Given a closed operatorA ∈ C(E, F)and a bounded operatorB∈ L(E, F), we know by lemma A.2.2 thatA+B∈ C(E, F)as well, whereD(A+B) =D(A). Theorem IV.2.17 of [Ka76] asserts that ifT, S∈ C(E, F)are closed operators andA∈ L(E, F)is bounded, then
dG(S+A, T+A)≤4(1 +kAk2)dG(S, T). (2.16) Our aim in this section is to generalise this theorem to the case thatS andT are perturbed by different bounded operators A and B. Moreover, this will lead to a result concerning the continuity of a certain type of canonical embeddings of L(E, F) with the norm topology into C(E, F). These embeddings appeared in the special case of selfadjoint Fredholm operators in the development of the spectral flow for unbounded operators some years ago. We will give details and references below.
We start by proving the main result of this section as announced above, which reads as follows:
2.2.1 Theorem. Let S, T ∈ C(E, F)be closed operators and A, B∈ L(E, F)bounded. Then
dG(T+A, S+B)≤4√ 2p
1 +kAk2p
1 +kBk2(dG(S, T) +kA−Bk).
Proof. Note at first that we can assume thatD(T),D(S)6={0}without loss of generality, because otherwisedG(T+A, S+B) =dG(T, S)and the assertion holds trivially.
Let nowϕ∈graph(S+B),kϕk= 1. Then there exists u∈ D(S), such thatϕ= (u,(S+B)u) andkuk2+k(S+B)uk2 =kϕk2 = 1. Settingr2:=kuk2+kSuk2 >0 we obtain, incidentally, the following inequality
r2=kuk2+k(S+B)u−Buk2≤ kuk2+ 2k(S+B)uk2+ 2kBuk2
≤2(kuk2+k(S+B)uk2
| {z }
=1
) + 2kBk2kuk2
| {z }
≤1
≤2(1 +kBk2), (2.17)
which we will need below. Moreover, by the choice ofr,r−1(u, Su)is an element of the unit sphere ofgraph(S). Hence we have for all δ0>δ(graph(S),ˆ graph(T))
d(r−1(u, Su),graph(T))≤ sup
w∈Sgraph(S)
d(w,graph(T))≤δ(graph(S),ˆ graph(T))< δ0, which impliesd((u, Su),graph(T))< rδ0 and so the existence ofv∈ D(T)such that
ku−vk2+kSu−T vk2< r2δ02. (2.18) Settingψ= (v,(T+A)v)∈graph(T +A), we obtain
kϕ−ψk2=k(u,(S+B)u)−(v,(T +A)v)k2=ku−vk2+kSu−T v+Bu−Avk2
≤ ku−vk2+ 2kSu−T vk2+ 2kBu−Avk2
≤2(ku−vk2+kSu−T vk2) + 2kBu−Avk2.
Now we use (2.18) twice and thatkuk ≤rby definition ofr, and get
kϕ−ψk2≤2r2δ02+ 2kBu−Avk2≤2r2δ02+ 2(kAv−Auk+kAu−Buk)2
≤2r2δ02+ 4kAk2kv−uk2+ 4kA−Bk2kuk2
≤2(1 + 2kAk2)r2δ02+ 4kA−Bk2kuk2
≤2(1 + 2kAk2)r2δ02+ 4r2kA−Bk2. Finally, by using (2.17), we conclude that
kϕ−ψk2≤4(1 + 2kAk2)(1 +kBk2)δ02+ 8(1 +kBk2)kA−Bk2
≤8(1 +kAk2)(1 +kBk2)δ02+ 8(1 +kAk2)(1 +kBk2)kA−Bk2
≤8(1 +kAk2)(1 +kBk2)(δ02+kA−Bk2)
≤8(1 +kAk2)(1 +kBk2)(δ0+kA−Bk)2 and infer
kϕ−ψk ≤2√ 2p
1 +kAk2p
1 +kBk2(δ0+kA−Bk). (2.19) Now we are almost done.
Let us point out that in (2.19)ϕ∈Sgraph(S+B) is arbitrary whereasψ∈graph(T+A)depends onϕ. Sinceψ∈graph(T+A)we obtain from (2.19)
d(ϕ,graph(T+A)) = inf
ψ∈graph(T˜ +A)
kϕ−ψk ≤ kϕ˜ −ψk
≤2√ 2p
1 +kAk2p
1 +kBk2(δ0+kA−Bk)
for anyϕ∈graph(S+B),kϕk= 1, and hence
Remembering thatδ0 is any fixed number greater thanδ(graph(S),ˆ graph(T)), we infer
δ(graph(S+B),graph(T+A))≤2√ 2p
1 +kAk2p
1 +kBk2(ˆδ(graph(S),graph(T)) +kA−Bk) and since the right hand side of this inequality is symmetric in T+AandS+B, we obtain by interchangingS+B andT+A
ˆδ(graph(T+A),graph(S+B))≤2√ 2p
1 +kAk2p
1 +kBk2(ˆδ(graph(S),graph(T)) +kA−Bk).
Finally we use lemma 2.1.3 and conclude
dG(T+A, S+B)≤4√ 2p
1 +kAk2p
1 +kBk2(dG(S, T) +kA−Bk).
Before going on, we want to state some remarks regarding theorem 2.2.1.
2.2.2 Remark. i) Note that if A = B, theorem 2.2.1 and (2.16) differ by a constant √ 2 which is of course not very important for applications.
ii) The proof of theorem 2.2.1 follows the argument of the proof of (2.16) as presented in [Ka76, IV.2.17]. Hence we doubt strongly that 2.2.1 is unknown, however, we could not find any reference for it in the literature.
iii) An alternative way to prove theorem 2.2.1 should be as follows: By using the triangle inequality, we obtain
dG(T+A, S+B)≤dG(T+A, T+B) +dG(T+B, S+B)
and the second term can be estimated according to (2.16). However, the proof of (2.16) is not much more elementary than the proof of theorem 2.2.1. Moreover, we do not know if there is a way to estimatedG(T+A, T+B)bykA−Bkother than just using the argument of 2.2.1 in the special case S=T.
We now want to consider a couple of applications of the foregoing theorem 2.2.1. The first one is just an obvious extension of lemma 2.1.8.
2.2.3 Corollary. Let α, β∈C,α6= 0. Then the map
C(E, F)→ C(E, F), T 7→αT +β is continuous.
Proof. IfT, S∈ C(E, F), we obtain by theorem 2.2.1 and lemma 2.1.8 that
dG(αT +β, αS+β)≤4√
2(1 +|β|2)dG(αT, αS)≤8√
2(1 +|β|2) max{|α|, 1
|α|}dG(T, S).
LetT ∈ C(E, F)be a fixed operator and consider the map
TT :L(E, F)→ C(E, F), C7→T+C
In the special case that E = F = H is a Hilbert space and T a selfadjoint operator, the continuity of the restriction of TT to the selfadjoint bounded operators in L(H) is a direct consequence of [Le05, 2.2]. Moreover, an alternative proof, using quite advanced functional calculus, is sketched in [BoFu98, 4.10].
We now obtain the continuity of TT on the whole domainL(E, F)in the general case whereE andF are Banach spaces andT ∈ C(E, F)as a direct consequence of theorem 2.2.1.
2.2.4 Corollary. TT :L(E, F)→ C(E, F)is continuous.
Proof. LetA∈ L(E, F)be fixed. For any B∈ L(E, F)such thatkA−Bk<1we have
kBk ≤ kA−Bk+kAk ≤1 +kAk and hence by theorem 2.2.1
dG(TT(B),TT(A))≤4√ 2p
1 +kAk2p
1 +kBk2kA−Bk
≤4√ 2p
1 +kAk2p
1 + (1 +kAk)2kA−Bk.
Note that in general it is a difficult task to show the continuity of a given family of closed operators because of the very definition of the gap metric. Accordingly, corollary 2.2.4 is an important result for constructing gap continuous families.
Finally, we want to take a further look at the mapTT and ask about the relation between the topology that the injectionTT induces on its image inC(E, F)and the subspace topology induced by the gap metric on imTT. Note thatTT is an embedding ifT ∈ L(E, F) because the norm topology and the gap topology coincide onL(E, F)as shown in corollary 2.1.7. Our next aim is to show that the topology induced on its image can also be strictly finer than the gap topology.
At first we need a technical result which is problem IV.1.2 in [Ka76].
2.2.5 Lemma. LetT andAbe operators acting between the normed linear spacesEandF such thatD(T)⊂ D(A)and
kAuk ≤akuk+bkT uk, u∈ D(T),
for somea≥0 and0≤b <1.3 Then the operator S=T+A,D(S) =D(T), satisfies
kAuk ≤akuk+bkT uk ≤ 1
1−b(akuk+bkSuk), u∈ D(T).
Proof. By assumption we have
−akuk −bkT uk ≤ −kAuk and hence for anyu∈ D(S) =D(T)
−akuk+ (1−b)kT uk ≤ kT uk − kAuk ≤ |kT uk − kAuk| ≤ kT u+Auk=kSuk.
Using this inequality, we find
akuk+bkT uk= 1
1−b(a(1−b)kuk+b(1−b)kT uk) = 1
1−b(akuk+b(−akuk+ (1−b)kT uk))
≤ 1
1−b(akuk+bkSuk).
The following result can also be obtained as a consequence of [Le05, 2.4]. There it is proved that the restriction ofTT to the selfadjoint elements inL(H)is not an embedding when viewed as a map into a certain space of unbounded selfadjoint operators which itself is continuously included in the subspace of selfadjoint elements in C(H). We do not want to introduce this intermediate space here and give a different proof in our setting instead.
3By definition, this just means thatAisT-bounded withT-bound smaller than1(cf. [Ka76, IV.1.1]).
2.2.6 Lemma. LetH be a separable Hilbert space andT a selfadjoint operator having a compact resolvent. Then
TT :L(H)→ C(H) is not an embedding.
Proof. SinceT is selfadjoint and has a compact resolvent, we obtain from [GGK90, XVI.5.1] the existence of a complete orthonormal system {en}n∈N of H and a sequence {λn}n∈N such that
Moreover, consider the sequence{An}n∈Nof bounded operators onH defined by
Anu=p
Now we consider the sequence of operators
Tn=T+An∈ C(H), D(Tn) =D(T), n∈N,
and our aim is to show thatTnconverges toTinC(H)with respect to the gap topology. Keep in mind that once we have shown this assertion we are done, that is,TT is not an embedding of L(H)intoC(H).
for alln≥n0as well as