV (S)=(−1,+1). Determine the corresponding eigenspaces.
5. Prove thatσ (S)= [−1,+1]. 6. DetermineSrandS.
7. Prove that for everyλ∈(−1,+1), the spacesR(Sr−λI )andR(S−λI )are closed. Give an explicit representation of these spaces.
[Hint: Apply Theorems 2.19 and 2.20.]
8. Prove that the spacesR(Sr±I )andR(S±I )are dense and that they are not closed.
Consider the multiplication operatorMdefined by Mx=(α1x1, α2x2, . . . , αnxn, . . . ), where(αn)is a bounded sequence inR.
9. DetermineEV (Sr◦M).
10. Assume thatαn→αasn→ ∞. Prove that
σ (Sr ◦M)= [−|α|,+|α|]. [Hint: Apply Theorem 6.6.]
11. Assume that for every integern, α2n=aandα2n+1=bwitha =b. Determine σ (Sr◦M).
[Hint: Compute(Sr ◦M)2and apply question 4 of Exercise 6.16.]
6.19 LetEbe a Banach space and letT ∈L(E).
1. Prove thatσ (T)=σ (T ).
2. Give examples showing that there is no general inclusion relation betweenEV (T ) andEV (T).
[Hint: Consider the right shift and the left shift.]
6.20 LetE=Lp(0,1)with 1≤p <∞. Givenu∈E, set T u(x)=
x
0
u(t )dt.
1. Prove thatT ∈K(E).
2. DetermineEV (T )andσ (T ).
3. Give an explicit formula for(T −λI )−1whenλ∈ρ(T ).
4. DetermineT.
6.21 Let V andH be two Banach spaces with norms and | | respectively, satisfying
V ⊂Hwith compact injection.
Letp(u)be a seminorm onV such thatp(u)+|u|is a norm onVthat is equivalent to .
Set
N = {u∈V;p(u)=0}, and
dist(u, N )= inf
v∈Nu−vforu∈V . 1. Prove thatNis a finite-dimensional space.
[Hint: Consider the unit ball inNequipped with the norm| |.]
2. Prove that there exists a constantK1>0 such that p(u)≤K1dist(u, N ) ∀u∈V . 3. Prove that there exists a constantK2>0 such that
6.4 Exercises for Chapter 6 177 K2dist(u, N )≤p(u) ∀u∈V .
[Hint: Argue by contradiction. Assume that there is a sequence(un)inV such that dist(un, N )=1 for allnandp(un)→0.]
6.22 LetE be a Banach space, and letT ∈ L(E). Given a polynomialQ(t ) = p
k=0aktkwithak∈R, letQ(T )=p
k=0akTk. 1. Prove thatQ(EV (T ))⊂EV (Q(T )).
2. Prove thatQ(σ (T ))⊂σ (Q(T )).
3. Construct an example inE=R2for which the above inclusions are strict.
In what follows we assume thatEis a Hilbert space (identified with its dual space H) and thatT=T.
4. Assume here that the polynomialQhas no real root, i.e.,Q(t ) = 0 ∀t ∈ R.
Prove thatQ(T )is bijective.
[Hint: Start with the case thatQis a polynomial of degree 2 and more specifically, Q(t )=t2+1.]
5. Deduce that foreverypolynomialQ, we have (i) Q(EV (T ))=EV (Q(T )),
(ii) Q(σ (T ))=σ (Q(T )).
[Hint: WriteQ(t )−λ=(t−t1)(t−t2)· · ·(t−tq)Q(t ), wheret1, t2, . . . , tqare the real roots ofQ(t )−λandQhas no real root.]
6.23 Spectral radius.
LetEbe a Banach space and letT ∈L(E). Set an=logTn, n≥1.
1. Check that
ai+j ≤ai+aj ∀i, j ≥1.
2. Deduce that
n→+∞lim (an/n)exists and coincides with inf
m≥1(am/m).
[Hint: Fix an integerm≥1. Given any integern≥1 writen=mq+r, where q= [mn]is the largest integer≤n/mand 0≤r < m. Note thatan≤ mnam+ar.]
3. Conclude thatr(T )=limn→∞Tn1/n exists and thatr(T )≤ T. Construct an example inE=R2such thatr(T )=0 andT =1.
The numberr(T )is called thespectral radiusofT.
4. Prove thatσ (T )⊂ [−r(T ),+r(T )]. Deduce that ifσ (T ) = ∅, then max{|λ|; λ∈σ (T )} ≤r(T ).
[Hint: Note that ifλ∈σ (T ), thenλn∈σ (Tn); see Exercise 6.22.]
5. Construct an example inE=R3such thatσ (T )= {0}, whiler(T )=1.
In what follows we takeE=Lp(0,1)with 1≤p≤ ∞. Consider the operator T ∈L(E)defined by
T u(t )= t
0
u(s)ds.
6. Prove by induction that forn≥2,
!Tnu"
(t )= 1 (n−1)!
t
0
(t−τ )n−1u(τ )dτ.
7. Deduce thatTn ≤ n1!.
[Hint: Use an inequality for the convolution product.]
8. Prove that the spectral radius ofT is 0.
[Hint: Use Stirling’s formula.]
9. Show thatσ (T )= {0}. Compare with Exercise 6.20.
6.24 Assume thatT ∈L(H )is self-adjoint.
1. Prove that the following properties are equivalent:
(i) (T u, u)≥0∀u∈H, (ii) σ (T )⊂ [0,∞).
[Hint: Apply Proposition 6.9.]
2. Prove that the following properties are equivalent:
(iii) T ≤1 and(T u, u)≥0∀u∈H, (iv) 0≤(T u, u)≤ |u|2∀u∈H,
(v) σ (T )⊂ [0,1],
(vi) (T u, u)≥ |T u|2∀u∈H.
[Hint: To prove that (v)⇒(vi) apply Proposition 6.9 to(T+εI )−1withε >0.]
3. Prove that the following properties are equivalent:
(vii) (T u, u)≤ |T u|2∀u∈H, (viii) (0,1)⊂ρ(T ).
[Hint: IntroduceU=2T −I.]
6.4 Exercises for Chapter 6 179 6.25 LetEbe a Banach space, and letK ∈K(E). Prove that there existM∈L(E), M∈L(E), and finite-rank projectionsP,Psuch that
(i) M◦(I+K)=I−P, (ii) (I+K)◦M=I−P .
[Hint: LetXbe a complement ofN (I+K)inE. Then(I+K)|Xis bijective from X ontoR(I +K). Denote by Mits inverse. Let Qbe a projection fromE onto R(I+K)and setM=M◦Q. Show that (i) and (ii) hold.]
6.26 From Brouwer to Schauder fixed-point theorems.
In this exercise we assume that the following result is known (for a proof, see, e.g., K. Deimling [1], A. Granas–J. Dugundji [1], or L. Nirenberg [2]).
Theorem (Brouwer).LetF be a finite-dimensional space, and let Q ⊂ F be a nonempty compact convex set. Letf :Q→Qbe a continuous map. Thenf has a fixed point, i.e., there existsp∈Qsuch thatf (p)=p.
Our goal is to prove the following.
Theorem (Schauder).LetE be a Banach space, and letC be a nonempty closed convex set inE. LetF :C→Cbe a continuous map such thatF (C)⊂K, where Kis a compact subset ofC. ThenF has a fixed point inK.
1. Givenε >0, consider a finite covering ofK, i.e.,K⊂ ∪i∈IB(yi, ε/2), where Iis finite, andyi ∈K∀i∈I. Define the functionq(x)=
i∈Iqi(x), where qi(x)=
i∈I
max{ε− F x−yi,0}.
Check thatqis continuous onCand thatq(x)≥ε/2∀x∈C.
2. Set
Fε(x)=
i∈Iqi(x)yi
q(x) , x∈C.
Prove thatFε :C→Cis continuous and that
Fε(x)−F (x) ≤ε, ∀x ∈C.
3. Show thatFεadmits a fixed pointxε∈C.
[Hint: LetQ=conv(∪i∈I{yi}). Check thatFε|Qadmits a fixed pointxε ∈Q.]
4. Prove that(xεn)converges to a limitx ∈C for some sequenceεn →0. Show thatF (x)=x.