We begin by presenting the result [Ka76, IV.2.20] which will be important for us in the sequel.
2.3.1 Lemma. Let T, S∈ C(E, F)be invertible. Then
dG(S, T) =dG(S−1, T−1).
Proof. We define the inverse graph of an operatorT :D(T)⊂E→F by
graph0(T) ={(T u, u)∈F×E:u∈ D(T)} ⊂F×E and note that
graph0(T−1) ={(T−1u, u)∈E×F :u∈F}={(u, T u)∈E×F :u∈ D(T)}= graph(T).
If nowT, S∈ C(E, F)are invertible, we obtain
d(graph(T−1),graph(S−1)) = sup
(u,T−1u)∈Sgraph(T−1 )
inf
(v,S−1v)∈Sgraph(S−1 )
k(u, T−1u)−(v, S−1v)k
= sup
(u,T−1u)∈Sgraph(T−1 )
inf
(v,S−1v)∈Sgraph(S−1 )
k(T−1u, u)−(S−1v, v)k
= sup
(T−1u,u)∈Sgraph0(T−1 )
inf
(S−1v,v)∈Sgraph0(S−1 )
k(T−1u, u)−(S−1v, v)k
=d(graph0(T−1),graph0(S−1)) =d(graph(T),graph(S)) and infer by interchangingS andT
dG(T−1, S−1) = ˆd(graph(T−1),graph(S−1)) = ˆd(graph(T),graph(S)) =dG(T, S).
We now turn to spectral theory and assume from now on thatE =F. Moreover, we define for anyz0∈C
Ωz0={T ∈ C(E) :z0∈/σ(T)}.
Even if the following simple result is not explicitly stated in [Ka76], it is at least used in the proof of theorem IV.2.23.
2.3.2 Lemma. Letz0∈CandS, T ∈Ωz0. Then
dG((S−z0)−1,(T−z0)−1)≤4√
2(1 +|z0|2)dG(S, T),
dG(S, T)≤8√
2(1 +|z0|2)k(S−z0)−1−(T−z0)−1k.
Proof. By using theorem 2.2.1 and lemma 2.3.1 we obtain
dG((S−z0)−1,(T−z0)−1) =dG(S−z0, T −z0)≤4√
2(1 +|z0|2)dG(S, T) and
dG(S, T) =dG(S−z0+z0, T −z0+z0)≤4√
2(1 +|z0|2)dG(S−z0, T −z0)
= 4√
2(1 +|z0|2)dG((S−z0)−1,(T−z0)−1)
≤8√
2(1 +|z0|2)k(S−z0)−1−(T −z0)−1k, where we additionally use lemma 2.1.5.
Note that the second inequality is of interest for proving the gap continuity of families of operators having non empty resolvent sets. Accordingly, one can check the continuity by consid-ering bounded operators in the operator norm.
An easy variation of the first inequality even allows to estimate the norm distance of the resolvents locally.
2.3.3 Corollary. Let z0∈CandT ∈Ωz0. Then for anyS∈Ωz0 such that
dG(S, T)< 1 8√
2(1 +|z0|2)(1 +k(T−z0)−1k2)−12 we have
k(S−z0)−1−(T−z0)−1k ≤8√
2(1 +|z0|2)(1 +k(T−z0)−1k2)dG(S, T).
Proof. Under the given assumptions we obtain by lemma 2.3.2 from above
dG((S−z0)−1,(T−z0)−1)≤4√
2(1 +|z0|2)dG(S, T)<1
2(1 +k(T−z0)−1k2)−12 Hence we can apply lemma 2.1.6 which yields
k(S−z0)−1−(T−z0)−1k ≤ (1 +k(T−z0)−1k2)dG((S−z0)−1,(T −z0)−1) 1−dG((S−z0)−1,(T−z0)−1)p
1 +k(T−z0)−1k2
≤2(1 +k(T−z0)−1k2)dG((S−z0)−1,(T−z0)−1)
≤8√
2(1 +|z0|2)(1 +k(T−z0)−1k2)dG(S, T), where we use once again lemma 2.3.2 in the last inequality.
Note that the map
dρ : Ωz0 →R, dρ(T, S) =k(T−z0)−1−(S−z0)−1k
defines a metric onΩz0 and by lemma 2.3.2 and corollary 2.3.3 dρ anddG induce the same topology onΩz0. This is also stated in an equivalent way in [Ka76, IV.2.25].
In the special case thatEis a Hilbert space the subsetCsa(E)⊂Ω−iconsisting of all selfadjoint operators was studied in [BLP05]. BesidesdG4 anddρ they defined a further metric by
dC(T, S) =kκ(T)−κ(S)k,
where κ(T) denotes the Cayley transform ofT. In [BLP05, 1.1] it is proved that all three metrics are mutually uniformly equivalent. Moreover, these constructions are used in [BLP05]
and [Le05] in order to study the space of selfadjoint Fredholm operators and the spectral flow with respect to the gap metric.
Below we will adapt some of the results of [BLP05] to the more general case of closed operators on Banach spaces at first. Afterwards we will use them in order to study the stability of spectra for closed operators.
In order to prove our first result we need the following stability theorem for Fredholm operators [Ka76, IV.5.17].
2.3.4 Theorem. Let T ∈ C(E, F) be Fredholm. There exists δ0 = δ0(T) > 0 such that if S∈ C(E, F)anddG(S, T)< δ0, thenS is Fredholm and
dim kerS≤dim kerT dim cokerS≤dim cokerT as well as
ind(S) = ind(T).
We obtain the following principle of stability of bounded invertibility which can also be proved independently in a more elementary way (see [Ka76, Theorem 2.21]).
2.3.5 Corollary. Let T ∈ C(E, F)be invertible. There exists δ0 >0 such that if S ∈ C(E, F) anddG(S, T)< δ0 thenS is invertible.
4Note that in [BLP05] a different but equivalent definition ofdG is used that we will consider below in a subsequent section.
Proof. IfS−1 exists, then it is closed by lemma A.2.2 and hence bounded by the closed graph theorem A.2.4. Accordingly, we just have to show that under the given assumptionsS:D(S)→ F is bijective. But, sinceT is invertible, bothkerT andcokerT are trivial. By theorem 2.3.4 we conclude that there existsδ0 >0 such that the same holds for anyS ∈ C(E, F)which is closer toT thanδ0.
2.3.6 Lemma. Forz0∈Cthe sets
Ωz0 ={T ∈ C(E) :z0∈/σ(T)}
Ωessz
0 ={T ∈ C(E) :z0∈/σess(T)}
are open in C(E).
Proof. At first,Ωz0 is open by corollary 2.3.5.
IfT ∈Ωessz0 ,T−z0 is by definition a Fredholm operator. Since by theorem 2.2.1
dG(S−z0, T −z0)≤4√ 2p
1 +|z0|2dG(S, T)
for any S ∈ C(E), we obtain from theorem 2.3.4 the existence of δ0 =δ0(T −z0) >0 such thatS−z0 is Fredholm if
dG(S, T)< δ0 4√
2p
1 +|z0|2.
2.3.7 Remark. The openness of the sets
{T ∈ Csa(H) :λ /∈σ(T)}
{T ∈ Csa(H) :λ /∈σess(T)}
of selfadjoint operators for λ∈R is proved in [BLP05, Prop.1.7] by using the Cayley trans-form instead of theorem 2.3.4.
2.3.8 Lemma. The map
Ωz03T 7→(T−z0)(T−z0)−1∈ L(E) is continuous for any z0∈C.
Proof. At first we note thatim(T−z0) =E and hence(T−z0)(T−z0)−1is indeed bounded by corollary A.2.6. Moreover, we can assume thatz0∈/ Rbecause otherwise(T−z0)(T−z0)−1=I for allT ∈Ωz0 and the assertion is trivial in this case.
So let us require thatz06=z0. Then we have
(T−z0)(T−z0)−1= (T−z0+ (z0−z0))(T−z0)−1=I+ (z0−z0)(T−z0)−1 and by theorem 2.2.1, lemma 2.1.8 and lemma 2.3.2 we obtain
dG((T−z0)(T−z0)−1,(S−z0)(S−z0)−1)
=dG(I+ (z0−z0)(T −z0)−1, I+ (z0−z0)(S−z0)−1)
≤8√
2dG((z0−z0)(T−z0)−1,(z0−z0)(S−z0)−1)
≤16√
2 max{|z0−z0|, 1
|z0−z0|}dG((T−z0)−1,(S−z0)−1)
≤256(1 +|z0|2) max{|z0−z0|, 1
|z0−z0|}dG(S, T)
Now the assertion follows since the norm- and the gap topology coincide onL(E)according to corollary 2.1.7.
The proof of the next result is precisely along the lines of its analogue for selfadjoint operators in [BLP05, Lemma 2.8]. Remind before that the composition ST of two operators
T :D(T)⊂E→E, S:D(S)⊂E→E
is defined usually on the domainD(ST) = T−1D(S). Note that if S and T are invertible, thenST is invertible as well and its inverse is given byT−1S−1:E→E.
2.3.9 Theorem. Let∅ 6=K⊂CandΩK ={T ∈ C(E) :K⊂ρ(T)}. Then the map
R:K×ΩK→ L(E), (λ, T)7→(T−λ)−1 is continuous.
Proof. We fixz0∈K and note that for(λ, T)∈K×ΩK we have
R(λ, T) = (T−λ)−1= ((T −z0)−(λ−z0))−1
= ((T−z0)(I−(λ−z0)(T −z0)−1))−1
= (I−(λ−z0)(T−z0)−1)−1(T−z0)−1=F(G(λ, T)),
(2.22)
where the mapsF andGare defined by5.
G:K×ΩK →K× {S ∈ L(E) : (K−z0)−1⊂ρ(S)}, (λ, T)7→(λ,(T−z0)−1)
and
F:K× {S∈ L(E) : (K−z0)−1⊂ρ(S)} → L(E), (λ, S)7→(I−(λ−z0)S)−1S,
respectively. Now Gis continuous by corollary 2.3.3. Furthermore, the continuity of F is a simple computation using the continuity of the inversion onGL(E) (cf.[Ka76, I.(4.24),III.3.1]).
As an important corollary we obtain the continuity of the spectral projections as introduced in A.3.3.
2.3.10 Corollary. Let 4 ⊂Cbe a bounded Cauchy domain with boundaryΓ and denote
Ω4={T ∈ C(E) : Γ⊂ρ(T)}.
Then the map
Ω4→ L(E), T 7→P4(T) = 1 2πi
Z
Γ
(λ−T)−1dλ is continuous.
Proof. For anyT, S∈Ω4 we have by [He92, (97.4)]
kP4(T)−P4(S)k ≤ 1
2π|Γ|max
λ∈Γk(λ−T)−1−(λ−S)−1k, (2.23) where|Γ|denotes the length ofΓ. The rest of the proof is a standard argument in calculus.
LetT ∈Ω4 andε >0. By theorem 2.3.9 there exists aδ(λ0)>0for anyλ0∈Γsuch that
k(λ−S)−1−(λ0−T)−1k< πε
|Γ|
if
5IfA⊂C\ {0}, we denoteA−1={1z ∈C:z∈A}
λ∈U(λ0, δ(λ0)) :={λ∈Γ :|λ−λ0|< δ(λ0)} anddG(S, T)< δ(λ0).
Since Γ is compact we can find λ1, . . . , λn ∈ Γ such that Sn
i=1U(λi, δ(λi)) = Γ and define δ:= min1≤i≤nδ(λi).
Now, for anyλ∈Γthere exists an1≤i≤nsuch thatλ∈U(λi, δ(λi))and hence we obtain for S∈Ω4,dG(S, T)< δ,
k(λ−T)−1−(λ−S)−1k ≤ k(λ−T)−1−(λi−T)−1k+k(λi−T)−1−(λ−S)−1k<2πε
|Γ|. We conclude by (2.23)
kP4(T)−P4(S)k ≤ 1
2π|Γ|max
λ∈Γk(λ−T)−1−(λ−S)−1k< ε for allS∈Ω4 such thatdG(S, T)< δ.
We now turn to the study of spectra of closed operators by means of the Cayley transform
κ: Ω−i→ L(X), κ(T) = (T−i)(T+i)−1
Note thatκis continuous by lemma 2.3.8. In addition, since we considerΩ−i we implicitly assume throughout that all operators have a non empty resolvent set.
Our first result shows how to reconstruct an operator inΩ−i from its Cayley transform. The proof does not differ from the corresponding statement for selfadjoint operators (cf. A.3.8).
2.3.11 Lemma. If T ∈Ω−i, then
T =i(I+κ(T))(I−κ(T))−1. (2.24)
Proof. Using the equality
κ(T) = (T−i)(T +i)−1= (T+i−2i)(T+i)−1=I−2i(T +i)−1 we obtain at first that
I−κ(T) = 2i(T+i)−1 (2.25)
and hence
(I−κ(T))−1= 1
2i(T+i) (2.26)
is defined onD(T).
We compute
i(I+κ(T))(I−κ(T))−1=i(I+κ(T))1 2i(T+i)
=1
2(I+ (T−i)(T+i)−1)(T+i)
=1
2(T+i+T−i) =T.
Below we will study the relation between the spectrum of a closed operator in Ω−i and the spectrum of its Cayley transform. Since1 ∈C is the only point having no counterimage in C under the classical Cayley transform
κ(λ) = λ−i λ+i, we have to treat this special case separately.
2.3.12 Lemma. Let T ∈Ω−i be densely defined. Then
i) 1∈ρ(κ(T))⇐⇒ D(T) =E and this is true if and only if T is bounded.
ii) 1∈σess(κ(T))⇐⇒ D(T)6=E and this is true if and only if T is unbounded.
Proof. The assertions regarding the boundedness and unboundedness ofT follow by A.2.1 and the assumption thatT is densely defined.
By (2.25) we have
I−κ(T) = 2i(T+i)−1∈ L(E)
mapping Ebijectively ontoD(T). Accordingly, if1∈ρ(κ(T)), we inferE= im(I−κ(T)) = D(T). Conversely, ifD(T) =E, then I−κ(T)mapsE bijectively onto E showing1∈ρ(κ(T)) by the closed graph theorem A.2.4. Hence assertioni)is proved.
In order to show ii) we note at first that byi), 1 ∈ σ(κ(T))if and only if D(T)6=E. Now it remains to show that if 1 ∈ σ(T), then we actually have 1 ∈ σess(T). But, if D(T) 6=E, we obtain that im(I−κ(T)) = D(T) is a proper dense subspace ofE and hence in particular not closed. Accordingly,I−κ(T)is not a Fredholm operator.
2.3.13 Lemma. If T ∈Ω−i andλ6=−i, then
λ−T = (λ+i)(κ(λ)−κ(T))(I−κ(T))−1. Proof. We obtain by lemma 2.3.11
λ−T =λ−i(I+κ(T))(I−κ(T))−1= (λ(I−κ(T))−i(I+κ(T)))(I−κ(T))−1
= (λ−λκ(T)−i−iκ(T))(I−κ(T))−1= ((λ−i)−(λ+i)κ(T))(I−κ(T))−1
= (λ+i)((λ−i)(λ+i)−1−κ(T))(I−κ(T))−1= (λ+i)(κ(λ)−κ(T))(I−κ(T))−1.
As a consequence we obtain the following important corollary, illustrating the relation between the spectrum of an operator in Ω−i and the spectrum of its Cayley transform.
2.3.14 Corollary. Let T ∈Ω−i. Ifλ6=−i, then
1. ker(λ−T)6={0} if and only ifker(κ(λ)−κ(T))6={0}. Moreover the dimensions of both spaces coincide.
2. im(λ−T) = im(κ(λ)−κ(T)).
In particular,
• λ∈ρ(T)⇐⇒κ(λ)∈ρ(κ(T))if6 λ6=−i
• λ∈σ(T)⇐⇒κ(λ)∈σ(κ(T))
• λ∈σp(T)⇐⇒κ(λ)∈σp(κ(T))
• λ∈σess(T)⇐⇒κ(λ)∈σess(κ(T)).
Proof. By the foregoing lemma 2.3.13 we know that ifλ6=−iandT ∈Ω−i, then
λ−T = (λ+i)(κ(λ)−κ(T))(I−κ(T))−1
and by (2.26) (I−κ(T))−1 maps D(T) bijectively onto H. This implies the assertions on ker(λ−T)andim(λ−T)and the remaining part of the corollary is an immediate consequence of them. Note that we do not need to exclude the case λ= −i at the results concerning the spectra, because by assumptionT ∈Ω−i and hence−i /∈σ(T).
2.3.15 Corollary. Let T ∈Ω−i be densely defined.
6Note that−i∈ρ(T)anyway, sinceT∈Ω−i
i) IfT is bounded, then κ(σ(T)) =σ(κ(T))
ii) If T is unbounded, thenκ(σ(T))∪ {1}=σ(κ(T)) Both conclusions hold true if we replaceσ byσess.
Proof. SinceκmapsC\ {−i}bijectively ontoC\ {1}, from the equivalences stated in corollary 2.3.14 it is clear that either κ(σ(ess)(T)) = σ(ess)(κ(T)) or κ(σ(ess)(T))∪ {1} = σ(ess)(κ(T)).
Now the assertions follow from lemma 2.3.12.
We finally can study the stability of spectra. But before that, we want to recall the corre-sponding results for bounded operators that we will subsequently extend toC(E)by the Cayley transform.
2.3.16 Theorem. Let A ∈ L(E) and Ω ⊂C be an open neighbourhood of σ(A). Then there exists ε > 0 such that σ(B) ⊂ Ω for any B ∈ L(E) with kA−Bk < ε. Moreover, the same conclusion holds true if we replaceσby σess.
Proof. The assertion on the spectrum is proved in [He92, 96.5] for any complex unital Banach algebraR. Hence the first assertion follows by settingR=L(E). The second assertion follows by setting R to be the Calkin algebra Cal(E) of E and the continuity of the quotient map q:L(E)→Cal(E).
The second theorem concerns the continuity of eigenvalues of finite type.
2.3.17 Theorem. Letσ be a finite set of eigenvalues of finite type ofA∈ L(E)and let4 be a Cauchy domain such thatσ⊂ 4and4 ∩(σ(A)\σ) =∅. Then there exists ε >0 such that for anyB ∈ L(E)withkA−Bk< ε:
i) σ(B)∩∂4=∅,
ii) σ(B)∩ 4is a finite set of eigenvalues of finite type, iii)
X
λ∈4
m(λ;B) =X
λ∈4
m(λ;A),
wherem(λ;B) = dim ker(λ−B)for anyB ∈ L(E).
Proof. [GGK90, II.4.2]
Next we transfer these results to closed operators.
2.3.18 Theorem. Let T ∈ C(E)be densely defined and unbounded. Moreover, letΩ⊂Cbe an open neighbourhood of∞ such thatσ(T)⊂Ω. Then there exists δ >0 such that σ(S)⊂Ωfor allS∈ C(E)withdG(T, S)< δ.
The same conclusion holds if we replaceσbyσessunder the additional assumption thatρ(T)6=∅.
Proof. First of all, we can assume without loss of generality thatρ(T)6=∅in both cases, because ifσ(T) =C, then we haveΩ =Cnecessarily and the first assertion is trivial.
We choose z0∈ ρ(T)and δ1 >0 such thatS ∈Ωz0 for all S ∈ C(E)such that dG(S, T)< δ1, where we use that Ωz0 ⊂ C(E)is open by lemma 2.3.6.We define
F : Ωz0→Ω−i, F(T) =T−z0−i
which is continuous by corollary 2.2.3. Thenκ◦F : Ωz0 → L(E)is defined and continuous as well becausez0∈ρ(T).
Now, since σ(ess)(T−z0−i) =σ(ess)(T)−z0−i⊂Ω−z0−i:= Ω0, we obtain from corollary 2.3.15 and our assumptions that
σ(ess)(κ(F(T)) =κ(σ(ess)(F(T)))∪ {1} ⊂κ(Ω0)∪ {1}
Note thatκ(Ω0)∪{1}is open inCbecauseΩ0is an open neighbourhood of∞and, accordingly, 1 is an interior point ofκ(Ω0)∪ {1}.
Now we use theorem 2.3.16 to obtain an ε >0 such that σ(ess)(A) is contained in κ(Ω0)∪ {1}
whenever A ∈ L(E) satisfies kA−κ(F(T))k < ε. Moreover, since κ◦F is continuous, there existsδ2>0such that
kκ(F(S))−κ(F(T))k< ε for allS∈Ωz0 such thatdG(S, T)< δ2.
Setting δ= min{δ1, δ2}, we finally obtain the assertion.
The following corollary is just an equivalent formulation of the theorem.
2.3.19 Corollary. Let T ∈ C(E) be densely defined and unbounded. Moreover, let K ⊂C be compact and z0∈C. Then the sets
{T ∈ C(E) :K⊂ρ(T)}
{T ∈Ωz0 :K⊂ρess(T)}
are open subsets ofC(E) with respect to the gap topology.
2.3.20 Remark. We expect our result 2.3.18 to be well known. At least, the stability of the whole spectrum is proved by different methods in [Ka76, IV.3.1]. On the other hand we could not find any reference concerning the stability of the essential spectrum with respect to the gap topology in this generality. Both results are proved in the special case of selfadjoint operators on a Hilbert space in [BLP05, Lemma 2.7].
Finally, we consider eigenvalues of finite type.
2.3.21 Theorem. LetT ∈ C(E)andσ⊂Cbe a finite set of eigenvalues of finite type ofT. We assume that there exists a bounded Cauchy domain4 such thatσ⊂ 4and4 ∩(σ(T)\σ) =∅.
Then there existsδ >0such that for any S∈C(E)withdG(T, S)< δ, we haveσ(S)∩∂4=∅, σ(S)∩ 4is a finite set of eigenvalues of finite type and
X
λ∈4
m(λ;T) = X
λ∈4
m(λ;S).
Proof. We use the same idea as in the proof of theorem 2.3.18 in order to reduce to the corre-sponding result for bounded operators.
By assumption we have ∂4 ∩σ(T) = ∅ and since σ(T)\σ is closed and 4 compact, we can choose a z0 ∈ ρ(T) such that z0 ∈ 4. By using lemma 2.3.6 we can find a/ δ1 > 0 such that S∈Ωz0 for allS∈ C(E)satisfyingdG(S, T)< δ1. Moreover, we set again
F: Ωz0 →Ω−i, F(T) =T−z0−i
and note that κ◦F : Ωz0 → L(E)is defined and continuous. Since z0 ∈ 4, we infer that/
−i /∈ 4 −z0−i =F(4) and henceκ(F(4)) ⊂C\ {1} is a bounded Cauchy domain having boundary∂κ(F(4)) =κ(F(∂4))and containingκ(F(σ)).
Now, by theorem 2.3.17 there exists anε >0such that for anyA∈ L(E)withkκ(F(T))−Ak< ε we haveσ(A)∩∂κ(F(4)) =∅,σ(A)∩κ(F(4))is a finite set of eigenvalues of finite multiplicity and
X
λ∈κ(F(4))
m(λ;κ(F(T))) = X
λ∈κ(F(4))
m(λ;A).
Due to the continuity ofκ◦F we can find δ2>0such that
kκ(F(S))−κ(F(T))k< ε
for all S ∈ Ωz0 such that dG(T, S) < δ2. Setting δ = min{δ1, δ2} we finally obtain the assertion from corollary 2.3.14.
2.3.22 Remark. i) A different proof of theorem 2.3.21 can be found in [Ka76, §IV.3.5].
ii) Note that under the assumptions of the foregoing theorem the map
S7→P4(S)
is continuous on the set of all S∈ C(E)such thatdG(S, T)< δ by corollary 2.3.10.