aa r X i v : . [ m a t h . SP ] O c t Spectral Sections
Marina Prokhorova
Abstract
The paper is devoted to the notion of a spectral section introduced by Melroseand Piazza. In the first part of the paper we generalize results of Melrose andPiazza to arbitrary base spaces, not necessarily compact. The second part containsa number of applications, including cobordism theorems for families of Diractype operators parametrized by a non-compact base space. In the third part ofthe paper we investigate whether Riesz continuity is necessary for existence of aspectral section or a generalized spectral section. In particular, we show that if agraph continuous family of regular self-adjoint operators with compact resolventshas a spectral section, then the family is Riesz continuous.
Contents Introduction Preliminaries I Riesz continuous families Generalized spectral sections Spectral sections Correcting operators Z -graded case Hilbert bundles II Applications Relatively compact deformations
Pseudodifferential operators
Cobordism theorems
Cobordism theorems: Z -graded case Tangential operators on a moving hypersurface III Graph continuous families Semibounded operators
Spectral sections
Generalized spectral sections
Introduction
In this paper we deal with families of regular (that is, closed and densely defined) lin-ear operators with compact resolvents acting between Hilbert spaces and parametri-zed by points of an arbitrary topological space X .Throughout the paper, “Hilbert space” always means a separable complex Hilbertspace of infinite dimension; “projection” always means an orthogonal projection, thatis, a self-adjoint idempotent; B ( H , H ′ ) denotes the space of bounded linear operatorsfrom H to H ′ with the norm topology, K ( H , H ′ ) denotes the subspace of B ( H , H ′ ) consisting of compact operators, and P ( H ) denotes the subspace of B ( H ) = B ( H , H ) consisting of projections. Spectral sections.
The notion of a spectral section was introduced by Melrose andPiazza in [MP ], in order to give a family version of a global boundary conditionof Atiyah-Patody-Singer type. It is convenient to split [MP , Definition ] into twoparts, as we do below.Let A be a regular self-adjoint operator with compact resolvents. A projection P iscalled an r -spectral section for A if( . ) Au = λu = ⇒ (cid:14) Pu = u if λ > rPu = if λ − r In other words,( . ) [ r , + ∞ ) ( A ) P (− r , + ∞ ) ( A ) .Here S denotes the characteristic function of the subset S ⊂ R , and we use thenatural partial order on the set of projections. We also say that P is a spectral sectionfor A with a cut-off parameter r .Let now A = ( A x ) x ∈ X be a family of regular self-adjoint operators with compactresolvents and r : X → R + be a continuous function. A norm continuous family P = ( P x ) x ∈ X of projections is called an r -spectral section for A if, for every x ∈ X , theprojection P x is an r x -spectral section for the operator A x . We also say that P is aspectral section for A with a cut-off function r .Point out that the requirement for the family ( P x ) of projections and for the cut-offfunction r to be continuous is an essential part of the definition of a spectral section.In ( . ) we replaced strict inequalities used by Melrose and Piazza with non-strictones, since it simplifies statements of our results. It influences only the cut-off func-tion and does not change the notion of a spectral section. Generalized spectral sections.
The notion of a generalized spectral section was in-troduced by Dai and Zhang in [DZ] in order to cover both usual spectral sections andthe Calderón projection. A projection P is called a generalized spectral section for aself-adjoint operator A if P − [ , + ∞ ) ( A ) is a compact operator. A norm continuous amily P = ( P x ) x ∈ X of projections is called a generalized spectral section for a family A = ( A x ) of self-adjoint operators if P x is a generalized spectral section for A x forevery x ∈ X . (We omit a requirement from [DZ, Definition . ] for projections P x tobe pseudodifferential, since we work in the general functional-analytic framework inthe most part of the paper.) Compact base spaces.
Let A be a family of first order elliptic self-adjoint differentialoperators over a closed manifold parametrized by points of a compact space X . Aswas shown by Melrose and Piazza in [MP , Proposition ], for such a family A theexistence of a spectral section is equivalent to the vanishing of the index of A in K ( X ) .Recall that the Riesz topology on the set R ( H , H ′ ) of regular operators from H to H ′ isinduced by the bounded transform map f : R ( H , H ′ ) → B ( H , H ′ ) , f ( A ) = A ( + A ∗ A ) − / ,from the norm topology on B ( H , H ′ ) . If a family of elliptic differential operators overa closed manifold has continuously changing coefficients, then the correspondingfamily of regular operators acting between the Hilbert spaces of square-integrablesections of corresponding vector bundles is Riesz continuous.The proof of Melrose and Piazza does not use the specifics of differential operators,so [MP , Proposition ] can be stated in a more general form: . Proposition.
Let A be a Riesz continuous family of self-adjoint regular operators withcompact resolvents parametrized by points of a compact space X . Then the following conditionsare equivalent: . A has a spectral section. . ind ( A ) = ∈ K ( X ) . Melrose and Piazza also proved a Z -graded analog of this result in [MP , Proposi-tion ]. The proofs of these results in [MP , MP ] depend crucially on the base spacebeing compact. Arbitrary base spaces.
The aim of the first part of this paper is to generalize afore-mentioned results of Melrose and Piazza to arbitrary base spaces, not necessarilycompact. In particular, we prove the following generalization of Proposition . . Theorem . . Let A = ( A x ) x ∈ X be a Riesz continuous family of regular self-adjoint opera-tors with compact resolvents acting on fibers of a numerable Hilbert bundle over a topologicalspace X . Then the following conditions are equivalent: . A has a spectral section. . A has a generalized spectral section. . A is homotopic to a family of invertible operators.If P is a generalized spectral section for A , then for every ε > a spectral section Q for A canbe chosen so that k Q − P k ∞ < ε and Q is homotopic to P as a generalized spectral section. e also prove a Z -graded version of this result, see Theorems . and . . It dealswith Cl ( ) spectral sections for odd self-adjoint operators and generalizes [MP , Pro-position ] in the same manner as Theorem . generalizes [MP , Proposition ]. Correcting operators of finite rank.
Melrose and Piazza showed in [MP , Lemma ] that if a self-adjoint family A over a compact base space has a spectral section P ,then there is a correcting family C of self-adjoint finite rank operators such that thecorrected family A ′ = A + C consists of invertible operators and P is the family ofpositive spectral projections for A ′ . They also proved a Z -graded analog in [MP ,Lemma ]. We generalize both these results to arbitrary base spaces in Section . Applications.
In Part II we present a variety of applications illustrating how theresults of Part I can be used. In particular, in Section we generalize the famousCobordism Theorem for Dirac operators to arbitrary, not necessarily compact, basespaces. We show in Theorem . that for an arbitrary family of Dirac type opera-tors over a compact manifold with boundary, the family of symmetrized boundaryoperators has a spectral section. Z / -graded case of this result is given by Theorem . . In Section we consider the family of symmetrized tangential operators ofa varying Dirac type operator along a varying hypersurface and show that such afamily parametrized by pairs (operator, hypersurface) has a spectral section. Graph continuous families.
The Riesz topology on the set of regular operators iswell suited for the theory of differential operators on closed manifolds. However, itis not quite adequate for differential operators on manifolds with boundary: it is un-known, except for several special cases, whether families of regular operators definedby boundary value problems are Riesz continuous. In the context of boundary valueproblems, the better suited topology is the graph topology .The graph topology on R ( H , H ′ ) is induced by the inclusion of R ( H , H ′ ) to P ( H ⊕ H ′ ) taking a regular operator to the orthogonal projection onto its graph. A family ofelliptic operators and boundary conditions with continuously varying coefficientsleads to a graph continuous family of regular operators [P , Appendix A. ].As we saw above, a spectral section of a Riesz continuous family always exists locally.There is a topological obstruction for existence of a global spectral section, which inthe case of a compact base space takes value in the K -group of the base. A graphcontinuous family, however, may admit no spectral section even locally . In fact, Rieszcontinuity is necessary for a local existence of a spectral section, as the followingresult shows. Theorem . . Let X be an arbitrary topological space and A : X → R sa K ( H ) be a graphcontinuous map admitting a spectral section. Then A is Riesz continuous. The situation with generalized spectral sections for graph continuous families is morecomplicated, as we show in Section . On the one hand, a generalized spectralsection does not necessarily exist even when a base space is an interval and/or afamily consists of invertible operators. On the other hand, existence of a generalizedspectral section does not imply Riesz continuity. Preliminaries
In this section we recall some basic facts about regular operators (for more detailedexposition see, for example, [Le, BLP, Ka]) and introduce some designations that areused in the rest of the paper.
Regular operators.
An unbounded operator A from H to H ′ is a linear operatordefined on a subspace D of H and taking values in H ′ ; the subspace D is called thedomain of A and is denoted by dom ( A ) . An unbounded operator A is called closedif its graph is closed in H ⊕ H ′ and densely defined if its domain is dense in H . Itis called regular if it is closed and densely defined. Let R ( H , H ′ ) denote the set of allregular operators from H to H ′ , and let R ( H ) = R ( H , H ) .The adjoint operator of an operator A ∈ R ( H , H ′ ) is an unbounded operator A ∗ from H ′ to H with the domaindom ( A ∗ ) = (cid:8) u ∈ H ′ | there exists v ∈ H such that h Aw , u i = h w , v i for all w ∈ H (cid:9) .For u ∈ dom ( A ∗ ) such an element v is unique and A ∗ u = v by definition. The adjointof a regular operator is itself a regular operator.An operator A ∈ R ( H ) is called self-adjoint if A ∗ = A (in particular, dom ( A ∗ ) = dom ( A ) ). Let R sa ( H ) ⊂ R ( H ) be the subset of self-adjoint regular operators. Operators with compact resolvents.
For a regular operator A ∈ R ( H , H ′ ) , the oper-ator + A ∗ A is regular, self-adjoint, and has dense range. Its densely defined inverse ( + A ∗ A ) − is bounded and hence can be extended to a bounded operator definedon the whole H . A regular operator A ∈ R ( H , H ′ ) is said to have compact resolvents if ( + A ∗ A ) − ∈ K ( H ) and ( + AA ∗ ) − ∈ K ( H ′ ) . We denote by R K ( H ) the subset of R ( H ) consisting of regular operators with compact resolvents.Let R sa K ( H ) = R sa ( H ) ∩ R K ( H ) be the subset of R ( H ) consisting of self-adjoint operatorswith compact resolvents. Equivalently, a self-adjoint regular operator A is an operatorwith compact resolvents if ( A + i ) − is a compact operator. Such an operator has adiscrete real spectrum. Bounded transform.
The bounded transform (or the Riesz map) A f ( A ) = A ( + A ∗ A ) − / defines the inclusion of the set R ( H , H ′ ) of regular operators to the unit ballin the space B ( H , H ′ ) of bounded operators, with the image f ( R ( H , H ′ )) = (cid:8) a ∈ B ( H , H ′ ) | k a k and − a ∗ a is injective (cid:9) .The inverse map is given by the formula f − ( a ) = a ( − a ∗ a ) − / .If A is self-adjoint, then so is f ( A ) . If A has compact resolvents, then a = f ( A ) isessentially unitary (that is, both − a ∗ a and − aa ∗ are compact operators). Essentially unitary operators.
Let B eu ( H , H ′ ) be the subspace of B ( H , H ′ ) consistingof essentially unitary operators. The bounded transform takes R K ( H , H ′ ) to f (cid:0) R K ( H , H ′ ) (cid:1) = (cid:8) a ∈ B eu ( H , H ′ ) | k a k and − a ∗ a is injective (cid:9) . et B saeu ( H ) be the the subspace of B ( H , H ′ ) consisting of self-adjoint essentially uni-tary operators.In Section and the first part of Section we prove our results simultaneously bothfor regular operators with compact resolvents and for essentially unitary operators.The notion of a generalized spectral section given in the Introduction works as wellfor self-adjoint essentially unitary operators. Equivalently, a projection P is a gener-alized spectral section for a ∈ B saeu ( H ) if ( P − ) − a ∈ K ( H ) (since [ , + ∞ ) ( a ) is acompact deformation of ( a + ) / for a ∈ B saeu ( H ) ).We call a projection P an r -spectral section for a ∈ B saeu ( H ) and r ∈ ( , ) if [ r , + ∞ ) ( a ) P (− r , + ∞ ) ( a ) . Let r : X → ( , ) be a continuous function; we call a norm continuousfamily of projections P = ( P x ) x ∈ X an r -spectral section (or simply a spectral section)for a family a = ( a x ) of self-adjoint essentially unitary operators if P x is an r x -spectralsection for a x for every x ∈ X .A family P = ( P x ) of projections is a generalized spectral section, resp. r -spectralsection for a family A = ( A x ) of self-adjoint regular operators with compact resolventsif and only if P is a generalized spectral section, resp. ( f ◦ r ) -spectral section for thefamily f ◦ A of self-adjoint essentially unitary operators. Two topologies on R ( H , H ′ ) . There are several natural topologies on the set ofregular operators. The two most useful of them are the Riesz topology and thegraph topology. They are induced by the inclusions of R ( H , H ′ ) to B ( H , H ′ ) andto P ( H ⊕ H ′ ) , respectively; see Introduction for details.We will always specify what topology (Riesz or graph) on R ( H , H ′ ) we consider.We will write r R ( H , H ′ ) or g R ( H , H ′ ) for the space of regular operators with Rieszor graph topology, respectively. Alternatively, we will write “Riesz continuous” or“graph continuous” for maps from or to R ( H , H ′ ) and for families of regular opera-tors.On the subset B ( H , H ′ ) ⊂ R ( H , H ′ ) both Riesz and graph topology coincide with theusual norm topology. Therefore, we always consider B ( H , H ′ ) as equipped with thenorm topology. Spectral projections.
The spectrum of an operator A ∈ R sa K ( H ) is discrete and real.For real numbers a < b , the spectral projection [ a , b ] ( A ) is defined as [ f ( a ) , f ( b )] ( f ( A )) ;its range is the subspace of H spanned by eigenvectors of A with eigenvalues inthe interval [ a , b ] . Similarly, [ a , + ∞ ) ( A ) is defined as [ f ( a ) , ] ( f ( A )) and (− ∞ , a ] ( A ) isdefined as [− , f ( a )] ( f ( A )) . The spectral projections for semi-open and open intervalsare defined in the same manner.Let Res ( A ) denote the resolvent set of A . For a compact subspace K ⊂ R , the subset V K = { A ∈ R sa K ( H ) | K ⊂ Res ( A ) } is open in both Riesz and graph topology on R sa K ( H ) .The map V { a , b } → P ( H ) taking A to [ a , b ] ( A ) is both Riesz-to-norm and graph-to- orm continuous. However, the spectral projection maps V a → P ( H ) correspondingto unbounded intervals, A (− ∞ , a ] ( A ) and A [ a , + ∞ ) ( A ) , are only Riesz-to-normcontinuous, but not graph-to-norm continuous. This is the major difference betweenthe two topologies in our context. Part I
Riesz continuous families
Throughout this part, all families of regular operators are supposed to be Riesz con-tinuous. Generalized spectral sections
Homotopy Lifting Property.
Recall that a continuous map G → Z is said to havethe Homotopy Lifting Property for a space X if for every homotopy h : X × [ , ] → Z ,every lifting ˜ h : X × { } → G of h can be continued to a lifting ˜ h : X × [ , ] → G of h .In this section the base space Z is either r R sa K ( H ) or B saeu ( H ) . For the most part of thesection, our reasoning works for both these cases simultaneously.Let Z be one of these two spaces. Let I denote the range of cut-off parameters for Z ,that is, I = R + if Z = R sa K ( H ) and I = ( , ) if Z = B saeu ( H ) .Let G denote the subspace of Z × P ( H ) consisting of pairs ( A , P ) such that P is ageneralized spectral section for A . We consider G as the total space of a fiber bundleover Z , with the projection π G : G → Z taking ( A , P ) to A . . Theorem.
The fiber bundle π G : G → Z is locally trivial and has the Homotopy LiftingProperty for all spaces. Proof.
The family ( Z r ) r ∈ I , Z r = { A ∈ Z | r ∈ Res ( A ) } , is an open covering of Z . Forevery r ∈ I , the map S + r : Z r → P ( H ) defined by the formula S + r ( A ) = [ r , + ∞ ) ( A ) iscontinuous. The restriction of G to Z r is( . ) G| Z r = (cid:8) ( A , P ) | A ∈ Z r and P − S + r ( A ) ∈ K ( H ) (cid:9) .Our next goal is to trivialize S + r locally. To this end, take an open covering ( V Q ) Q ∈ P ( H ) of P ( H ) , where V Q = { P ∈ P ( H ) | k P − Q k < } . By [WO, Proposition . . ] for every Q ∈ P ( H ) there is a continuous map g Q : V Q → U ( H ) such that( . ) P = g Q ( P ) Q ( g Q ( P )) ∗ for every P ∈ V Q .The inverse images Z r , Q = ( S + r ) − ( V Q ) ⊂ Z r , with r running I and Q running P ( H ) ,form an open covering ( Z r , Q ) of Z . We claim that the restriction G r , Q of G to Z r , Q is a rivial bundle with the fiber F Q = { P ∈ P ( H ) | P − Q ∈ K ( H ) } .Indeed, fix an arbitrary pair ( r , Q ) and consider the map g = g Q ◦ S + r : Z r , Q → U ( H ) .It follows from ( . ) that ( g ( A )) ∗ S + r ( A ) g ( A ) = Q does not depend on A ∈ Z r , Q .Together with ( . ) this implies that the map Φ : G r , Q → Z r , Q × F Q , Φ ( A , P ) = ( A , g ( A ) ∗ P g ( A )) is a trivializing bundle isomorphism, which proves the claim.For both Z = r R sa K ( H ) and Z = B saeu ( H ) the topology of Z is induced by the embeddingof Z to B ( H ) , so Z is a metric space. This implies paracompactness of Z [St, Corollary ]. Thus π G is a locally trivial fiber bundle with a paracompact base space. By [Hu,Uniformization Theorem], π G has the Homotopy Lifting Property for all spaces. Thiscompletes the proof of the Proposition. (cid:3) Generalized spectral sections.
Using Theorem . , we now can prove the followingresult. . Theorem.
Let X be an arbitrary topological space and Z be either r R sa K ( H ) or B saeu ( H ) .Let A : X → Z be a continuous map. Then the following conditions are equivalent: . A has a generalized spectral section. . A is homotopic (as a map from X to Z ) to a family of invertible operators. Proof. ( ⇒ ) Let h : X × [ , ] → Z be a homotopy between A = h and an invertiblefamily h . Since h is invertible, it has a spectral section P ( x ) = [ , + ∞ ) ( h ( x )) . Thenthe map ˜ h : X → G given by the formula ˜ h ( x ) = ( h ( x ) , P ( x )) covers h . By Theorem . , ˜ h can be continued to a map ˜ h : X × [ , ] → G covering h . Restriction of ˜ h to X × { } gives the map ˜ h : X → G covering A . The composition of ˜ h with the naturalprojection G → P ( H ) is a generalized spectral section for A .( ⇒ ) Let P : X → P ( H ) be a generalized spectral section for A : X → Z . Then T : X → B ( H ) , T ( x ) = P ( x ) − , is a continuous family of symmetries (that is, self-adjoint unitaries). . Consider first the case Z = B saeu ( H ) . Then A ( x ) − T ( x ) ∈ K ( H ) for every x ∈ X .Therefore, h : X × [ , ] → B saeu ( H ) , h t ( x ) = ( − t ) A ( x ) + tT ( x ) is a homotopy from h = A to h = T , with T ( x ) invertible for every x ∈ X . . Let now Z = r R sa K ( H ) . The composition a = f ◦ A : X → B saeu ( H ) is continuousand a ( x ) − T ( x ) ∈ K ( H ) for every x ∈ X . One is tempted to apply f − to the linearhomotopy between a and T , as above. But the image of f ,( . ) f ( R sa K ( H )) = { b ∈ B saeu ( H ) | k b k and − b is injective } ,does not contain T ( x ) , so this naive idea does not work. To fix it, we replace T byits compact deformation T ′ lying in the image of f . Let us fix a strictly positive ompact operator K ∈ K ( H ) of norm less than . For example, one can identify H with l ( N ) and take the diagonal operator K = diag ( , , , . . . ) . Then T ′ ( x ) =( − K ) T ( x )( − K ) is self-adjoint and invertible, k T ′ ( x ) k , and T ′ ( x ) − T ( x ) ∈ K ( H ) .Let a t ( x ) = ( − t ) a ( x ) + tT ′ ( x ) be the linear homotopy between a = a and a = T ′ .By definition, a ( x ) = f ( A ( x )) lies in the image of f . For every t ∈ ( , ] , x ∈ X ,and ξ ∈ H \ { } we get k a t ( x ) ξ k < k ξ k , so − a t ( x ) is injective and thus a t ( x ) liesin the image of f . Applying f − to a t ( x ) , we obtain a homotopy h : X × [ , ] → R sa K ( H ) , h t ( x ) = f − ( a t ( x )) , connecting h = A with an invertible family f − ( T ′ ) . Thiscompletes the proof of the theorem. (cid:3) Spectral sections
Fiber homotopy equivalence.
Let Z be the space R sa K ( H ) equipped with the Riesztopology, and let S be the subspace of Z × P ( H ) × R + consisting of triples ( A , P , r ) such that P is an r -spectral section for A . We consider S as the total space of a fiberbundle over Z , with the projection π S : S → Z taking ( A , P , r ) to A . . Theorem.
The bundle map ι : S → G taking ( A , P , r ) to ( A , P ) is a fiber homotopyequivalence. Moreover, for every ε > , a fiber-homotopy inverse bundle map ϕ = ϕ ε : G → S can be chosen so that k Q − P k < ε for every ( A , P ) ∈ G , ( A , Q ) = ι ◦ ϕ ( A , P ) . As an immediate corollary of this theorem we get the following result. . Corollary.
The fiber bundle π S : S → Z has the Weak Homotopy Lifting Property for allspaces (that is, for any homotopy h : X × [ , ] → Z and for any lifting ˜ h : X × { } → S of h , there is a lifting X × [ , ] → S of h whose restriction to X × { } is vertically homotopic to ˜ h ). Proof of Theorem . . Let us fix δ ∈ ( , ) such that δ ε/ .Let ( A , P ) be an arbitrary element of G . For r > , define the spectral projections( . ) S − r ( A ) = (− ∞ , r ] ( A ) , S ◦ r ( A ) = (− r , r ) ( A ) , and S + r ( A ) = [ r , + ∞ ) ( A ) .We approximate P by bounded self-adjoint operators( . ) T r ( A , P ) = S + r ( A ) + S ◦ r ( A ) P S ◦ r ( A ) .Thus defined map T r is continuous on the open set { ( A , P ) ∈ G | ± r ∈ Res ( A ) } . Equal-ity ( . ) can be written equivalently as( . ) T r ( A , P ) = S + ( A ) + S ◦ r ( A ) (cid:0) P − S + ( A ) (cid:1) S ◦ r ( A ) .Consider the family( . ) U = ( U r ) r> , U r = { ( A , P ) ∈ G | ± r ∈ Res ( A ) and k T r ( A , P ) − P k < δ } f open subsets of G . We claim that U is an open covering of G . Indeed, let ( A , P ) bean arbitrary point of G . Since P is a generalized spectral section for A , the difference P − S + ( A ) is a compact operator. The net { S ◦ r ( A ) } r> is an approximate unit for K ( H ) .Therefore, the second summand in the right hand side of ( . ) has a limit P − S + ( A ) ,and T r ( A , P ) − P → as r → + ∞ . It follows that ( A , P ) lies in U r for r large enough,and thus U covers G .The topology of Z × P ( H ) is induced by its embedding to B ( H ) × B ( H ) , so Z × P ( H ) ,as well as its subspace G , is a metric space. It follows from [St, Corollary ] that G is paracompact. Hence there is a partition of unity ( u i ) subordinated to U , withsupp ( u i ) ⊂ U r i . (By a partition of unity we always mean locally finite partition.) Forevery ( A , P ) ∈ G we define a bounded self-adjoint operator T ( A , P ) by the formula( . ) T ( A , P ) = X u i ( A , P ) · T r i ( A , P ) .The restriction of T r to U r is continuous, so all the summands in ( . ) are continuous,and thus T itself is continuous as a map from G to B ( H ) .If u i ( A , P ) = , then ( A , P ) ∈ U r i and k T r i ( A , P ) − P k < δ . Therefore,( . ) k T ( A , P ) − P k < δ for every ( A , P ) ∈ G .The spectrum of P is { , } , so the last inequality implies that the spectrum of T ( A , P ) lies in Λ = [− δ , δ ] ∪ [ − δ , + δ ] . Since δ < , these two intervals are disjoint. Thefunction [ , + ∞ ) is continuous (and even smooth) on Λ , so the projection( . ) Q ( A , P ) = [ , + ∞ ) ( T ( A , P )) continuously depends on ( A , P ) . Moreover, k Q ( A , P ) − T ( A , P ) k δ , which togetherwith ( . ) provides an estimate k Q ( A , P ) − P k < ε for every ( A , P ) ∈ G .Let ( A , P ) ∈ G and r = max { r i | u i ( A , P ) = } . Then T ( A , P ) = ⊕ T ◦ ( A , P ) ⊕ andthus Q ( A , P ) = ⊕ Q ◦ ( A , P ) ⊕ with respect to the orthogonal decomposition( . ) H = H − ⊕ H ◦ ⊕ H + , where H α = Im ( S αr ( A )) for α ∈ { + , − , ◦ } .In other words, Q ( A , P ) is an r -spectral section for A .For every point ( A , P ) ∈ G choose its neighbourhood V A , P ⊂ G intersecting only afinite number of inverse images u − i ( , ] . Let ( v j ) be a partition of unity subordinatedto the covering V = ( V A , P ) of G . Let R j = max (cid:10) r i | u − i ( , ] ∩ v − j ( , ] = ∅ (cid:11) .We define a continuous map R : G → R + by the formula( . ) R ( A , P ) = X v j ( A , P ) · R j .Then R ( A , P ) > max { r i | u i ( A , P ) = } for every ( A , P ) ∈ G , so Q ( A , P ) is an R ( A , P ) -spectral section for A . inally, we define the map ϕ : G → S by the formula ϕ ( A , P ) = ( A , Q ( A , P ) , R ( A , P )) .It remains to show that ϕ is homotopy inverse to ι as a bundle map. To this end,consider the map h : G × [ , ] → P ( H ) given by the formula( . ) h t ( A , P ) = [ , + ∞ ) ( tP + ( − t ) T ( A , P )) .Since k ( tP + ( − t ) T ( A , P )) − P k k T ( A , P ) − P k < δ < / , the same argument asin the proof of continuity of ( . ) shows that h is continuous. Obviously, h ( A , P ) = Q ( A , P ) and h ( A , P ) = P . Thus the formula ( A , P ) ( A , h t ( A , P )) defines a bundlehomotopy between ι ◦ ϕ and Id G .To construct a homotopy between ϕ ◦ ι and Id S , we use three auxiliary homotopies.We will write Q and R instead of Q ( A , P ) and R ( A , P ) for brevity. The first homotopy h ′ t ( A , P , r ) = ( A , Q , R + tr ) connects ϕ ◦ ι ( A , P , r ) = ( A , Q , R ) with ( A , Q , R + r ) . Thesecond one h ′′ t ( A , P , r ) = ( A , h t ( A , P ) , R + r ) connects ( A , Q , R + r ) with ( A , P , R + r ) .The third one h ′′′ t ( A , P , r ) = ( A , P , ( − t ) R + r ) connects ( A , P , R + r ) with ( A , P , r ) .The concatenation of h ′ , h ′′ , and h ′′′ is a desired bundle homotopy between ϕ ◦ ι andId S . This completes the proof of the theorem. (cid:3) Spectral sections.
Using Theorem . , we immediately obtain the following result. . Theorem.
Let X be an arbitrary topological space and A : X → R sa K ( H ) be a Riesz con-tinuous map. Then the following conditions are equivalent: . A has a spectral section. . A has a generalized spectral section. . A is homotopic, via a Riesz continuous homotopy X × [ , ] → R sa K ( H ) , to a family ofinvertible operators.If P is a generalized spectral section for A and ε > , then a spectral section Q for A can bechosen so that k Q − P k ∞ < ε and Q is homotopic to P as a generalized spectral section. Proof. ( ⇒ ) is trivial.( ⇔ ) follows from Theorem . .( ⇒ ) and the last part of the theorem follow from Theorem . . Indeed, if ε > and P is a generalized spectral section for A , then the map x ϕ ε ( A x , P x ) =: ( A x , Q x , r x ) defines an r -spectral section Q for A . Moreover, k Q x − P x k < ε for every x ∈ X .Since ι ◦ ϕ ε and Id G are vertically homotopic, Q and P are homotopic as generalizedspectral sections. This completes the proof of the theorem. (cid:3) . Remark.
For a self-adjoint family A x parametrized by points of a compact space X ,Melrose and Piazza showed in [MP , Proposition ] that if the set of spectral sectionsfor A is non-empty, then it contains “arbitrary small” and “arbitrary large” spectralsections, in the following sense: for every given s ∈ R , there are spectral sections P and Q such that P x S x Q x for every x ∈ X , where S x = [ s , + ∞ ) ( A x ) . This propertyis no longer true in the general case of a non-compact base space, as the followingsimple example shows. . Example.
Fix A ∈ R sa K ( H ) and consider the family A x = A + x parametrized byreal numbers x ∈ X = R . The family A admits a spectral section. Indeed, the constantmap taking every x ∈ X to [ , + ∞ ) ( A ) is an r -spectral section for A , where r : X → R + is an arbitrary function satisfying condition r ( x ) > | x | .Suppose now that A is unbounded from above. Then, for any given s ∈ R , A hasno spectral section P dominated by the family S = ( S x ) as above. Indeed, supposethat P : X → P ( H ) is such a spectral section. Then P x [ s , + ∞ ) ( A + x ) = [ s − x , + ∞ ) ( A ) .For x s , let P ′ x be the restriction of P x to the range H ′ of [ , + ∞ ) ( A ) . Then ( P ′ x ) is acontinuous one-parameter family of projections in H ′ parametrized by points of theray (− ∞ , s ] . The kernels of P ′ x are finite-dimensional; by continuity, their dimensionsshould be independent of x . On the other hand, the dimension of Ker ( P ′ x ) is boundedfrom below by the rank of [ , s − x ) ( A ) , which goes to infinity as x → − ∞ . Thiscontradiction shows that such a spectral section P does not exist.Similar argument shows that if A is unbounded from below, then A has no spectralsection dominating the family S . Correcting operators
Melrose and Piazza showed in [MP , Lemma ] that if a self-adjoint family A over acompact base space admits a spectral section P , then A admits a finite rank correctionto an invertible family A ′ such that P is the family of positive spectral projectionsfor A ′ . They also proved a Z -graded analog of this result in [MP , Lemma ].Their proofs are based on the existence of “arbitrarily small” and “arbitrarily large”spectral sections in the sense of Remark . . However, for a non-compact base spacethere may be no such spectral sections, as Example . shows. In this section wegeneralize [MP , Lemma ] and [MP , Lemma ] to arbitrary base spaces using adifferent method. Correcting operators.
It is convenient to have a special term for operators consid-ered by Melrose and Piazza. Let A be a self-adjoint regular operator with compactresolvents. We say that a self-adjoint bounded operator C is an r -correcting operator (or simply a correcting operator) for A if the sum A + C is invertible and the range of C lies in the range of (− r , r ) ( A ) . We also say that C is r -correcting the operator A .Obviously, if C is an r -correcting operator for A , then( . ) P = [ , + ∞ ) ( A + C ) is an r -spectral section for A . We say that a correcting operator C agrees with a spectralsection P if ( . ) holds.These notions are generalized to the family case in a natural way. We say that a normcontinuous family C : X → B sa ( H ) is correcting a family A : X → R sa K ( H ) , or that C isa correcting family for A , if there is a continuous function r : X → R + such that C x s r x -correcting the operator A x for every x ∈ X . We also call such a family C an r -correcting family for A . We say that a correcting family C agrees with a spectral section P if P x = [ , + ∞ ) ( A x + C x ) for every x ∈ X . Correcting operators and spectral sections.
Let D be the subspace of r R sa K ( H ) × B sa ( H ) × R + consisting of triples ( A , C , r ) such that C is an r -correcting operator for A .Recall that S denotes the subspace of r R sa K ( H ) × P ( H ) × R + consisting of triples ( A , P , r ) such that P is an r -spectral section for A . There is a natural map( . ) π : D → S , π ( A , C , r ) = ( A , P , r ) , where P = [ , + ∞ ) ( A + C ) .The fibers of π considered as subsets of B sa ( H ) are convex. . Proposition.
The map π : D → S is continuous. Proof.
The map S : D → P ( H ) taking ( A , C , r ) to (− r , r ) ( A ) is continuous on D r = { ( A , C , r ) ∈ D | ± r ∈ Res ( A ) } . The projection P = [ , + ∞ ) ( A + C ) can be written asan orthogonal sum [ , + ∞ ) ( SAS + C ) + [ r , + ∞ ) ( A ) . The operator SAS + C is boundedand continuously depends on ( A , C , r ) ∈ D r . Therefore, P depends continuously on ( A , C , r ) ∈ D r . Since the open sets D r cover D , ( . ) is continuous. (cid:3) . Theorem.
The map π : D → S has a section. Such a section ( A , P , r ) ( A , γ ( A , P , r ) , r ) can be chosen so that k γ ( A , P , r ) k < r for every ( A , P , r ) ∈ S . Proof.
Let us fix a continuous function ψ : R → [ , ] which is equal to on (− ∞ , ] and to on [ , + ∞ ) . Our construction of the map γ : S → B sa ( H ) depends on thechoice of such a function ψ .Let ( A , P , r ) ∈ S . We take Q = − P , A + = PAP , and A − = QAQ . The operators r + A + and r − A − are positive and invertible, and the sum A + + A − is orthogonal.The difference( . ) C ′ = ( A + + A − ) − A = − PAQ − QAP is a self-adjoint operator of norm less than r with the range of C ′ lying in the range H ◦ of (− r , r ) ( A ) . Let( . ) C + = P ψ (cid:0) r − A + (cid:1) P , C − = − Q ψ (cid:0) − r − A − (cid:1) Q , and C ′′ = r (cid:0) C + + C − (cid:1) .The range of C ′′ lies in H ◦ and k C ′′ k r , so the range of C = C ′ + C ′′ lies in H ◦ and k C k < r .We define γ by the formula γ ( A , P , r ) = C = C ′ + C ′′ , where C ′ and C ′′ are definedby formulas . and . .The function Ψ ( t ) = t + ψ ( t ) is strictly positive on (− , + ∞ ) . The sum A + C can bewritten as A + C = ( A + + A − ) + r (cid:0) C + + C − (cid:1) = r P Ψ ( r − A + ) P − r Q Ψ (− r − A − ) Q , o it is invertible and [ , + ∞ ) ( A + C ) = P .It remains to show that γ is continuous. For λ > , let S λ = { ( A , P , r ) ∈ S | ± λ ∈ Res ( A ) and r < λ } .The projection S = (− λ , λ ) ( A ) and the cut-off operator B = SAS = f − ( Sf ( A ) S ) ∈ B sa ( H ) continuously depend on ( A , P , r ) ∈ S λ . The projections S and P commute,so C ′ = SC ′ S = PBQ + QBP , and thus the restriction of C ′ to S λ is continuous. Therestriction of C ′′ to S λ can be written in a similar manner: C ′′ = r S (cid:2) Pψ ( r − PBP ) P − Qψ (− r − QBQ ) Q (cid:3) S ,which provides continuity of C ′′ on S λ . Open sets S λ cover S when λ runs R + , so both C ′ and C ′′ are continuous on the whole S . Therefore, the sum γ = C ′ + C ′′ is alsocontinuous. This completes the proof of the first two statements of the theorem. (cid:3) . Theorem.
Let X be an arbitrary topological space and A : X → R sa K ( H ) be a Riesz con-tinuous map. Suppose that A admits an r -spectral section P . Then A admits an r -correctingfamily C : X → B sa ( H ) that agrees with P and satisfies k C x k < r x . Proof.
Let γ : S → B sa ( H ) be a map satisfying conditions of Theorem . . Thenthe formula C x = γ ( A x , P x , r x ) determines an r -correcting family for A satisfyingconditions of the theorem. (cid:3) . Theorem.
Let P be a spectral section for a Riesz continuous map A : X → R sa K ( H ) . Thenthe space of all correcting families for A that agree with P is convex and therefore contractible. Proof.
Let C i be an r i -correcting family for A , i = , . Then both C and C are r -correcting families for A , where r = max ( r , r ) . Let C = t C + ( − t ) C , t ∈ [ , ] .Then the range of C ( x ) lies in the range of (− r x , r x ) ( A ( x )) . The sum A + C = t ( A + C ) + ( − t )( A + C ) commutes with P since both A + C and A + C commute with P .The operator A ( x ) + C ( x ) is strictly positive on the range of P ( x ) and strictly negativeon the kernel of P ( x ) . Therefore, C is an r -correcting family for A . This completes theproof of the proposition. (cid:3) Z -graded case Throughout this section H = H ⊕ H will be a Z -graded Hilbert space. Let σ =( − ) ∈ B ( H ) be the symmetry defining the grading. We denote by R K ( H ) thesubset of R sa K ( H ) consisting of odd operators, that is, operators anticommuting with σ . Similarly, we denote by B ( H ) the subspace of B sa ( H ) consisting of odd operators. Spectral sections for odd operators.
The natural inclusion R K ( H ) ֒ → R sa K ( H ) admitsa spectral section P : R K ( H ) → P ( H ) . Indeed, fix a continuous even function ψ : R → R supported on [− r , r ] which does not vanish at zero. For every A ∈ R K ( H ) , theself-adjoint finite rank operator C A = σ ψ ( A ) is r -correcting the operator A , since A + C A ) = A + ψ ( A ) is invertible. Therefore, P A = [ , + ∞ ) ( A + C A ) is an r -spectralsection for A . Moreover, the maps A C A and A P A are continuous on R K ( H ) and thus determine an r -correcting family and an r -spectral section for the inclusion R K ( H ) ֒ → R sa K ( H ) .It follows that, by a trivial reason, every family of odd self-adjoint operators withcompact resolvents has a spectral section. Hence the notion of a spectral section isnot very relevant for such operators. Instead, one should consider spectral sectionsbehaving well with respect to the grading. Such a notion of a Cl ( ) spectral sectionwas introduced by Melrose and Piazza in [MP ].Cl ( ) spectral sections. A norm continuous family of projections P : X → P ( H ) iscalled a Cl ( ) spectral section for a family A : X → R K ( H ) if P is a spectral section for A and satisfies additionally the anticommutation property( . ) σPσ − = − P .Let P ( H ) denote the subspace of P ( H ) consisting of projections P satisfying ( . ).Equivalently, a projection P lies in P ( H ) if the symmetry P − anticommutes with σ . P ( H ) is naturally homeomorphic to the space of unitary operators U ( H , H ) ; thecorresponding homeomorphism( . ) ν : P ( H ) → U ( H , H ) takes P ∈ P ( H ) to v ∈ U ( H , H ) such that P − = (cid:0) v ∗ v (cid:1) . Generalized Cl ( ) spectral sections. Again, we consider two cases in parallel: . Z = r R sa K ( H ) , Z ′ = r R K ( H , H ) , and I = R + ; . Z = B saeu ( H ) , Z ′ = B eu ( H , H ) , and I = ( , ) .Let Z denote the subspace of Z consisting of odd operators. The formula( . ) A ˆ A = (cid:18) A ∗ A (cid:19) defines a natural homeomorphism Z ′ → Z .Both Z ′ and Z are empty if one of H i is finite-dimensional (recall that H itself isinfinite-dimensional). So we will always suppose that both H and H are infinite-dimensional.We define a generalized Cl ( ) spectral section for ˆ A : X → Z as a generalized spectralsection P for ˆ A satisfying ( . ). Equivalently, a norm continuous map P : X → P ( H ) is a generalized Cl ( ) spectral section for ˆ A if a x is a compact deformation of theunitary ν ( P x ) for every x ∈ X , where a x = A x for Z = B saeu ( H ) and a x = f ( A x ) for Z = R sa K ( H ) . n element a ∈ B eu ( H , H ) is a compact deformation of a unitary if and only if theindex of a vanishes. For A ∈ R K ( H , H ) the indices of A and f(A) coincide. Therefore,the index of A ∈ Z ′ vanishes if and only if the operator ˆ A given by formula ( . ) hasa generalized Cl ( ) spectral section. We denote by ¯ Z ′ the subspace of Z ′ consisting ofoperators with vanishing index and by ¯ Z the subspace of Z consisting of operatorsadmitting a generalized Cl ( ) spectral section.Since the index is locally constant on B eu ( H , H ) and thus also on r R K ( H , H ) , ¯ Z ′ isan open and closed subspace of Z ′ (in fact, it is a connected component of Z ′ , but wedo not use its connectedness). Homeomorphism ( . ) takes ¯ Z ′ to ¯ Z , so ¯ Z is an openand closed subspace of Z . . Proposition.
For every A ∈ Z ′ , the following conditions are equivalent: . The index of A vanishes. . The signature of the restriction of σ to the kernel of ˆ A vanishes. . ˆ A has a Cl ( ) spectral section. . ˆ A has a generalized Cl ( ) spectral section. Proof. ( ⇒ ) is trivial.( ⇔ ) is explained above.( ⇔ ) follows from the equalitysign (cid:0) σ | Ker ˆ A (cid:1) = dim (cid:0) H ∩ Ker ˆ A (cid:1) − dim (cid:0) H ∩ Ker ˆ A (cid:1) = dim ( Ker A ) − dim ( Ker A ∗ ) = ind ( A ) .( ⇒ ) If dim Ker A = dim Ker A ∗ , then there is a unitary v ∈ U ( Ker A , Ker A ∗ ) .Let S ∈ P ( H ) be the orthogonal projection onto the kernel of ˆ A and P = (cid:0) v ∗ v (cid:1) ∈ P ( Ker ˆ A ) . Then P = P S + ( , + ∞ ) ( ˆ A ) is a Cl ( ) spectral section for ˆ A . (cid:3) . Proposition.
For every A ∈ Z and r ∈ I , the signatures of the restrictions of σ to Ker ( A ) and to the range of (− r , r ) ( A ) coincide. Proof.
The range V of (− r , r ) ( A ) can be decomposed into the orthogonal sum V = V − ⊕ Ker ( A ) ⊕ V + corresponding to the decomposition of the interval (− r , r ) =(− r , ) ∪ { } ∪ ( , r ) . Since σ anticommutes with A , σ takes V − to V + and vice versa.Therefore, the signature of the restriction of σ to V − ⊕ V + vanishes, and so the signa-tures of σ | V and σ | Ker ( A ) coincide. (cid:3) Homotopy Lifting Property.
Let G denote the subspace of Z × P ( H ) consisting ofpairs ( A , P ) such that P is a generalized spectral section for A . Let U K ( H ) = { u ∈ U ( H ) | u − ∈ K ( H ) } be the group of unitaries which are compact deformation of the identity. . Theorem.
The natural projection G → ¯ Z is a locally trivial principal U K ( H ) -bundle.Both G → ¯ Z and G → Z have the Homotopy Lifting Property for all spaces. roof. . We define the action µ of U K ( H ) on the product ¯ Z × P ( H ) by the formula µ u ( A , P ) = (cid:0) A , ν − ( ν ( P ) u ) (cid:1) , u ∈ U K ( H ) ,where ν : P ( H ) → U ( H , H ) is homeomorphism ( . ). For every P , P ′ ∈ P ( H ) , P − P ′ ∈ K ( H ) is equivalent to ν ( P ) − ν ( P ′ ) ∈ K ( H , H ) , which in turn is equivalentto ν ( P ′ ) − ν ( P ) − ∈ K ( H ) . Therefore, G is fixed by the action of µ , and µ actstransitively on the fibers of G → ¯ Z . Obviously, this action is free. It follows that G → ¯ Z is a principal U K ( H ) -bundle. . The next step of the proof is local triviality of G → ¯ Z . Since this bundle is prin-cipal, it is sufficient to show that it allows a local section over a neighbourhood of anarbitrary point A ∈ ¯ Z . Fix r > such that ± r ∈ Res ( A ) . Let S ◦ r ( A ) and S + r ( A ) bethe projections defined by formulae ( . ). They are continuous on the neighbourhood V r = (cid:8) A ∈ ¯ Z | ± r ∈ Res ( A ) (cid:9) of A . The projections ± σ S ◦ r ( A ) are also continuouson V r , so their ranges are locally trivial vector bundles over V r . Let V ⊂ V r be aneighbourhood of A over which these two vector bundles are trivial; denote theirrestrictions to V by E + and E − . The ranks of E + and E − are equal to the dimen-sions of their fibers over A ; since A ∈ ¯ Z , these ranks coincide. Choose a unitarybundle isomorphism v : E + → E − . Let T = (cid:0) v ∗ v (cid:1) be the corresponding odd sym-metry and P = ( T + ) / ∈ P ( E + ⊕ E − ) the bundle projection. Then the formula A ( A , P ( A ) S ◦ r ( A ) + S + r ( A )) defines a section of G over V . This completes the proofof the second step. . The same reasoning as in the end of the proof of Theorem . shows that a locallytrivial bundle G → ¯ Z has the Homotopy Lifting Property for all spaces. Since ¯ Z isclosed and open in Z , the same is true for G → Z . This completes the proof of thetheorem. (cid:3) . Theorem.
Let X be an arbitrary topological space and Z be either r R sa K ( H ) or B saeu ( H ) .Let A : X → Z be a continuous map. Then the following conditions are equivalent: . A has a generalized Cl ( ) spectral section. . A is homotopic, as a map from X to Z , to a family of invertible operators. Proof.
The proof reproduces completely the proof of Theorem . , with (generalized)spectral sections replaced by (generalized) Cl ( ) spectral sections and Theorem . used instead of Theorem . . The only additional care is needed for the choice of acompact operator K : it should commute with σ . Then T ′ ( x ) is odd and the homotopy h consists of odd operators. (cid:3) Fiber homotopy equivalence.
From now on till the end of the section Z denotes thespace R K ( H ) equipped with the Riesz topology. . Theorem.
The bundle map ι : S → G taking ( A , P , r ) to ( A , P ) is a fiber homotopyequivalence. Moreover, for every ε > , a fiber-homotopy inverse bundle map ϕ = ϕ ε : G → S can be chosen so that k Q − P k < ε for every ( A , P ) ∈ G and ( A , Q ) = ι ◦ ϕ ( A , P ) . Proof.
We will show that the bundle map ϕ from the proof of Theorem . mapsthe subspace G of G to the subspace S of S , and that the restriction of ϕ to G ishomotopy inverse, as a bundle map, to the restriction of ι to S . e use the designations from the proof of Theorem . . It will be convenient touse the following convention: if B is a self-adjoint operator, then we write ˜ B as anabbreviation for B − . We will also use the “sign function” ρ : R \ { } → { − , } , ρ = ( , + ∞ ) − (− ∞ , ) .Let ( A , P ) be an arbitrary element of G . Equality ( . ) can be written equivalently as˜ T r ( A , P ) = ( S + r ( A ) − S − r ( A )) + S ◦ r ( A ) ˜ P S ◦ r ( A ) .Since A and ˜ P are odd, ˜ T r ( A , P ) is also odd. It follows that ˜ T ( A , P ) and ˜ Q ( A , P ) = ρ (cid:0) ˜ T ( A , P ) (cid:1) are odd. Therefore, ϕ takes G to S .Equality ( . ) can be written equivalently as ˜ h t ( A , P ) = ρ (cid:0) t ˜ P + ( − t ) ˜ T ( A , P ) (cid:1) . If ˜ P and ˜ T are odd, then ˜ h t ( A , P ) is also odd. It follows that h maps G × [ , ] to P ( H ) .The reasoning in the rest of the proof of Theorem . shows that the restriction of ϕ to G is homotopy inverse, as a bundle map, to the restriction of ι to S . This completesthe proof of the theorem. (cid:3) . Theorem.
Let X be an arbitrary topological space and A : X → R K ( H ) be a Riesz contin-uous map. Then the following conditions are equivalent: . A has a Cl ( ) spectral section. . A has a generalized Cl ( ) spectral section. . A is homotopic, via a Riesz continuous homotopy X × [ , ] → R K ( H ) , to a family ofinvertible operators.If P is a generalized Cl ( ) spectral section for A and ε > , then a Cl ( ) spectral section Q for A can be chosen so that k Q − P k ∞ < ε and Q is homotopic to P as a generalized Cl ( ) spectral section. Proof. ( ⇒ ) is trivial.( ⇔ ) follows from Theorem . .( ⇒ ) and the last part of the theorem follow from Theorem . , in the same manneras in the proof of Theorem . . (cid:3) Correcting operators.
Recall that in the previous section we denoted by D thesubspace of r R sa K ( H ) × B sa ( H ) × R + consisting of triples ( A , C , r ) such that C is an r -correcting operator for A . Let D be the subspace of D consisting of triples ( A , C , r ) with A and C odd operators. In other words, D is the subspace of r R K ( H ) × B ( H ) × R + consisting of triples ( A , C , r ) such that C is an r -correctingoperator for A .If ( A , C , r ) ∈ D , then P = [ , + ∞ ) ( A + C ) is a Cl ( ) spectral section for A . Therefore,the restriction of π : D → S to D defines a natural projection π : D → S . . Theorem.
The map π : D → S has a section ( A , P , r ) ( A , γ ( A , P , r ) , r ) . It can bechosen so that k γ ( A , P , r ) k < r for every ( A , P , r ) ∈ S . roof. Let γ : S → B sa ( H ) be the map constructed in the proof of Theorem . . Then γ takes S to P ( H ) and thus defines a section satisfying conditions of the theorem.Indeed, if A is odd and P is a Cl ( ) spectral section for A , then the conjugation by σ takes A + to − A − and vice versa. Therefore, both C ′ and C ′′ are odd, and thus C = C ′ + C ′′ is also odd for every ( A , P , r ) ∈ S . (cid:3) . Theorem.
Let X be an arbitrary topological space and A : X → R K ( H ) be a Riesz con-tinuous map. Suppose that A admits a Cl ( ) spectral section P with a cut-off function r .Then A admits an odd r -correcting family C : X → B ( H ) that agrees with P and satisfies k C x k < r x . Proof.
Let γ : S → B sa ( H ) be a map satisfying conditions of Theorem . . Then theformula C x = γ ( A x , P x , r x ) determines an odd r -correcting family for A satisfyingconditions of the theorem. (cid:3) . Theorem.
Let P be a Cl ( ) spectral section for a Riesz continuous map A : X → R K ( H ) .Then the space of all odd correcting families for A that agree with P is convex and thereforecontractible. Proof.
This space is the intersection, inside the vector space C ( X , B sa ( H )) , of thevector subspace C ( X , B ( H )) with the subset of all correcting families for A . The lastsubset is convex by Theorem . , so their intersection is also convex. (cid:3) Hilbert bundles
Let H → X be a Hilbert bundle over X (that is, a locally trivial fiber bundle over X with the fiber a separable Hilbert space H and the structure group U ( H ) ).If a base space X is paracompact, then H is trivial. More generally, every numerableHilbert bundle is trivial. (Recall that a fiber bundle is called numerable if it admitsa local trivialization over a numerable open cover of the base space, that is, an opencover admitting a subordinate partition of unity.) Indeed, for every contractible group G , a principal G -bundle over a point is a universal G -bundle [Do, Theorem . ]. Inparticular, every numerable principal G -bundle E is trivial, and so every fiber bundleassociated with E is trivial. By the Kuiper theorem [Ku], the unitary group U ( H ) iscontractible. Therefore, every numerable Hilbert bundle is trivial.This allows to reformulate results of the previous sections in terms of families ofoperators acting between fibers of Hilbert bundles.A map A : X → R sa K ( H ) is now replaced by a family A = ( A x ) x ∈ X such that A x ∈ R sa K ( H x ) . A (generalized) spectral section for such a family A is a norm continuousfamily P = ( P x ) x ∈ X of projections P x ∈ P ( H x ) such that P x is a (generalized) spectralsection for A x for every x ∈ X . All the other notions are carried over to the Hilbertbundle case in a similar manner. . Theorem.
Let A = ( A x ) x ∈ X be a Riesz continuous family of regular self-adjoint operatorswith compact resolvents acting on fibers of a numerable Hilbert bundle H over a topologicalspace X . Then the following conditions are equivalent: . A has a spectral section. . A has a generalized spectral section. . A is homotopic to a family of invertible operators.If P is a generalized spectral section for A and ε > , then a spectral section Q for A can bechosen so that k Q − P k ∞ < ε and Q is homotopic to P as a generalized spectral section. Proof.
This follows immediately from Theorem . and triviality of H . (cid:3) A Z -graded Hilbert bundle H = H ⊕ H is a locally trivial fiber bundle with thefiber a Z -graded Hilbert space H = H ⊕ H and the structure group U ( H ) ⊕ U ( H ) .Equivalently, a Z -graded Hilbert bundle can be defined as a Hilbert bundle H equipped with a continuous family σ = ( σ x ) of gradings on the fibers H x of H .Cl ( ) spectral sections and generalized Cl ( ) spectral sections for families of oddself-adjoint operators acting on sections of H are defined as in Section . . Theorem.
Let A = ( A x ) x ∈ X be a Riesz continuous family of regular self-adjoint oddoperators with compact resolvents acting on fibers of a Z -graded numerable Hilbert bundle H over a topological space X . Then the following conditions are equivalent: . A has a Cl ( ) spectral section. . A has a generalized Cl ( ) spectral section. . A is homotopic to a family of invertible operators.If P is a generalized Cl ( ) spectral section for A and ε > , then a Cl ( ) spectral section Q for A can be chosen so that k Q − P k ∞ < ε and Q is homotopic to P as a generalized Cl ( ) spectral section. Proof.
As before, we can suppose that both H and H have infinite rank. Then theyboth are trivial, and the statement of the theorem follows from Theorem . . (cid:3) Correcting operators.
Theorems . and . can be formulated in terms of Hilbertbundles as follows. . Theorem.
Let A = ( A x ) x ∈ X be a Riesz continuous family of regular self-adjoint operatorswith compact resolvents acting on fibers of a numerable Hilbert bundle H over a topologicalspace X . Suppose that A admits an r -spectral section P . Then A admits an r -correcting family C = ( C x ) that agrees with P and satisfies k C x k < r x . . Theorem.
Let A = ( A x ) x ∈ X be a Riesz continuous family of regular self-adjoint oddoperators with compact resolvents acting on fibers of a Z -graded numerable Hilbert bundle H over a topological space X . Suppose that A admits a Cl ( ) spectral section P with a cut-off function r . Then A admits an odd r -correcting family C = ( C x ) that agrees with P andsatisfies k C x k < r x . Non-self-adjoint operators.
Passing from odd self-adjoint operators to their chiralcomponents, we obtain the following result. . Theorem.
Let H and H be numerable Hilbert bundles over a topological space X , andlet A = ( A x ) x ∈ X be a Riesz continuous family of regular operators with compact resolventsacting from fibers of H to fibers of H . Then the following conditions are equivalent: . The bounded transform f ◦ A is a compact deformation of a norm continuous family ofunitaries. . A is Riesz homotopic to a family of invertible operators.If this is the case, then there is a norm continuous family C = ( C x ) of finite rank operatorsacting from fibers of H to fibers of H such that A x + C x is invertible for every x ∈ X ,the range of C x lies in the range of [ , r x ) ( A x A ∗ x ) , and the kernel of C x contains the range of [ r x , + ∞ ) ( A ∗ x A x ) for some continuous function r : X → R + . Proof.
The equivalence of conditions ( ) and ( ) follows from the part ( ⇔ ) ofTheorem . applied to the family ˆ A = (cid:0) A ∗ A (cid:1) of regular odd self-adjoint operatorswith compact resolvents acting on fibers of the Z -graded Hilbert bundle H = H ⊕ H .Suppose now that A is Riesz homotopic to a family of invertible operators. Then, byTheorem . , ˆ A has a Cl ( ) spectral section; let R : X → R + be its cut-off function.Applying Theorem . to ˆ A , we get a norm continuous family ˆ C = (cid:0) C ∗ C (cid:1) of oddself-adjoint operators such that ˆ A + ˆ C is invertible and the range of ˆ C lies in the rangeof (− R , R ) ( ˆ A ) (we omit the subscript x for brevity). Since (− R , R ) ( ˆ A ) = [ , R ) ( ˆ A ) = [ , R ) ( A ∗ A ⊕ AA ∗ ) ,the family C and the function r = √ R satisfy conditions of the theorem. (cid:3) Part II
Applications
In this part we present a number of applications illustrating how the results of theprevious sections can be used. Relatively compact deformations
Let H and H ′ be Hilbert spaces. Deformations of a single operator.
We give here only two of possible examples.Clearly, one can write a Z -analog of Theorem . , using Theorems . and . insteadof Theorems . and . ; we omit it since it is quite straightforward. . Theorem.
For a regular operator B : H → H ′ with compact resolvents, let Z B = (cid:8) A ∈ R K ( H , H ′ ) | f ( A ) − f ( B ) ∈ K ( H , H ′ ) (cid:9) . hen there are Riesz-to-norm continuous maps α : Z B → B ( H , H ′ ) and r : Z B → R + suchthat A + α ( A ) is invertible, the range of α ( A ) lies in the range of [ , r x ) ( AA ∗ ) , and the kernelof α ( A ) contains the range of [ r x , + ∞ ) ( A ∗ A ) for every A ∈ Z B . Proof.
The bounded transform f ( B ) is a compact deformation of some unitary u ∈ U ( H , H ′ ) . Therefore, the composition of the bounded transform with the inclusion Z B ֒ → R K ( H , H ′ ) is a compact deformation of a norm continuous (even constant!)family of unitaries Z B ∋ A u . It remains to apply Theorem . . (cid:3) . Theorem.
For a regular self-adjoint operator B ∈ R sa K ( H ) with compact resolvents, let ( . ) Z sa B = Z B ∩ R sa K ( H ) = { A ∈ R sa K ( H ) | f ( A ) − f ( B ) ∈ K ( H ) } ⊂ R sa K ( H ) be the subspace of R sa K ( H ) equipped with the Riesz topology. Let P = [ , + ∞ ) ( B ) be the positivespectral projection of B . Then inclusion ( . ) admits both a spectral section and a correctingfamily. Moreover, for every ε > a spectral section Q can be chosen so that k Q ( A ) − P k < ε for every A ∈ Z sa B . Proof.
The constant map A P is a generalized spectral section for inclusion ( . ).It remains to apply Theorems . and . . (cid:3) Essentially self-adjoint operators.
Recall that a bounded operator a is called essen-tially self-adjoint if a − a ∗ is a compact operator. . Theorem.
Let X be the subspace of R K ( H ) consisting of operators A whose boundedtransform is an essentially self-adjoint operator. Then there are Riesz-to-norm continuousmaps α : X → B ( H ) and r : X → R + such that A + α ( A ) is invertible, the range of α ( A ) liesin the range of [ , r x ) ( AA ∗ ) , and the kernel of α ( A ) contains the range of [ r x , + ∞ ) ( A ∗ A ) forevery A ∈ X . Proof.
Let A ∈ X and a = f ( A ) . Then b = ( a + a ∗ ) / is a self-adjoint operatorof norm and u = u A = b + i √ − b is a unitary. Moreover, both b and u are compact deformations of a . The map X → U ( H ) taking A to u A is Riesz-to-norm continuous. Therefore, the composition of the bounded transform with theembedding X ֒ → R K ( H ) is a compact deformation of a norm continuous family ofunitaries. It remains to apply Theorem . . (cid:3) Essentially odd operators.
Let H be a Z -graded Hilbert space, with the gradinggiven by the symmetry σ . A bounded operator a ∈ B ( H ) is called essentially odd if σa + aσ is a compact operator. . Theorem.
Let X be the subspace of r R sa K ( H ) consisting of operators A whose boundedtransform is an essentially odd operator. Then the natural embedding X ֒ → r R sa K ( H ) admitsboth a spectral section X → P ( H ) and a correcting family X → B sa ( H ) . Proof.
Let A ∈ X and a = f ( A ) . Then b = ( a − σaσ ) / is an odd self-adjoint operatorof norm , u = b + σ √ − b is a symmetry, and P = P A = ( u + ) / is a projection.Moreover, both b and u are compact deformations of a . The map X → P ( H ) taking A to P A is Riesz-to-norm continuous. Therefore, this map is a generalized spectralsection for the embedding X ֒ → r R sa K ( H ) . It remains to apply Theorems . and . . (cid:3) Pseudodifferential operators
We show here several examples of applications of our results to pseudodifferentialoperators over closed manifolds. For simplicity, we restrict ourselves by operatorsacting on sections of a fixed vector bundle over a fixed manifold. Again, we omit the Z -graded case here.Let M be a closed smooth manifold equipped with a smooth positive measure, andlet E , E ′ be smooth Hermitian bundles over M . Let Ψ d ( E , E ′ ) denote the space ofpseudodifferential operators of order d > acting from sections of E to sections of E ′ . We equip it with the topology induced by the inclusion Ψ d ( E , E ′ ) ֒ → B ( H d ( E ) , L ( E ′ )) × B ( H d ( E ′ ) , L ( E )) taking a pseudodifferential operator A to the pair ( A , A t ) , where A t is the operatorformally adjoint to A . By [Le, Proposition . ], the natural inclusion of the subspace Ψ ell d ( E , E ′ ) ⊂ Ψ d ( E , E ′ ) of elliptic operators to R ( L ( E ) , L ( E ′ )) is Riesz continuous. . Theorem.
Let X be a topological space and A : X → Ψ ell d ( E , E ′ ) be a continuous familyof elliptic operators of order d > . Suppose that A is homotopic to a family of invertibleoperators. Then there is a norm continuous family α = ( α x , B ) of smoothing finite rankoperators parametrized by pairs ( x , B ) ∈ X × Ψ d − ( E , E ′ ) such that A x + B + α x , B is invertiblefor every x ∈ X and every B ∈ Ψ d − ( E , E ′ ) . . Remark.
In this theorem, A is homotopic to a family of invertible operators, inparticular, in each of the following cases: . X is contractible. . X is compact and ind ( A ) = ∈ K ( X ) . . The kernel and cokernel of A x have locally constant ranks, and the correspond-ing vector bundles Ker ( A ) and Coker ( A ) over X are isomorphic. Proof.
Let Y = X × Ψ d − ( E , E ′ ) , H= L ( E ) , and H ′ = L ( E ′ ) . The map ˜ A : Y → Ψ ell d ( E , E ′ ) taking ( x , B ) to A x + B is continuous. Therefore, the composed map˜ A : Y → Ψ ell d ( E , E ′ ) ֒ → R K ( H , H ′ ) is Riesz continuous. It is Riesz homotopic to the map Y → R K ( H , H ′ ) taking ( x , B ) to A x via the homotopy ( x , B , t ) A x + tB . Since A is homotopic to a family ofinvertible operators, the same is true for ˜ A .By Theorem . , there is a continuous map α : Y → B ( H , H ′ ) such that ˜ A + α is aninvertible family, the range of α x , B lies in the range V of [ , r ) ( ˜ A x , B ˜ A ∗ x , B ) , and theorthogonal complement of the kernel of α x , B lies in the range V ′ of [ , r ) ( ˜ A ∗ x , B ˜ A x , B ) for some r = r ( x , B ) . Since ˜ A x , B is an elliptic operator of positive order, both V and V ′ are spanned by a finite number of C ∞ -sections. Therefore, α x , B is a smoothingoperator of finite rank. This completes the proof of the theorem. (cid:3) Self-adjoint case.
For M and E as above, let Ψ sa d ( E ) denote the subspace of Ψ d ( E ) consisting of symmetric operators, and let Ψ ell,sa d ( E ) = Ψ ell d ( E ) ∩ Ψ sa d ( E ) . . Theorem.
Let X be a topological space and A : X → Ψ ell,sa d ( E , E ′ ) be a continuous familyof symmetric elliptic operators of order d > . Suppose that A is homotopic to a family ofinvertible operators. Then the map ˜ A : Y = X × Ψ sa d − ( E ) → Ψ ell,sa d ( E ) , ( x , B ) A x + B admits both a spectral section P : Y → Ψ sa ( E ) and a correcting family with smoothing cor-recting operators. Proof.
The proof is completely similar to the proof of Theorem . ; one only needs touse Theorems . and . instead of Theorem . . (cid:3) . Remark.
In this theorem, A is homotopic to a family of invertible operators, inparticular, in each of the following cases: . X is contractible. . X is compact and ind ( A ) = ∈ K ( X ) . . The kernel of A x has locally constant rank. Cobordism theorems
Calderón projection.
Let M be a smooth compact Riemannian manifold with non-empty boundary ∂M and E , E ′ be smooth Hermitian vector bundles over M . Denoteby E ∂ and E ′ ∂ the restrictions of E and E ′ to ∂M .Let A be a first order elliptic differential operator over M acting from sections of E to sections of E ′ . The space of Cauchy data of A is the closure in H = L ( ∂M ; E ∂ ) ofthe subspace consisting of restrictions to ∂M of all smooth solutions of the equation Au = . The orthogonal projection Q = Q ( A ) ∈ P ( H ) onto the Cauchy data space iscalled the (orthogonal) Calderón projection of A ; it is a pseudodifferential operator ofzeroth order.At the points of the boundary the operator A can be written as( . ) A = − iJ ( ∂ z + B ) ,where z is the normal coordinate, J ∈ Iso ( E ∂ , E ′ ∂ ) is the conormal symbol of A , and B is a first order elliptic differential operator over ∂M acting on sections of E ∂ . Suchan operator B is called the tangential operator of A along the boundary , or simply the boundary operator of A .Suppose that the principal symbol of B is self-adjoint, that is B − B t is a bundle en-domorphism (here B t denotes the operator formally adjoint to B ). Then the Calderónprojection Q ( A ) has the same principal symbol as the positive spectral projection [ , + ∞ ) ( B + B t ) . In other words, Q ( A ) is a generalized spectral section for the sym-metrized tangential operator ˜ B = ( B + B t ) / . eneral cobordism theorem. Let M → X be a locally trivial fiber bundle over aparacompact Hausdorff space X , with the typical fiber a smooth compact connectedmanifold M with non-empty boundary. Let E and E ′ be complex vector bundles over M . We suppose that every fiber M x is equipped with a Riemannian metric and that E x and E ′ x are equipped with structures of smooth Hermitian vector bundles over M x for every x ∈ X . We also suppose that these structures and metrics continuouslydepend on x ∈ X . We consider E and E ′ as two families ( E x ) , ( E ′ x ) of vector bundlesover a family ( M x ) of manifolds parametrized by points x ∈ X . . Theorem.
Let A = ( A x ) x ∈ X be a family of first order elliptic differential operatorsparametrized by points of a paracompact Hausdorff space X , with A x acting from sections of E x to sections of E ′ x . Suppose that the principal symbols b x of the boundary operators B x areself-adjoint, as above. Suppose, moreover, that both b x and the Calderón projection Q ( A x ) continuously depend on x . Let ˜ B = ( ˜ B x ) be a continuous family of first order symmetricoperators over ∂ M x having b x as their principal symbols. Then ˜ B admits both a spectralsection and a correcting family, with smoothing correcting operators. Proof.
The operators ˜ B x act on the fibers H x = L ( ∂ M x ; E x ) of the Hilbert bundle H over X . The family ˜ B is Riesz continuous. The Calderón projection Q x = Q ( A x ) is a generalized spectral section for ˜ B x . Since the map x Q x is continuous, thefamily ( Q x ) of projections is a generalized spectral section for ˜ B . It remains to applyTheorems . and . . Since the range of every correcting operator lies in the span ofa finite number of C ∞ -sections (namely, eigenvectors of ˜ B x ), all correcting operatorsare smoothing. (cid:3) Unique Continuation Properties.
There are different criteria for continuity of thefamily of Calderón projections for different classes of operators. One of such criteriaconvenient for our purposes is [BLZ, Corollary . ] of Booss-Bavnbek, Lesch, andZhu. It concerns families A = ( A x ) of first order elliptic operators satisfying, togetherwith their formally adjoints, the Weak inner Unique Continuation Property (weakinner UCP). Recall that a first order operator A is said to have weak inner UCP if theonly solution of the equation Au = vanishing on the boundary is the trivial solution u = .Another useful property is the Weak Unique Continuation Property (weak UCP). Anoperator A over a connected manifold M is said to have weak UCP if any solution ofthe equation Au = which vanishes on an open subset of M vanishes on the whole M .Let M be a smooth connected Riemannian manifold, not necessarily compact, and E , E ′ be smooth Hermitian vector bundles over M . Denote by W ( E , E ′ ) the set offirst order elliptic differential operators A : C ∞ ( M ; E ) → C ∞ ( M ; E ′ ) whose principalsymbol a satisfies the following condition:The fiber endomorphism ia ( ξ ) − a ( η ) ∈ End ( E x ) is self-adjointfor every pair of orthogonal cotangent vectors ξ , η ∈ T ∗ x M , x ∈ M .( . ) . Proposition.
Every A ∈ W ( E , E ′ ) has the Weak Unique Continuation Property. roof. We follow the line of the proof of weak UCP for perturbed Dirac type operatorsin [BBB, Theorem . ], but write it in more detail.We can suppose without loss of generality that M has no boundary (otherwise replace M by M \ ∂M ). Suppose that a nontrivial solution u of Au = vanishes on a non-empty open subset of M . Let V be the union of all open subsets of M where u vanishes, and let M ′ = supp ( u ) = ∅ be the complement of V in M . . We claim that there is a point p ∈ V such that the injectivity radius of p is greaterthan the distance from p to M ′ , inj ( p ) > dist ( p , M ′ ) . To show this, choose x ∈ M ′ ∩ V ,and let r = inj ( x ) . Since the injectivity radius function is lower-semicontinuous, thereis a δ ∈ ( , r/ ) such that the injectivity radius is greater than r/ for all points ofthe open ball B δ ( x ) = { y ∈ M | dist ( x , y ) < δ } . Since x lies in the boundary of V ,the intersection V ∩ B δ ( x ) is non-empty; let p be a point in this intersection. Thendist ( p , M ′ ) dist ( p , x ) < r/ and inj ( p ) > r/ . . Let p ∈ V be such a point that r = inj ( p ) > dist ( p , M ′ ) = d . Then the openball B d ( p ) is contained in V and the larger open ball B r ( p ) can be equipped with(geodesical) spherical coordinates. It follows from [BB, Lemmata and ] that u vanishes on some intermediate ball B R ( p ) with d < R < r , that is B R ( p ) ⊂ V . On theother hand, the radius of B R ( p ) is greater than the distance from p to M ′ , so B R ( p ) intersects M ′ . This contradiction shows that A satisfies weak UCP, which completesthe proof of the proposition. (cid:3) Dirac type operators.
Recall that a first order operator A with the principal symbol a is called a Dirac type operator if a ( ξ ) ∗ a ( ξ ) = k ξ k · Id = a ( ξ ) a ( ξ ) ∗ for every ξ ∈ T ∗ M .Every Dirac type operator acting from sections of E to sections of E ′ is an element of W ( E , E ′ ) , but not vice versa.In the first version of this preprint [P ], we used Proposition . to prove a cobor-dism theorem for families of operators of the class W ( E , E ′ ) , see [P , Theorem . ].As it happens, that result is not really more general than a cobordism theorem forDirac type operators. Indeed, composing A with a bundle automorphism of E ′ doesnot affect the boundary operator of A . The following proposition shows that everyoperator A ∈ W ( E , E ′ ) can be obtain from a Dirac type operator by such a composi-tion. . Proposition.
Let A be a first order operator acting from sections of E to sections of E ′ .Then the following two conditions are equivalent: . A ∈ W ( E , E ′ ) , . A = T D , where T is a bundle automorphism of E ′ and D is a Dirac type operator. Proof.
Condition ( . ) can be equivalently written as follows:( . ) a ( ξ ) a ( η ) ∗ + a ( η ) a ( ξ ) ∗ = for every ξ ⊥ η ∈ T ∗ x M , x ∈ M .Left hand side of ( . ) is an End ( E ′ x ) -valued symmetric bilinear form on T ∗ x M , α ( ξ , η ) = a ( ξ ) a ( η ) ∗ + a ( η ) a ( ξ ) ∗ . ⇔ ) If A = T D , then α ( ξ , η ) = h ξ , η i T T ∗ satisfies ( . ), so A ∈ W ( E , E ′ ) .( ⇔ ) Let A ∈ W ( E , E ′ ) . For arbitrary non-zero ξ , η ∈ T ∗ x M , we write ξ = ξ ′ + tη with ξ ′ orthogonal to η . This gives( . ) α ( ξ , η ) = tα ( η , η ) = h ξ , η i S ( η ) ,where S ( η ) = α ( η , η ) / k η k = a ( η ) a ( η ) ∗ / k η k ∈ End ( E ′ x ) is a homogenous function on S ∗ x M \ { } of degree . Since α ( ξ , η ) = α ( η , ξ ) , identity( . ) impies S ( ξ ) = S ( η ) for every pair of non-orthogonal vectors ξ , η ∈ S ∗ x M . Forevery non-zero ξ , η ∈ T ∗ x M there is a third vector ζ ∈ S ∗ x M which is non-orthogonal toboth ξ and η , so that S ( ξ ) = S ( ζ ) = S ( η ) . Therefore, S ( η ) is independent of η ∈ S ∗ x M and depends only on x , S ( η ) = S x . Moreover, S x is positive for every x ∈ M . Let T x be the positive square root of S x / . Then T is a smooth bundle automorphism of E ′ . The equality a ( ξ ) a ( ξ ) ∗ = k ξ k T x T ∗ x implies ( T − x a ( ξ ))( T − x a ( ξ )) ∗ = k ξ k · Id forevery ξ ∈ T ∗ x M . Since T − x a ( ξ ) is invertible for ξ = , this implies the second identity ( T − x a ( ξ )) ∗ ( T − x a ( ξ )) = k ξ k · Id. Therefore, T − A is a Dirac type operator. (cid:3) . Proposition.
Let M be a smooth connected Riemannian manifold with non-emptyboundary and A be a Dirac type operator over M . Then A has the weak inner UniqueContinuation Property. Proof.
The operator A admits an extension across the boundary, that is, A is therestriction to M of some Dirac type operator ˜ A over ˜ M , where ˜ M is a smooth Rie-mannian manifold without boundary containing M as a smooth submanifold of codi-mension zero. Indeed, let A act from sections of E to sections of E ′ . The symbol a of A determines the structure of a Clifford module over T ∗ M ⊕ R on E ⊕ E ′ , withthe cotangent vector ξ acting as ˆ a ( ξ ) = (cid:16) a ( ξ ) ∗ a ( ξ ) (cid:17) and the unit vector in the addi-tional R -direction acting as Id E ⊕ (− Id E ′ ) . Such a structure can be smoothly extendedacross the boundary of M , to an external collar neighbourhood of ∂M equipped witha metric in a compatible way. Such an extension gives rise to the symbol ˜ a of a Diractype operator extending a and acting from sections of ˜ E to sections of ˜ E ′ , where ˜ E and ˜ E ′ are smooth Hermitian vector bundles over ˜ M extending E and E ′ . Given anextension of the symbol, the whole operator A can be smoothly extended to ˜ M usinga partition of unity.Let u ∈ L ( M ; E , E ′ ) be in the kernel of A and let ˜ u ∈ L ( ˜ M ; ˜ E , ˜ E ′ ) be the extension of u to ˜ M \ M by zero. Suppose that u vanishes on the boundary ∂M . Then the Greenformula for A implies that ˜ u is a weak solution of ˜ A . Since ˜ A is elliptic, ˜ u is smooth.By Proposition . , the operator ˜ A has weak UCP. Since ˜ A ˜ u = and ˜ u vanishes onthe open subset ˜ M \ M , we get ˜ u ≡ and thus u = ˜ u | M ≡ . Therefore, A has weakinner UCP. (cid:3) Cobordism theorem for Dirac type operators.
The boundary operator of a Diractype operator is again a Dirac type operator; moreover, it has a self-adjoint principalsymbol. Applying Theorem . to this situation, we obtain the following generaliza-tion of [MP , Section , Corollary]. . Theorem.
Let X be a paracompact Hausdorff space and M , E , E ′ be as in Theorem . . Let A = ( A x ) x ∈ X be a family of Dirac type operators such that, in every local chart,all the coefficients of the operator A x , together with their first derivatives in the M x -direction,continuously depend on x ∈ X . Then the family ˜ B = ( ˜ B x ) of symmetrized boundary operatorsadmits both a spectral section and a correcting family, with smoothing correcting operators. Proof.
The formal conjugate of a Dirac type operator is again a Dirac type operator.By Proposition . , both A x and A tx have weak inner UCP. The family ( A x , iJ x ) iscontinuous by x ∈ X with respect to the strong metric of [BLZ, Definition . ]. By[BLZ, Corollary . ], the Calderón projections Q ( A x ) continuously depend on x . Itremains to apply Theorem . . (cid:3) . Remark.
While the statement of Theorem . does not distinguish betweeneven- and odd-dimensional manifolds M x , the theorem is really interesting only inthe even-dimensional case.If a manifold is odd-dimensional, then its boundary is a closed manifold of evendimension. At the same time, an arbitrary family of symmetric Dirac type operators B = ( B x ) over a closed oriented manifold N of dimension k (or, more generally, overa family N = ( N x ) of such manifolds parametrized by X ) admits a spectral sectionand a correcting family, regardless of whether B is cobordant to zero or not.Indeed, let B be a symmetric Dirac type operator over N with the symbol b . Let σ be the “normalized orientation”, σ y = i k b ( ξ ) · . . . · b ( ξ k ) , where ( ξ , . . . , ξ k ) is apositively oriented orthonormal basis of T ∗ y N , y ∈ N . Then σ is a bundle symmetryanticommuting with b , so Theorem . can be applied. In more concrete terms, theoperators ¯ B = ( B − σBσ ) / and B ′ = ¯ B + σ are symmetric and have the same symbolas B . Moreover, ¯ B anticommutes with σ , so ( B ′ ) = ¯ B + is invertible and thus B ′ itself is invertible. Therefore, every family B = ( B x ) of symmetric Dirac typeoperators over a family N = ( N x ) of closed even-dimensional manifolds is homotopicto a family B ′ = ( B ′ x ) of invertible operators and thus admits a spectral section and acorrecting family by Theorems . and . .A relevant cobordism theorem in this case should take into account the grading σ and state the existence of a Cl ( ) spectral section for the family ¯ B of odd operators.We perform this in the next section, see Theorem . . Cobordism theorems: Z -graded case General cobordism theorem.
Let A be a first order symmetric operator acting onsections of E , B be its boundary operator, and J = a ( n ) be the conormal symbol of A .Then J is self-adjoint and the operator JB + B t J has zeroth order (that is, JB + B t J isa bundle endomorphism). Suppose, in addition, that B has self-adjoint symbol . Thenboth B − B t and JB + BJ are bundle endomorphisms.Let σ be a self-adjoint unitary bundle automorphism of E ∂ defined by the formula( . ) σ = J · | J | − = J · ( J ) − / . hen σB + Bσ is also a bundle endomorphism. Indeed, the symbol b of B satisfiesthe anticommutation relation( . ) J ( y ) b ( ξ ) + b ( ξ ) J ( y ) = for every y ∈ ∂M and ξ ∈ T ∗ y ∂M .Thus the positive operator T = J ( y ) commutes with b ( ξ ) . Multiplying ( . ) by T − / , we get σ ( y ) b ( ξ ) + b ( ξ ) σ ( y ) = , which implies σB + Bσ ∈ End ( E ) .Instead of the symmetrized boundary operator ˜ B = ( B + B t ) / , we now consider thesupersymmetrized operator( . ) ¯ B = ( ˜ B − σ ˜ B σ ) / = ( B + B t − σBσ − σB t σ ) / ,which also differs from B by a bundle endomorphism.In such a way, every first order symmetric elliptic operator A acting on E , whoseboundary operator has self-adjoint symbol, determines a grading σ = σ A of E ∂ andan odd symmetric elliptic operator ¯ B = ¯ B A acting on E ∂ .By [BLZ, Theorem . .I], the space Λ of Cauchy data of A is a Lagrangian subspace ofthe symplectic Hilbert space ( H , iJ ) , H = L ( ∂M ; E ∂ ) . In other words, the orthogonalprojection Q of H onto Λ (the Calderón projection) satisfies the anticommutationrelation J ( Q − ) + ( Q − ) J = . Reasoning as above, we obtain σ ( Q − ) + ( Q − ) σ = . Therefore, Q is a generalized Cl ( ) spectral section both for ¯ B and for anyother odd symmetric operator with the symbol b .Applying our previous results to this situation, we obtain a graded version of thegeneral cobordism theorem. . Theorem.
Let X , M , E = E ′ , A , and b be as in Theorem . . Suppose, in addition,that all the operators A x are symmetric. Let the grading on E x | ∂ M x be defined by the unitarypart of the conormal symbol of A x , as in ( . ) , and let ¯ B = ( ¯ B x ) be a continuous family ofodd symmetric operators over ∂ M x having b x as the principal symbol. Then ¯ B admits both a Cl ( ) spectral section and a correcting family of odd smoothing correcting operators. Proof.
By the condition of the theorem, the Calderón projection Q x = Q ( A x ) contin-uously depend on x . Since Q x is a generalized Cl ( ) spectral section for ¯ B x for every x ∈ X , the family ( Q x ) is a generalized Cl ( ) spectral section for ¯ B . It remains toapply Theorems . and . . The same reasoning as in the proof of the Theorem . shows that all correcting operators are smoothing. (cid:3) Dirac type operators. If A is a symmetric Dirac type operator, then the conormalsymbol of A is unitary and thus coincides with the grading σ defined by ( . ).Applying Theorem . to this situation, we immediately obtain the following gener-alization of [MP , Corollary ]. . Theorem.
Let X , M , E = E ′ , and a family A of Dirac type operators be as in Theorem . . Suppose, in addition, that all the operators A x are symmetric. Then the family ¯ B = ( ¯ B x ) of supersymmetrized boundary operators admits both a Cl ( ) spectral section and a correctingfamily with odd smoothing correcting operators, with respect to the grading over ∂ M x givenby the conormal symbol of A x . roof. As was shown in the proof of Theorem . , the Calderón projection Q ( A x ) continuously depends on x . By the definition of the supersymmetrized boundaryoperator, ¯ B x continuously depends on x and its symbol is b x . It remains to applyTheorem . . (cid:3) Tangential operators on a moving hypersurface
Let M be a smooth connected Riemannian manifold without boundary, not necessar-ily compact. Let E and E ′ be smooth Hermitian vector bundles over M . Tangential operators.
Let Σ be a smooth cooriented hypersurface in M and E Σ bethe restriction of E to Σ . Every first order elliptic differential operator A : C ∞ ( M ; E ) → C ∞ ( M ; E ′ ) determines the tangential operator B = B A , Σ of A along Σ acting on sec-tions of E Σ . It is defined in exactly the same manner as the tangential operatorof A along the boundary, see formula ( . ). Let ˜ B A , Σ = ( B A , Σ + B tA , Σ ) / be thesymmetrized tangential operator. Then ( ˜ B A , Σ ) is a family of symmetric operatorsparametrized by pairs ( A , Σ ) . The space of parameters.
With a cobordism theorem in mind, we choose a differentparametrization of the appropriate part of the family ( ˜ B A , Σ ) .Let D ( E , E ′ ) denote the set of all Dirac type operators acting from sections of E tosections of E ′ . We equip D ( E , E ′ ) with the topology induced by the family of C -metrics on coefficients of operators restricted to small (that is, lying in local charts)compact subsets K ⊂ M . If A ∈ D ( E , E ′ ) , then B A , Σ ∈ D ( E ∂ ) has self-adjoint principalsymbol.Let S ( M ) be the set of all smooth compact submanifolds of M of codimension zero,excluding M itself. For every N ∈ S ( M ) its boundary ∂N is a smooth coorientedhypersurface in M , so the symmetrized tangential operator ˜ B A , ∂N is well defined.Let S ∗ M be the cosphere bundle of M , that is the subbundle of T ∗ M consisting of unitvectors. A smooth cooriented submanifold L ⊂ M of codimension one is naturallylifted to S ∗ M ; the corresponding embedding j : L ֒ → S ∗ M takes x ∈ L to ( x , n x ) ∈ S ∗ M , where n x is the conormal to L at x . Let d be the Hausdorff distance betweenclosed subsets of M and d be the Hausdorff distance between closed subsets of S ∗ M .We equip S ( M ) with the topology induced by the metric d ( N , N ′ ) = d ( N , N ′ ) + d ( j ( ∂N ) , j ( ∂N ′ )) .This topology does not depend on the Riemannian metric on M , since all N ∈ S ( M ) are compact. . Theorem.
The family ˜ B = ( ˜ B A , ∂N ) of symmetrized tangential operators parametrized bythe product D ( E , E ′ ) × S ( M ) admits a spectral section and a correcting family, with smooth-ing correcting operators. roof. We have a continuous family ( ∂N ) of closed manifolds parametrized by N ∈ S ( M ) and a continuous family ( ˜ B A , ∂N ) of symmetric elliptic operators over ∂N parametrized by ( A , N ) ∈ X = D ( E , E ′ ) × S ( M ) .By Proposition . , the restriction of A to N has weak inner UCP. By [BLZ, Corollary . ], the Calderón projection Q ( A | N ) continuously depends on ( A , N ) , so the family ( Q ( A | N )) is a generalized spectral section for the family ( ˜ B A , ∂N ) .The operator ˜ B A , ∂N considered as an unbounded operator acts on the Hilbert space H N = L ( ∂N ; E ) . The corresponding Hilbert bundle over X is the lifting to X of theHilbert bundle H over S ( M ) , where the fiber of H over N ∈ S ( M ) is H N . The base S ( M ) of H is a metric space and thus paracompact. Therefore, H is a numerableHilbert bundle. Its lifting to X is also a numerable Hilbert bundle.By Theorems . and . , the family ˜ B admits a spectral section and a correctingfamily. The same reasoning as in the proof of Theorem . shows that all correctingoperators are smoothing. This completes the proof of the theorem. (cid:3) Z -graded case. If A is a symmetric Dirac type operator over M and Σ is a smoothcooriented hypersurface in M , then the conormal symbol of A defines the grading σ A , Σ on E Σ . The supersymmetrized tangential operator ¯ B A , Σ given by formula ( . )is odd with respect to this grading.Let D sa ( E ) denote the subspace of D ( E ) consisting of symmetric operators. . Theorem.
The family ¯ B = ( ¯ B A , ∂N ) of supersymmetrized tangential operatorsparametrized by the product D sa ( E ) × S ( M ) admits both a Cl ( ) spectral section and a cor-recting family with odd smoothing correcting operators. Proof.
As was shown in the previous section, the Calderón projection Q ( A | N ) isa generalized Cl ( ) spectral section for the operator ¯ B A , ∂N . As was shown in theproof of the previous theorem, Q ( A | N ) continuously depends on ( A , N ) , so the family Q = ( Q ( A | N )) is a generalized Cl ( ) spectral section for ¯ B . The operators ¯ B A , Σ acton the fibers of the lifting of H to D sa ( E ) × S ( M ) , where H is the Hilbert bundle over S ( M ) defined in the proof of the previous theorem. Since H is numerable, the liftedHilbert bundle is also numerable. By Theorems . and . , ¯ B admits a Cl ( ) spectralsection and a correcting family with odd correcting operators. The same reasoningas in the proof of Theorem . shows that all correcting operators are smoothing. (cid:3) Part III
Graph continuous families
Throughout this part, all families of regular operators are supposed to be graphcontinuous.Recall that the Cayley transform of a regular self-adjoint operator A is the unitary perator defined by the formula κ ( A ) = ( A − i )( A + i ) − . The Cayley transform κ : R sa ( H ) → U ( H ) is a homeomorphism on the image [BLP, Theorem . ]. Therefore,the graph topology on the subspace R sa ( H ) of R ( H ) can be equivalently described asthe topology induced by the inclusion κ : R sa ( H ) ֒ → U ( H ) . We will use this fact below. Semibounded operators
Positive operators.
Let R + ( H ) denote the subspace of R sa ( H ) consisting of positiveoperators. . Proposition.
The restrictions of the graph topology and the Riesz topology to R + ( H ) coincide. Proof.
For a = f ( A ) the identity ( + A ) − = − a implies κ ( A ) = A − iA + i = a − i √ − a a + i √ − a = (cid:0) a − i √ − a (cid:1) = ˜ κ ( a ) ,where ˜ κ : [− , ] → U ( C ) = { z ∈ C | | z | = } is a continuous function given by theformula ˜ κ ( a ) = (cid:0) a − i √ − a (cid:1) . Thus the Cayley transform factors through thebounded transform: κ = ˜ κ ◦ f . The function ˜ κ is not invertible. However, its re-striction to the interval [ , ] is invertible: it is a homeomorphism from [ , ] to thebottom half of the unite circle Γ = (cid:8) e it | t ∈ [− π , ] (cid:9) . The inverse homeomorphism ϕ : Γ → [ , ] is given by the formula ϕ ( e it ) = cos ( t/ ) . Hence the restriction of thebounded transform f to R + ( H ) coincides with the composition ϕ ◦ κ and thus is con-tinuous. It follows that the restriction of the graph topology to R + ( H ) coincides withthe Riesz topology. (cid:3) Semibounded operators.
By [CL, Addendum, Theorem ], the restriction of thegraph topology to the subspace of bounded operators coincides with the usual normtopology, and thus with the Riesz topology. The following result generalizes thisproperty to semibounded operators. It is given here for the sake of completeness; wedo not use it in the rest of the paper.Let R sb ( H ) denote the subspace of R sa ( H ) consisting of semibounded operators, thatis, operators bounded from below or above (bounded operators included). . Theorem.
The restrictions of the graph topology and the Riesz topology to R sb ( H ) coincide. Proof.
For c running R , the subsets V c = { A ∈ R sa ( H ) | Res ( A ) ⊃ (− ∞ , c ] } and V ′ c = { A ∈ R sa ( H ) | Res ( A ) ⊃ [ c , + ∞ ) } form an open covering of R sb ( H ) in both graph and Riesz topologies. It is sufficientto show that the restriction of the graph topology to each of these subsets coincideswith the Riesz topology. Let us prove this for V c ; the proof for V ′ c is quite similar. e work as in the proof of Proposition . , with the same designations. The function˜ κ defines a homeomorphism from the interval [ f ( c ) , ] to the arc Γ c = (cid:8) e it | t ∈ [ t c , ] (cid:9) ,where t c ∈ (− π , ) , e it c = κ ( c ) . The inverse homeomorphism ϕ : Γ c → [ f ( c ) , ] is given by the same formula as before, ϕ ( e it ) = cos ( t/ ) . The restriction of thebounded transform f to V c coincides with the composition ϕ ◦ κ and thus is contin-uous. It follows that the restriction of the graph topology to V c coincides with theRiesz topology. This completes the proof of the theorem. (cid:3) Spectral sections
Arbitrary base spaces.
It follows from Theorem . that a Riesz continuous familyalways has local spectral sections locally, and the only obstruction for existence of aglobal spectral section is a topological one. In contrast with this, a graph continuousfamily may have no spectral section even locally. In fact, Riesz continuity is necessary for a local existence of a spectral section, as the following result shows. . Theorem.
Let X be an arbitrary topological space and A : X → R sa K ( H ) be a graphcontinuous map having a spectral section. Then A is Riesz continuous. Proof.
Let P be a spectral section for A and r : X → R + be the corresponding cut-offfunction. Let x ∈ X . Choose a positive constant r and a neighborhood V of x suchthat ± r ∈ Res ( A x ) and r > r ( x ) for all x ∈ V .The finite rank projection S x = (− r , r ) ( A x ) continuously depends on x ∈ V andcommutes with P x . Hence( . ) S x , S + x = ( − S x ) P x , and S − x = ( − S x )( − P x ) are mutually orthogonal projections continuously depending on x ∈ V . Decreasing V if necessary, one can find a continuous map g : V → U ( H ) such that the conjugationby g takes these three projection-valued maps to constant projections S , S + , and S − .Indeed, one can first find a neighborhood V ⊂ V of x and a map g : V → U ( H ) suchthat g ( x ) S x g ( x ) ∗ ≡ S [WO, Proposition . . ]. Next, one can find a neighbourhood V ⊂ V of x and a map g : V → U ( H ) such that g ( x ) is equal to the identity on therange of S and the conjugation by g ( x ) takes g ( x ) S + x g ( x ) ∗ to S + . Then g = g g isa desired trivialization of projections ( . ) over V .Let H ◦ , H + , and H − be the ranges of projections S , S + , and S − . Then( . ) g x A x g ∗ x = A − x ⊕ A ◦ x ⊕ A + x with respect to the orthogonal decomposition H = H − ⊕ H ◦ ⊕ H + . The map g A g ∗ : V → R sa K ( H ) is graph continuous, so its components A − , A ◦ , A + are also graphcontinuous. Since A ◦ acts on the finite-dimensional space H ◦ , it is norm continuous.Both A + and − A − are bounded from below by a positive constant r , so by Propo-sition . they are Riesz continuous. Substituting this to ( . ), we see that g A g ∗ isRiesz continuous, and thus the restriction of A to V is also Riesz continuous. Since ∈ X was chosen arbitrarily, A is Riesz continuous on the whole X . This completesthe proof of the theorem. (cid:3) Compact base spaces.
Taking together Theorem . and Proposition . , we imme-diately obtain the following result. . Theorem.
Let A = ( A x ) x ∈ X be a graph continuous family of regular self-adjoint opera-tors with compact resolvents acting on fibers of a Hilbert bundle H over a compact space X .Then the following conditions are equivalent: . A has a spectral section. . A is Riesz continuous and ind ( A ) = ∈ K ( X ) . . Remark.
Theorems . and . show that the straightforward transfer of [MP ,Proposition ] and [MP , Proposition ] to elliptic operators on manifolds withboundary does not work; one needs to be very careful using spectral sections inthis framework. For example, Yu applies [MP , Proposition ] to families of Diracoperators with local boundary conditions in [Yu]. However, one needs to ensure firstthat the corresponding families of unbounded operators are Riesz continuous. Generalized spectral sections
In the previous section we showed that a graph continuous family admitting a spec-tral section has to be Riesz continuous. Situation with generalized spectral sections,however, is more ambiguous. We give here several illustrating examples.
Graph continuous but Riesz discontinuous family with generalized spectral sec-tion.
The following example shows that existence of a generalized spectral sectiondoes not imply Riesz continuity. . Example.
Let X = N ∪ { ∞ } be the one-point compactification of N . Let ( e n ) n ∈ N be an orthonormal basis of H . Consider a family A : X → R sa K ( H ) of diagonal (in thechosen basis) operators given by the formulae A x ( e n ) = (cid:14) n , n = x − n , n = x for x ∈ N ; A ∞ ( e n ) = n .Since k κ ( A ∞ ) − κ ( A x ) k = k κ ( x ) − κ (− x ) k → as x → ∞ , the family A is graphcontinuous. On the other side, k f ( A ∞ ) − f ( A x ) k = f ( x ) → as x → ∞ , so A is Rieszdiscontinuous at ∞ .The constant function P : X → P ( H ) taking every x ∈ X to the identity is a generalizedspectral section for A . Moreover, P x is even an r x -spectral section for A x , where r : X → R + is an arbitrary function such that r n > n for n ∈ N . However, every suchfunction r is discontinuous at ∞ , so A has no global spectral section. (Otherwise, ofcourse, we would have a contradiction with Theorem . .) raph continuous family without generalized spectral sections. A generalizedspectral section does not necessarily exist even for a contractible base space (see Ex-ample . below) or for a family of invertible operators (see Example . below).Note that, in contrast with Example . , we cannot construct such graph continuousfamilies from semibounded operators, in view of Theorem . . . Example.
The space R sa K ( H ) equipped with the graph topology is path connected[Jo]. Let A : [ , ] → R sa K ( H ) be a graph continuous path connecting a negative opera-tor A with a positive operator A . (Such a path A can even be chosen consisting ofinvertible operators, but we do not explore it here for simplicity.) Then A has no gen-eralized spectral section. Indeed, a generalized spectral section P for A should becompact, while a generalized spectral section P for A should have compact comple-ment − P . Any two such projections P and P lie in the different connected com-ponents of the space P ( H ) , so they cannot be connected by a path P : [ , ] → P ( H ) . . Example.