On the $L^2-$Poincar\'e duality for incomplete riemannian manifolds: a general construction with applications

0.58 MB·

by  Francesco Bei

On the L2-Poincar´e duality for incomplete
riemannian manifolds: a general construction
with applications
Francesco Bei
Institute fu¨r Mathematik, Humboldt Universit¨at zu Berlin,
E-mail addresses: [email protected] [email protected]
5
1
Abstract
0
2 Let(M,g)beanopen,orientedandincompleteriemannianmanifoldofdimensionm.
UndersomegeneralconditionsweshowtheexistenceofaHilbertcomplex(L2Ωi(M,g),d )
n M,i
suchthatitscohomologygroups,labeledwithHi (M,g),satisfythefollowingproperties:
u 2,M
J • Hi (M,g)=ker(d )/im(d )
2,M max,i min,i

8 • Hi (M,g)∼=Hm−i(M,g) (Poincar´e duality holds)
2,M 2,M
] • There exists a well defined and non degenerate pairing:
G
(cid:90)
D Hi (M,g)×Hm−i(M,g)−→R, ([ω],[η])(cid:55)−→ ω∧η
2,M 2,M
M
.
h
• If (L2Ωi(M,g),d ) is a Fredholm complex then every closed extension of the de
t M,i
a Rham complex (Ωi(M),d ) is a Fredholm complex and, for each i = 0,...,m, the
m c i
quotient D(d )/D(d ) is a finite dimensional vector space.
max,i min,i
[

3
v
Keywords: L2-cohomology, Poincar´e duality, Incomplete riemannian manifolds, Fredholm
6
6 complexes.
7
2
.
1
Mathematics subject classification: 58J10, 14F40, 14F43.
0
4
1
Introduction
:
v
i
X Poincar´e duality is one of the best known and most important properties of the de Rham
cohomology on a closed and oriented smooth manifold M. Using the pairing induced by the
r
a wedge product we have:
(cid:90)
Hi (M)×Hm−i(M)−→R, ([ω],[η])(cid:55)−→ ω∧η. (1)
dR dR
M
Poincar´e duality says that (1) induces an isomorphism between Hi (M) and (Hm−i(M))∗ for
dR dR
all i=0,...,m, where m is the dimension of M . Besides the previous isomorphism, putting a
riemannian metric g on M and using the results coming from Hodge theory, we have also the
following isomorphisms:
Hi (M)∼=ker(∆ )∼=ker(∆ )∼=Hm−i(M) (2)
dR i m−i dR
where ∆ := d ◦δ +δ ◦d is the i−th Hodge Laplacian acting on Ωi(M). As it is well
i i−1 i−1 i i
known (1) and (2) are not longer true when M is not compact.
In this case two natural and important variations of the de Rham cohomology are provided
1
bytheL2-deRhamcohomologyandbythereducedL2-deRhamcohomology. Werecallbriefly
thatthereducedmaximalL2-cohomology,Hi (M,g),isdefinedasker(d )/im(d )
2,max max,i max,i−1
whilethemaximalL2-cohomology,Hi (M,g),isdefinedasker(d )/im(d )where
2,max max,i max,i−1
d : L2Ωi(M,g) → L2Ωi+1(M,g) is the distributional extension of d : Ωi(M) → Ωi(M).
max,i i c c
Analogously the reduced minimal L2-cohomology, Hi (M,g), is defined taking the quo-
2,min
tient ker(d )/im(d ) while the minimal L2-cohomology, Hi (M,g), is defined as
min,i min,i−1 2,min
ker(d )/im(d ) where d : L2Ωi(M,g) → L2Ωi+1(M,g) is the graph closure of
min,i min,i−1 min,i
d : Ωi(M) → Ωi(M). In the non compact setting they are an important tool and indeed
i c c
they have been the subject of many studies during the last decades. In this case, as it is well
known, the completeness of (M,g) plays a fundamental role. When (M,g) is complete, the
Laplacian ∆ , with domain given by the smooth and compactly supported forms Ωi(M), is an
i c
essentiallyself-adjointoperatoronL2Ωi(M,g). InparticularthisimpliesthatPoincar´eduality
holds for the reduced L2-cohomology of (M,g). Therefore, when the L2-cohomology is finite
dimensional, it coincides with the reduced L2-cohomology and so it satisfies Poincar´e dual-
ity. All these properties in general fail when (M,g) is incomplete. Generally in this case the
differential d acting on smooth i−forms with compact support admits several different closed
i
extensionswhenwelookatitasanunboundedoperatorbetweenL2Ωi(M,g)andL2Ωi+1(M,g).
Therefore, depending on the closed extensions considered, we will get different L2-cohomology
groups and L2-reduced cohomology groups for which, in general, Poincar´e duality does not
hold. However open and incomplete riemannian manifolds appear naturally in the context of
riemannian geometry and in that of global analysis, in particular when we deal with spaces
with ”singularities” such as stratified pseudomanifolds or singular complex (or real) algebraic
varieties. Therefore it is an interesting question to investigate some general constructions for
the L2-cohomology of (M,g), when g is incomplete, such that suitable versions of (1) and (2)
are satisfied or, briefly, such that Poincar´e duality holds. In the literature other papers have
dealt with this question: for example we mention [1], [5] and [7].
Thispaperisorganizedinthefollowingway: ThefirstchapterisdevotedtoHilbertcomplexes.
Asexplained byBru¨ningandLeschin[7] thisisthenatural frameworktodescribe thegeneral
propertiesofanellipticcomplexfromanL2 pointofview. Westartrecallingfrom[7]themain
properties and definitions and then we prove some abstract results about Poincar´e duality for
Hilbert complexes.
Inthesecondsection,afterrecalledthenotionofL2-deRhamcohomology,weapplytheresults
ofthefirstchaptertothecaseoftheL2-deRhamcomplex. Wecansummarizeourmainresults
in the following way:
Theorem 0.1. Let (M,g) be an open, oriented and incomplete riemannian manifold of di-
mension m. Then, for each i=0,...,m, we have the following isomorphism:
ker(d )/im(d )∼=ker(d )/im(d ).
max,i min,i−1 max,m−i min,m−i−1
Assume now that, for each i = 0,...,m, im(d ) is closed in L2Ωi+1(M,g). Then there
min,i
exists a Hilbert complex (L2Ωi(M,g)),d ) which satisfies the following properties for each
M,i
i=0,...,m:
• D(d ) ⊆ D(d ) ⊆ D(d ), that is d is an extension of d which is an
min,i M,i max,i max,i M,i
extension of d .
min,i
• im(d ) is closed in L2Ωi+1(M,g).
M,i
• If we call Hi (M,g) the cohomology of the Hilbert complex (L2Ωi(M,g),d ) then we
2,M M,i
have:
Hi (M,g)=ker(d )/im(d )
2,M max,i min,i
and
Hi (M,g)∼=Hm−i(M,g).
2,M 2,M
• There exists a well defined and non degenerate pairing:
(cid:90)
Hi (M,g)×Hm−i(M,g)−→R, ([ω],[η])(cid:55)−→ ω∧η.
2,M 2,M
M
2
Then we prove that:
Theorem 0.2. Under the assumptions of Theorem 0.1. Consider the Hilbert complexes:
0←L2(M,g)δm←ax,0 L2Ω1(M,g)δm←ax,1 L2Ω2(M,g)δm←ax,2 ...δma←x,n−1 L2Ωn(M,g)←0, (3)
and
0←L2(M,g)δm←in,0 L2Ω1(M,g)δm←in,1 L2Ω2(M,g)δm←in,2 ...δmi←n,n−1 L2Ωn(M,g)←0 (4)
Let
0←L2(M,g)δ←M,0 L2Ω1(M,g)δ←M,1 L2Ω2(M,g)δ←M,2 ...δM←,n−1 L2Ωn(M,g)←0 (5)
be the intermediate complex, which extends (4) and which is extended by (3), built according
to Theorem 0.1. Then, for each i=0,...,m, we have:
d∗ =δ =±∗d ∗ (6)
M,i M,i M,i
where d∗ is the adjoint of d and ∗ is the Hodge star operator.
M,i M,i
Moreover we prove the following result:
Theorem 0.3. Let (M,g) be an open, oriented and incomplete riemannian manifold of di-
mension m. Suppose that, for each i = 0,...,m, im(d ) is closed in L2Ωi+1(M,g). Let
min,i
(L2Ωi(M,g),d )betheHilbertcomplexbuiltinTheorem0.1. Assumethat(L2Ωi(M,g),d )
M,i M,i
is a Fredholm complex. Then:
1. Every closed extension (L2Ωi(M,g),D ) of (Ωi(M),d ) is a Fredholm complex.
i c i
2. For every i = 0,...,m the quotient of the domain of d with the domain of d ,
max,i min,i
that is
D(d )/D(d )
max,i min,i
is a finite dimensional vector space.
According to Theorem 0.3, we define the following number associated to (M,g):
m
(cid:88)
ψ (M,g):= (−1)idim(D(d )/D(d )) (7)
L2 max,i min,i
i=0
and we prove the following formula:
Theorem 0.4. Under the hypotheses of Theorem 0.3. The following formula holds:
(cid:26)
0 dim(M) is even
ψ (M,g)=χ (M,g)−χ (M,g)= (8)
L2 2,M 2,m 2χ (M,g) dim(M) is odd
2,M
where χ (M,g) and χ (M,g) are the Euler characteristics associated respectively to the
2,M 2,m
complexes (L2Ωi(M,g),d ) and (L2Ωi(M,g),d ).
max,i min,i
In the remaining part of the second chapter and in the third one we prove other results for
the complex (L2Ωi(M,g),d ). In particular we prove a Hodge type theorem for the coho-
M,i
mology groups Hi (M,g), we introduce the L2−Euler characteristic χ (M,g) associated
2,M 2,M
to (L2Ωi(M,g),d ) and the L2−signature σ (M,g) for (M,g) when dim(M) = 4l. Then
M,i 2,M
weshowthattheyaretheindexofsomesuitableFredholmoperatorsarisingfromthecomplex
(L2Ωi(M,g),d ). Finallythelastpartofthepapercontainssomeexamplesandapplications
M,i
of the previous results.
We conclude this introduction mentioning that in a subsequent paper we plan to come back
againonthissubjectinvestigatingsometopologicalpropertiesofthevectorspacesHi (M,g)
2,M
with particular attention to the cases when they are finite dimensional.
Acknowledgments. IwishtothankJochenBru¨ningformanyinterestingdiscussionsand
helpful hints. I also wish to thank Pierre Albin, Erich Leichtnam, Rafe Mazzeo and Paolo
Piazza for interesting comments and emails. This research has been financially supported by
the SFB 647 : Raum-Zeit-Materie.
3
1 Hilbert Complexes
We start the section recalling the notion of Hilbert complex and its main properties. For a
complete development of the subject we refer to [7].
Definition 1.1. A Hilbert complex is a complex, (H ,D ) of the form:
∗ ∗
0→H D→0 H D→1 H D→2 ...D→n−1 H →0, (9)
0 1 2 n
where each H is a separable Hilbert space and each map D is a closed operator called the
i i
differential such that:
1. D(D ), the domain of D , is dense in H .
i i i
2. im(D )⊂D(D ).
i i+1
3. D ◦D =0 for all i.
i+1 i
ThecohomologygroupsofthecomplexareHi(H ,D ):=ker(D )/im(D ). Ifthegroups
∗ ∗ i i−1
Hi(H ,D ) are all finite dimensional we say that the complex is a Fredholm complex.
∗ ∗
Given a Hilbert complex there is a dual Hilbert complex
0←H D←0∗ H D←1∗ H D←2∗ ...D←n∗−1 H ←0, (10)
0 1 2 n
defined using D∗ : H → H , the Hilbert space adjoint of the differential D : H → H .
i i+1 i i i i+1
The cohomology groups of (H ,(D )∗), the dual Hilbert complex, are
j j
Hi(H ,(D )∗):=ker(D∗ )/im(D∗ ).
j j n−i−1 n−i
An important self-adjoint operator associated to (9) is the following one: let us label H :=
(cid:76)n H and let
i=0 i
D+D∗ :H →H (11)
be the self-adjoint operator with domain
n
(cid:77)
D(D+D∗)= (D(D )∩D(D∗ ))
i i−1
i=0
and defined as
n
(cid:77)
D+D∗ := (D +D∗ ).
i i−1
i=0
Moreover, for all i, there is also a Laplacian ∆ = D∗D +D D∗ which is a self-adjoint
i i i i−1 i−1
operator on H with domain
i
D(∆ )={v ∈D(D )∩D(D∗ ):D v ∈D(D∗),D∗ v ∈D(D )} (12)
i i i−1 i i i−1 i−1
and nullspace:
Hi(H ,D ):=ker(∆ )=ker(D )∩ker(D∗ ). (13)
∗ ∗ i i i−1
The following propositions are well known. The first result is the weak Kodaira decompo-
sition:
Proposition 1.1. [[7], Lemma 2.1] Let (H ,D ) be a Hilbert complex and (H ,(D )∗) its dual
i i i i
complex, then:
H =Hi⊕im(D )⊕im(D∗). (14)
i i−1 i
The reduced cohomology groups of the complex are:
i
H (H ,D ):=ker(D )/(im(D )).
∗ ∗ i i−1
4
By the above proposition there is a pair of weak de Rham isomorphism theorems:
(cid:40) Hi(H ,D )∼=Hi(H ,D )
j j j j (15)
Hi(H ,D )∼=Hn−i(H ,(D )∗)
j j j j
where in the second case we mean the cohomology of the dual Hilbert complex.
Thecomplex(H ,D )iscalledweaklyFredholmifHi(H ,D )isfinitedimensionalforeachi.
∗ ∗ ∗ ∗
By the next propositions we get immediately that each Fredholm complex is a weak Fredholm
complex.
Proposition 1.2. [[7], corollary 2.5] If the cohomology of a Hilbert complex (H ,D ) is finite
∗ ∗
dimensional then, for all i, im(D ) is closed and Hi(H ,D )∼=Hi(H ,D ).
i−1 ∗ ∗ ∗ ∗
Proposition 1.3. The following properties are equivalent:
1. (9) is a Fredholm complex.
2. Theoperatordefinedin (11)isaFredholmoperatoronitsdomainendowedwiththegraph
norm.
3. For all i=0,...,n ∆ :D(∆ )→H is a Fredholm operator on its domain endowed with
i i i
the graph norm.
Proof. See [7] Theorem 2.4
Proposition 1.4. [[7], corollary 2.6] A Hilbert complex (H ,D ), j = 0,...,n is a Fredholm
j j
complex (weakly Fredholm) if and only if its dual complex, (H ,D∗), is Fredholm (weakly Fred-
j j
holm). In the Fredholm case we have:
Hi(H ,D )∼=Hi(H ,D )∼=Hn−i(H ,(D )∗)∼=Hn−i(H ,(D )∗). (16)
j j j j j j j j
Analogously in the weak Fredholm case we have:
Hi(H ,D )∼=Hi(H ,D )∼=Hn−i(H ,(D )∗)∼=Hn−i(H ,(D )∗). (17)
j j j j j j j j
Now we recall some definitions from [5]. We refer to the same paper for more properties
and comments.
Definition 1.2. Consider a pair of Hilbert complexes (H ,D ) and (H ,L ) with i = 0,...,n.
i i i i
The pair (H ,D ) and (H ,L ) is said to be complementary if the following property is
i i i i
satisfied:
• for each i there exists an isometry φ : H −→ H such that φ (D(D )) = D(L∗ )
i i n−i i i n−i−1
and L∗ ◦φ = C (φ ◦D ) on D(D ) where L∗ : H −→ H is the
n−i−1 i i i+1 i i n−i−1 n−i n−i−1
adjoint of L :H −→H and C (cid:54)=0 is a constant which depends only on i.
n−i−1 n−i−1 n−i i
We call the maps φ duality maps.
i
We have the following proposition:
Proposition 1.5. Let (H ,D ) and (H ,L ) be complementary Hilbert complexes. Then:
i i i i
1. Also (H ,L ) and (H ,D ) are complementary Hilbert complexes. Moreover if {φ } are
i i i i i
the duality maps which make (H ,D ) and (H ,L ) complementary then {φ∗}, the family
i i i i i
obtained taking the adjoint maps, are the duality maps which make (H ,L ) and (H ,D )
i i i i
complementary.
2. Each φ induces an isomorphism between Hj(H ,D ) and Hn−j(H ,L ).
j ∗ ∗ ∗ ∗
3. The complexes (H ,D ) and (H ,L∗) have isomorphic cohomology groups and isomorphic
i i i i
reduced cohomology groups. In the same way the complexes (H ,L ) and (H ,D∗) have
i i i i
isomorphic cohomology groups and isomorphic reduced cohomology groups.
4. The following isomorphism holds: Hj(H ,D )∼=Hn−j(H ,L ).
∗ ∗ ∗ ∗
5
Proof. See [5] Prop. 5.
Finally, given a pair of Hilbert complexes (H ,D ) and (H ,D(cid:48)), we will write (H ,D )⊆
j j j j j j
(H ,D(cid:48)) if, for each j, D(cid:48) extends D . We will write (H ,D ) ⊂ (H ,D(cid:48)) if D (cid:54)= D(cid:48) for at
j j j j j j j j j j
least one j. We are now in position to prove the main results of this section:
Theorem 1.1. Let (H ,D )⊆(H ,L ) be a pair of complementary Hilbert complexes. Then,
j j j j
for every j =0,...,n, we have the following isomorphism:
ker(L )/(im(D ))∼=ker(L )/(im(D )). (18)
j j−1 n−j n−j−1
Proof. The fact that L is an extension of D implies that, the complex below is well defined
j j
for each j =0,...,n
0→H D→0 H D→1 ...D→j−1 H →Lj ...L→n−1 H →0. (19)
0 1 j n
The dual Hilbert complex is clearly:
0←H D←0∗ ...D←j∗−1 H ←L∗j ...L←∗n−1 H ←0, (20)
0 j n
Therefore, by (13) and (15), we get that:
ker(L )/(im(D ))∼=ker(L )∩ker(D∗ ). (21)
j j−1 j j−1
By Def. 1.2 and Prop. 1.5 we know that φ induces an isomorphism between ker(L ) and
n−j j
ker(D∗ ) and between ker(L ) and ker(D∗ ). Therefore it induces an isomorphism
n−j−1 n−j j−1
between ker(L )∩ker(D∗ ) and ker(D∗ )∩ker(L ). In this way, using (21), we get:
j j−1 n−j−1 n−j
ker(L )/(im(D ))∼=ker(L )∩ker(D∗ )∼=
j j−1 j j−1
∼=ker(D∗ )∩ker(L )∼=ker(L )/(im(D ))
n−j−1 n−j n−j n−j−1
and this completes the proof.
Theorem 1.2. Let (H ,D ) ⊆ (H ,L ), j = 0,...,n, be a pair of Hilbert complexes. Suppose
j j j j
that, for each j, im(D ) is closed in H . Then there exists a third Hilbert complex (H ,P )
j j+1 j j
such that:
1. (H ,D )⊆(H ,P )⊆(H ,L ) and the image of P is closed for each j.
j j j j j j j
2. Hj(H ,P )=ker(L )/(im(D )).
∗ ∗ j j−1
3. If (H ,D )⊆(H ,L ) are complementary then:
j j j j
Hj(H ,P )∼=Hn−j(H ,P ).
∗ ∗ ∗ ∗
Proof. To prove the first part of the proposition we have to exhibit a Hilbert complex which
satisfies the assertions of the statement. To do this consider the following Hilbert space
(D(L ),(cid:104) , (cid:105) )
j G
which is by definition the domain of L endowed with the graph scalar product, that is for
j
each pair of elements u,v ∈D(L ) we have
j
(cid:104)u,v(cid:105) :=(cid:104)u,v(cid:105) +(cid:104)L u,L v(cid:105) .
G Hj j j Hj+1
During the rest of the proof we will work with this Hilbert space and therefore all the direct
sums that will appear and all the assertions of topological type are referred to this Hilbert
space (D(L ),(cid:104) , (cid:105) ). We can decompose (D(L ),(cid:104) , (cid:105) ) in the following way:
j G j G
(D(L ),(cid:104) , (cid:105) )=ker(L )⊕V (22)
j G j j
where V ={α∈D(L )∩im(L∗)} and it is immediate to check that these subspaces are both
j j j
closed in (D(L ),(cid:104) , (cid:105) ).
j G
6
Consider now (D(D ),(cid:104) , (cid:105) ); it is a closed subspace of (D(L ),(cid:104) ,(cid:105) ) and we can decompose
j G j G
it as
(D(D ),(cid:104) , (cid:105) )=ker(D )⊕A . (23)
j G j j
BytheassumptionontherangeofD wegetthatalsotherangeofD∗isclosed. So,analogously
j j
tothepreviouscase,A ={α∈D(D )∩im(D∗)}andobviouslythesesubspacesarebothclosed
j j j
in(D(D ),(cid:104), (cid:105) ). Clearlyifker(D )=ker(L )thentheHilbertcomplex(H ,D )satisfiesthe
j G j j j j
first two properties of the statement, that is defining (H ,P ) as (H ,D ), we have (H ,D )⊆
j j j j j j
(H ,P )⊆(H ,L ),theimageofP isclosedforeachj andHj(H ,P )=ker(L )/(im(D )).
j j j j j ∗ ∗ j j−1
So we can suppose that ker(D ) is properly contained in ker(L ). Let π be the orthogonal
j j 1,j
projection of A onto ker(L ) and analogously let π be the orthogonal projection of A onto
j j 2,j j
V . We have the following properties:
j
1. π is injective
2,j
2. im(π ) is closed.
2,j
The first property follows from the fact that ker(π )=A ∩ker(L ). But L is an extension
2,j j j j
of D ; therefore if an element α lies in A ∩ker(L ) then it lies also in ker(D ) and so α = 0
j j j j
becauseker(Dj)∩Aj ={0}. Forthesecondpropertyconsiderasequence{γm}m∈N ⊂Aj such
that π (γ ) converges to γ ∈V . We recall that we are in (D(L ),(cid:104) , (cid:105) ) and therefore this
2,j m j j G
means that
lim π (γ )=γ in H and lim L (π (γ ))=L (γ) in H .
2,j m j j 2,j m j j+1
m→∞ m→∞
Then
lim D (γ )= lim L (γ )= lim L (π (γ ))=L (γ).
j m j m j 2,j m j
m→∞ m→∞ m→∞
This implies that
lim D (γ )=L (γ)
j m j
m→∞
and therefore the limit exists. So by the assumptions about the range of D we get that there
j
exists an element η ∈A such that
j
lim D (γ )=D (η).
j m j
m→∞
Moreover L (γ) = D (η) = L (η) = L (π (η)). This implies that L (π (η)−γ) = 0 and
j j j j 2,j j 2,j
therefore π (η) = γ because π (η),γ ∈ V and L is injective on V . In this way we have
2,j 2,j j i j
shown that im(π ) is closed.
2,j
Now define N as the range of π . Finally define W as the vector space generated by the
j 2,j j
sum of ker(L ) and N . By the fact that ker(L ) and N are orthogonal to each other we have
j j j j
W =ker(L )⊕N and therefore W is closed in (D(L ),(cid:104) , (cid:105) ). Finally define P as
j j j j j G j
P :=L | (24)
j j Wj
By the fact that W is closed in D(L ) and that π (A ),π (A ) ⊂ W we get that P is a
j j 1,j j 2,j j j j
closed extension of D which is in turn extended by L . Moreover, by the construction, it is
j j
clear that ker(P ) = ker(L ). Finally, again by the definition of P and its domain, we have
j j j
im(P )=L (π (A ))=im(D ). Therefore we got that im(P ) is closed and that
j j 2,j j j j
ker(P )/im(P )=ker(L )/imD .
j j−1 j j−1
This completes the proof of the first two statements.
Finally,combiningthesecondstatementofthisTheoremwithTheorem1.1,thethirdstatement
follows.
For the dual complex of (H ,P ) we have the following description:
i i
Theorem 1.3. Under the hypotheses of Theorem 1.2. Assume moreover that the image of L ,
i
im(L ), is closed for each i=0,...,n. Consider the Hilbert complexes:
i
0←H D←0∗ H D←1∗ H D←2∗ ...D←n∗−1 H ←0, (25)
0 1 2 n
7
and
0←H ←L∗0 H ←L∗1 H ←L∗2 ...L←∗n−1 H ←0. (26)
0 1 2 n
Let
0←H ←S0 H ←S1 H ←S2 ...S←n−1 H ←0 (27)
0 1 2 n
be the intermediate complex, which extends (26) and which is extended by (25), constructed
according to Theorem 1.2 (and its proof). Then, for each i=0,...,n, we have:
P∗ =S . (28)
i i
Furthermore assume that (H ,D ) and (H ,L ) are complementary (in this case the fact that
i i i i
im(D ) is closed implies that im(L ) is closed). Let {φ } be the duality maps and suppose that
i i i
φ−1 =±φ . Then we have:
i n−i
S =C−1φ−1◦P ◦φ =±C−1φ ◦P ◦φ . (29)
i i i n−i−1 i+1 i n−i n−i−1 i+1
In order to prove Theorem 1.3 we need the following proposition:
Proposition 1.6. Let H and K be two Hilbert spaces and let T : H → K be a closed and
densely defined operator. Let S :H →K be another closed and densely defined operator which
extends T. Assume that ker(T)=ker(S) and im(T)=im(S). Then:
T =S
that is D(T)=D(S) and T(u)=S(u) for each u∈D(S).
Proof. Consider the Hilbert space (D(S),(cid:104) , (cid:105) ). As in the proof of Theoren 1.2 we can
G
decompose it as (D(S),(cid:104) , (cid:105) ) = ker(S)⊕A where A = D(S)∩im(S∗). Analogously, if we
G
consider (D(T),(cid:104) , (cid:105) ), then we have (D(T),(cid:104) , (cid:105) )=ker(T)⊕B where B =D(T)∩im(T∗).
G G
By the fact that D(T)⊆D(S) and ker(S)=ker(T) we get that B ⊆A. Now let u∈A. Then
there exists v ∈ B such that S(u) = T(v). Therefore, by the fact that S extends T, we have
S(u−v) = 0 and this implies that u = v because (u−v) ∈ ker(S)∩A. So we can conclude
that S =T.
Proof. (of Theorem 1.3). First of all we remark that we can apply Theorem 1.2 to the pair of
complexes (25) and (26). Clearly (25) extends (26); moreover, having assumed that im(L ) is
i
closed,itfollowsthatim(L∗)isclosed. InthiswaytheassumptionsofTheorem1.2arefulfilled.
i
Now,byTheorem1.2anditsproof,weknowthatim(P )isclosedforeachi=0,...,n. Therefore
i
also im(P∗) is closed for each i = 0,...,n and we have im(P∗) = (ker(P ))⊥ = (ker(L ))⊥ =
i i i i
im(L∗). In the same way ker(P∗) = (im(P ))⊥ = (im(D ))⊥ = ker(D∗). Now, if we consider
i i i i i
S , again by Theorem 1.2 and its proof, we have im(S ) = im(L∗), ker(S ) = ker(D∗) and
i i i i i
in particular im(S ) is closed. Therefore, according to Prop. 1.6, in order to prove (28) it is
i
enough to show that P∗ extends S . To do this we have to show that:
i i
(cid:104)P (u),v(cid:105) =(cid:104)u,S (v)(cid:105) (30)
i Hi+1 i Hi
for each u ∈ D(P ) and for each v ∈ D(S ). We start observing that we can decompose u
i i
as u +u where u ∈ ker(P ) and u ∈ D(P )∩im(P∗). Analogously v = v +v where
1 2 1 i 2 i i 1 2
v ∈ ker(S ) and v ∈ D(S )∩im(S∗). So we get (cid:104)P (u),v(cid:105) = (cid:104)P (u ),v (cid:105) because
1 i 2 i i i Hi+1 i 2 2 Hi+1
0 = P (u ) and P (u ) ∈ im(D ) which is orthogonal to ker(D∗) = ker(S ). By the proof of
i 1 i 2 i i i
Theorem 1.2 we know that P (u )=D (w) for a unique element w ∈D(D )∩im(D∗). There-
i 2 i i i
fore (cid:104)P (u ),v (cid:105) =(cid:104)D (w),v (cid:105) =(cid:104)w,D∗(v )(cid:105) =(cid:104)w,S (v )(cid:105) because v ∈D(S )⊂
i 2 2 Hi+1 i 2 Hi+1 i 2 Hi i 2 Hi 2 i
D(D∗) and D∗| = S . Now, using the projections π and π defined in the proof of
i i D(Si) i 1,i 2,i
Theorem 1.2 we can decompose w as π (w)+π (w) where π (w) is the projection of w on
1,i 2,i 1,i
ker(L ) and π (w) is the projection of w on D(L )∩im(L∗).
i 2,i i i
We have that π (w) = u because π (w)−u ∈ ker(L )∩(D(L )∩im(L∗)) = {0}. There-
2,i 2 2,i 2 i i i
fore we get (cid:104)w,S (v )(cid:105) = (cid:104)π (w)+u ,S (v )(cid:105) . Now by the fact that S (v ) = 0 and
i 2 Hi 1,i 2 i 2 Hi i 1
(cid:104)z,S (v )(cid:105) =0 for each z ∈ker(L ) we get (cid:104)π (w)+u ,S (v )(cid:105) =(cid:104)u ,S (v )+S (v )(cid:105)
i 2 Hi i 1,i 2 i 2 Hi 2 i 2 i 1 Hi
=(cid:104)u +u ,S (v )+S (v )(cid:105) =(cid:104)u,S (v)(cid:105) . Summarizing all the passages we have:
1 2 i 2 i 1 Hi i Hi
(cid:104)P (u),v(cid:105) =(cid:104)P (u ),v (cid:105) =(cid:104)D (w),v (cid:105) =(cid:104)w,D∗(v )(cid:105) =(cid:104)w,S (v )(cid:105) =
i Hi+1 i 2 2 Hi+1 i 2 Hi+1 i 2 Hi i 2 Hi
8
=(cid:104)π (w)+π (w),S (v )(cid:105) =(cid:104)π (w)+u ,S (v )(cid:105) =(cid:104)u ,S (v )+S (v )(cid:105) =
1,i 2,i i 2 Hi 1,i 2 i 2 Hi 2 i 2 i 1 Hi
=(cid:104)u +u ,S (v )+S (v )(cid:105) =(cid:104)u,S (v)(cid:105)
1 2 i 2 i 1 Hi i Hi
and this completes the proof of (28).
Now we prove (29). We start recalling that
D(S )=ker(D∗)⊕π (D(L∗)∩im(L )) (31)
i i 2,i+1 i i
and
D(P )=ker(L )⊕π (D(D )∩im(D∗ )) (32)
n−i−1 n−i−1 2,n−i−1 n−i−1 n−i−1
where, as defined in the proof of Theorem 1.2, in (31) π is the projection
2,i+1
π :D(L∗)∩im(L )−→D(D∗)∩im(D ) (33)
2,i+1 i i i i
and in (32) π is the projection
2,n−i−1
π :D(D )∩im(D∗ )−→D(L )∩im(L∗ ). (34)
2,n−i−1 n−i−1 n−i−1 n−i−1 n−i−1
Therefore,inordertoavoidanyconfusion,duringtherestoftheproofwewilllabelwithπS
2,i+1
the projection (33) and with πP the projection (34). First of all, in order to establish
2,n−i−1
(29), we need to prove that
φ (D(S ))=D(P ).
i+1 i n−i−1
This is equivalent to show that
φ (ker(D∗))=ker(L ) (35)
i+1 i n−i−1
and that
φ (πS (D(L∗)∩im(L )))=πP (D(D )∩im(D∗ )) (36)
i+1 2,i+1 i i 2,n−i−1 n−i−1 n−i−1
By the fact that C (φ ◦D )=L∗ ◦φ we get
i i+1 i n−i−1 i
C (D∗◦φ−1 )=φ−1◦L (37)
i i i+1 i n−i−1
and this implies immediately (35).
Nowtoestablish(36)weneedtoprovethatφ (D(L∗)∩im(L ))=D(D )∩im(D∗ )
i+1 i i n−i−1 n−i−1
and that πP ◦φ =φ ◦πS .
2,n−i−1 i+1 i+1 2,i+1
Consider again C (φ ◦D )=L∗ ◦φ . It follows immediately that
i i+1 i n−i−1 i
φ (D(D ))=D(L∗ ). (38)
i i n−i−1
ThisimpliesthatD(D )=φ−1(D(L∗ ))whichinturnimpliesthatD(D )=φ (D(L∗ ))
i i n−i−1 i n−i n−i−1
or equivalently D(D )=φ (D(L∗)).
n−i−1 i+1 i
Taking again C (φ ◦D ) = L∗ ◦φ , we get C (D∗◦φ ) = ±φ ◦L that is
i i+1 i n−i−1 i i i n−i−1 n−i n−i−1
C (D∗ ◦φ ) = ±φ ◦L . In this way we get that φ (im(L )) = im(D∗ ). So
n−i−1 n−i−1 i i+1 i i+1 i n−i−1
we can conclude that:
φ (D(L∗)∩im(L ))=D(D )∩im(D∗ ).
i+1 i i n−i−1 n−i−1
Now,tocompletetheproofof(36),wehavetoshowthat(φ ◦πS )(u)=(πP ◦φ )(u)
i+1 2,i+1 2,n−i−1 i+1
for each i=0,...,n and for each u∈D(L∗)∩im(L ). Let u∈D(L∗)∩im(L ). Then:
i i i i
φ (u)=πP (φ (u))+πP (φ (u)) (39)
i+1 1,n−i−1 i+1 2,n−i−1 i+1
where, as defined in the proof of Theorem 1.2, πP is the projection
1,n−i−1
πP :D(D )∩im(D∗ )−→ker(L ).
1,n−i−1 n−i−1 n−i−1 n−i−1
On the other hand:
φ (u)=φ (πS (u)+πS (u)) (40)
i+1 i+1 1,i+1 2,i+1
9
where now πS is the projection
1,i+1
πS :D(L∗)∩im(L )−→ker(D∗).
1,i+1 i i i
Now if we look at (37) we get:
φ (D(D∗))=D(L ) and φ (ker(D∗))=ker(L )
i+1 i n−i−1 i+1 i n−i−1
while from Def. 1.2 we get:
φ (im(D ))=im(L∗ ).
i+1 i n−i−1
Therefore, if we consider again (40), we have:
φ (πS (u))∈ker(L ) because πS (u)∈ker(D∗)
i+1 1,i+1 n−i−1 1,i+1 i
and
φ (πS (u))∈D(L )∩im(L∗ ) because πS (u)∈im(D )∩D(D∗).
i+1 2,i+1 n−i−1 n−i−1 2,i+1 i i
In this way we can conclude that:
πP (φ (u))=φ (πS (u)) and πP (φ (u))=φ (πS (u))
1,n−i−1 i+1 i+1 1,i+1 2,n−i−1 i+1 i+1 2,i+1
because
πP (φ (u))+πP (φ (u))=φ (πS (u)+πS (u))=φ (u)
1,n−i−1 i+1 2,n−i−1 i+1 i+1 1,i+1 2,i+1 i+1
πP (φ (u))−φ (πS (u))∈ker(L ),
1,n−i−1 i+1 i+1 1,i+1 n−i−1
πP (φ (u))−φ (πS (u))∈D(L )∩im(L∗ )
2,n−i−1 i+1 i+1 2,i+1 n−i−1 n−i−1
and
ker(L )∩im(L∗ )={0}.
n−i−1 n−i−1
Thus we proved (35) and (36) and this means that
φ (D(S ))=D(P ).
i+1 i n−i−1
Now, in order to complete the proof of (29), we have to show that
S (v)=±C−1(φ ◦P ◦φ )(v)
i i n−i n−i−1 i+1
for each v ∈D(S ). Let v ∈D(S ). Then, according to (31),
i i
v =v +v
1 2
with v ∈ ker(D∗), v ∈ πS (D(L∗) ∩ im(L )) and we have φ (S (v)) = φ (D∗(v )) =
1 i 2 2,i+1 i i i i i i 2
φ (D∗(πS (w))) where w ∈ D(L∗)∩im(L ) is unique because, as proved in the proof of
i i 2,i+1 i i
Theorem 1.2, πS is injective. So we get:
2,i+1
φ (D∗(πS (w)))=C−1L (φ (πS (w)))=C−1L (πP (φ (w)))=(because
i i 2,i+1 i n−i−1 i+1 2,i+1 i n−i−1 2,n−i−1 i+1
πP (φ (w))∈D(P ))=C−1P (πP (φ (w)))=
2,n−i−1 i+1 n−i−1 i n−i−1 2,n−i−1 i+1
C−1P (φ (πS (w)))=C−1P (φ (v ))=C−1P (φ (v +v ))=
i n−i−1 i+1 2,i+1 i n−i−1 i+1 2 i n−i−1 i+1 1 2
=C−1P (φ (v)).
i n−i−1 i+1
Thus we proved that
φ (S (v))=C−1P (φ (v))
i i i n−i−1 i+1
for each v ∈D(S ) and therefore we can conclude that:
i
S =C−1φ−1◦P ◦φ =±C−1φ ◦P ◦φ .
i i i n−i−1 i+1 i n−i n−i−1 i+1
10