Realisations of Elliptic Operators on Compact Manifolds with Boundary 3

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[26, 27, 58, 60], and were generalised to an extended Boutet de Monvel calculus by Schulze-Seiler [63].

In this paper we take a new approach to higher order elliptic operators on manifolds with boundary. The approach is based on work of Bär-Ballmann [5]. The paper [5] consists of a coherent overview of first order boundary value problems (admitting a self-adjoint adapted boundary operator) based on an analytic description of the Cauchy data space as well as a novel graphical decomposition of regular realisations of first order operators. The results were later extended to general first order problems by the first listed author and Bär in [6]. Although this approach was first fully utilised and made explicit in [5], the ideas have been around since the ’60s and was implicitly contained in a paper by Seeley [65, top of page 782]. They were visible already in Agranovich-Dynin’s classical works on index theory for elliptic boundary value problems of general order. Similar approaches predating Bär-Ballmann’s have been studied in the abstract in [13], been applied to Dirac-Schrödinger operators in [8] as well as having been used in the study of Maxwell’s equations on Lipschitz domains [20, 48]. Related ideas can also be seen in for instance [4, 19, 34, 37, 41, 43]. Since its appearance, the coherence of Bär-Ballmann’s work [5] has paved the way for numerous applications [7, 9, 17, 21, 66].

The aim of this paper is to present a perspective similar to [5], providing an overview of the theory for general order elliptic boundary value problems from the vantage point of the Cauchy data space. As such, the main novelty lies in the framework and the methods. Indeed, several of the results in this paper are, as stand alone results, at best mild generalisations of known results and are unlikely to surprise experts in the ﬁeld of boundary value problems. A key feature in several of the results is that they can treat all (or at least the regular extensions with the occasional restriction to the pseudo-local case) closed extensions of an elliptic diﬀerential operator on an equal footing with the special classes of boundary conditions studied classically. We have attempted to the best of our abilities to indicate where the results we are generalising can be found in the abundance of literature on boundary value problems.

The approach we take lies close in spirit to the ideas of noncommutative geometry [22] whose methods have proven their worth in index theory, relating to recent work on boundaries in noncommutative geometry [28]. It takes a more abstract perspective than the semiclassical methods ordinarily deployed to study boundary value problems, e.g. in [32, 37, 50, 57, 61, 69], and lies closer to methods for order one elliptic operators seen in [5, 6, 13, 15, 18, 41, 43] rather than the abstract methods of, for instance, [10,23,34,36]. The level of abstraction makes the method feasible for further development of index theory and applications to a larger class of problems than elliptic diﬀerential operators on manifolds with boundary. We anticipate this in- cludes elliptic operators on singular manifolds, hypoelliptic operators (cf. the subelliptic boundary conditions in [25]) or geometric operators in Lorentzian geometry (cf. [7]). These facts raises opti- mism for demystifying the general index formulas appearing in the (extended) Boutet de Monvel calculus relying on abstract homotopies [16,60], non-explicit inverses to isomorphisms in K-theory [58] or Chern characters of operator valued symbols [26, 27].

The method of Bär-Ballmann for first order operators can be summarised as describing boundary value problems by means of the Cauchy data space, i.e. the range of the corresponding trace map onto a mixed Sobolev space on the boundary. For higher order operators we apply the same ideas to the full trace map (taking into account all boundary traces up to the order of the operator) into a mixed Sobolev space on the boundary. The range of the full trace map – the Cauchy data space – characterises the realisation defining the boundary value problem when topologising it so that the full trace map is a quotient. The analytic details involved in this procedure produces precise information about regularity and Fredholm properties of the realisations in terms of properties of the boundary condition relative to the range of the full trace mapping on the maximal domain of the elliptic operator. The general theme in this framework is to reduce properties of realisations, and their proofs, to neat functional analysis arguments and Fredholm theory. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 3

The description of the possible realisations of an elliptic operator as subspaces of a Hilbert space of functions on the boundary relies on a precise description of the Cauchy data space. Abstractly, the Cauchy data space is isomorphic to the quotient of the maximal domain by the minimal domain. By classical results of Lions-Magenes [56, 57] the Cauchy data space is a concrete subspace of a mixed order Sobolev space on the boundary. For an elliptic first order differential operator this is described in detail in [6] using an adapted boundary operator. To describe the Cauchy data space of a general order elliptic differential operator, we employ the Calderón projection of Seeley [65]. Although our setup is based on the work of Bär-Ballmann, similar ideas date back much further as discussed above.

We emphasise that this paper, despite allowing for elliptic differential operators of general order, by no means supersedes [5, 6]. A significant novelty distinguishing [6] from this paper is that [6] links analysis on the Cauchy data space to the H∞-functional calculus, which in [6] forms the key to obtaining the estimates required to topologise the Cauchy data space. In contrast, in this paper we use Seeley’s work on Calderón projectors to analyse the Cauchy data space.

1.1. Main results and overview of paper. Let M be a compact manifold with boundary Σ := ∂M. We ﬁx a smooth measure µ on M. In our convention, Σ M and the interior of M ⊂ is denoted by M˚ = M Σ. We also choose a smooth interior pointing vectorﬁeld T~ transversal to \ Σ= ∂M. The smooth measure on Σ induced by µ and T~ will be denoted by ν.

In this paper, we are mainly concerned with elliptic diﬀerential operators D : C∞(M; E) C∞(M; F ) acting between hermitian vector bundles (E,hE ) M and (F,hF ) M. The hermit-→ ian metric will be implicit in the notations. We let m denote→ the order of D. The→ maximal realisa- 2 2 2 tion Dmax of D on L is given by dom(Dmax) := f L (M; E): Df L (M; F ) where D acts in a distributional sense on L2(M; E) as a consequence{ ∈ of the presence∈ of a unique} formal adjoint † ∞ ∞ D :C (M; F ) C (M; E). The minimal realisation Dmin is deﬁned from the domain obtained ∞→˚ m from closing Cc (M; E) in the graph norm of D; it is readily seen that dom(Dmin) = H0 (M; E) (cf. Proposition 2.1, on page 9). Many of the general results we prove only rely on abstract properties of the minimal and maximal realisations using the machinery of Appendix B (see page 72). For s R, we use the notation ∈ m−1 Hs(Σ; E Cm) := Hs−j(Σ; E). ⊗ j=0 M As described in Subsection 3.1, the space Hs(Σ; E Cm) is easiest thought of as a graded Sobolev space with respect to a certain grading on E Cm⊗. ⊗

A (closed) realisation of D is a (closed) extension De of Dmin such that De Dmax. In other m ⊆ 2 words, a realisation of D is an extension of D acting on H0 (M; E) to a domain on L (M; E) where it acts in the ordinary distributional sense.

The following theorem summarises how the Bär-Ballmann machinery applies in the setting of higher order elliptic differential operators on manifolds with boundary. Theorem 1. Let D be an elliptic differential operator of order m> 0 as in the preceding paragraph.

m m− 1 m j m−1 H 2 C (i) The full trace mapping γ : H (M; E) (Σ; E ), u (∂xn u Σ)j=0 extends to → − 1 ⊗ m 7→ | a continuous trace mapping γ : dom(Dmax) H 2 (Σ; E C ) with kernel dom(Dmin)= m → ⊗ H0 (M; E) and range being the Cauchy data space − 1 m m− 1 m H(ˇ D)= P H 2 (Σ; E C ) (1 P )H 2 (Σ; E C ), C ⊗ − C ⊗ where PC is a Calder´on projection (forM more details on PC , see Section 3). Equipping − 1 m H(ˇ D) with the Hilbert space topology making the projectors onto H 2 (Σ; E C ) and m− 1 m ⊗ H 2 (Σ; E C ), respectively, into partial isometries, the canonical isomorphism H(ˇ D) = ⊗ ∼ dom(Dmax)/dom(Dmin) is a Banach space isomorphism. 4 LASHIBANDARA, MAGNUS GOFFENG, ANDHEMANTHSARATCHANDRAN

(ii) There is a one-to-one correspondence between (closed) realisations of D and (closed) subspaces B H(ˇ D) as follows. A (closed) subspace B H(ˇ D) uniquely determines a (closed) ⊆ ⊆ realisation DB of D deﬁned by dom(D )= f dom(D ): γ(f) B , B { ∈ max ∈ } and any (closed) realisation Dˆ uniquely determines a (closed) subspace B := γdom(Dˆ) with DB = Dˆ. Moreover, ∗ † (DB) = DB∗ , where B∗ is the adjoint boundary condition (for more details see Subsection 4.3, starting on page 36, and Appendix B.2, starting on page 74).

(iii) A realisation DB of D is semi-regular, that is we have the domain inclusion dom(D ) Hm(M; E), B ⊆ m− 1 m ∗ if and only if B H 2 (Σ; E C ). A realisation DB is regular (i.e. DB and (DB) are ⊆ ⊗ m− 1 m ∗ m− 1 m semi-regular) if and only if B H 2 (Σ; E C ) and B H 2 (Σ; E C ) where B∗ is the adjoint boundary condition.⊆ ⊗ ⊆ ⊗

The reader can find this theorem spread out in the bulk of the paper as follows. Part (i) can be found in Theorem 3.18 (on page 27). Part (ii) is proven in much larger generality in Appendix B, more precisely in Proposition B.5 on page 74. Part (iii) is proven in Proposition 2.5, see page 11. The precise definition of regularity can be found in Definition 2.3 (page 11). In [5, 6], regular boundary conditions are called elliptic, we discuss our choice of terminology further in Remark 2.4 (page 11). We remark that Theorem 1 concerns local statements at the boundary and therefore1 extends to the case that M is noncompact with compact boundary when D is complete.

Based on Theorem 1, a boundary condition for D is deﬁned to be a closed subspace B H(ˇ D) and we write D for the associated realisation. We say that a boundary condition ⊆ B B is (semi-) regular if DB is (semi-) regular.

Theorem 1 has direct consequences to the well-posedness of PDEs. If D is an elliptic differential operator of order m > 0 with a semi-regular boundary condition B such that ker(DB) = 0, then the partial differential equation Du = f, γ(u) B ∈ † ⊥ is well-posed for f ran(DB) = ker(DB∗ ) . For more details, see Proposition 2.20 on page 17. In particular, if D is a∈ Dirac type operator (so m = 1), the partial differential equation Du = f, γ(u)=0 is well-posed for f ran(D ). See more in Corollary 2.22 on page 17. ∈ min The previous theorem has the following consequence on the spectral theory of the maximal and minimal realisations of an elliptic operator. We include this result because we have noticed some confusion surrounding it in the community. The result is known to experts in the field (cf. the discussion on [45, page 60-61]). Theorem 2. Let D : C∞(M; E) C∞(M; E) be an elliptic differential operator of order m > 0 acting between sections on a Hermitian→ vector bundle E M over a compact manifold with 2 → boundary with dim(M) > 1. The spectrum of Dmax on L (M; E) is

spec(Dmax) = specpt(Dmax)= C.

That is, the spectrum of Dmax is purely discrete and each generalised eigenspace is of inﬁnite 2 dimension. Moreover, the spectrum of Dmin on L (M; E) is

spec(Dmin)= C.

1Using for instance Lemma 4.1 on page 33 or Proposition 4.3 on page 34 REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 5

If m = 1 and D is a Dirac type operator then spec(Dmin) = specres(Dmin) = C; i.e., it consists solely of residual spectrum.

The reader can ﬁnd the ﬁrst part of this theorem on general order operators as Proposition 3.12 (see page 26) in the bulk of the text. The statement concerning spec(Dmin) for Dirac-type operators can be found in Corollary 3.14 (see page 26). The property distinguishing Dirac type operators from general elliptic operators is the so-called unique continuation property which undermines the existence of eigenvalues for Dmin. See the discussion in Remark 3.13 on page 26.

We now turn to some results on the index theory of realisations of elliptic diﬀerential operators. We say that a boundary condition B H(ˇ D) is Fredholm if the associated realisation D is Fredholm. ⊆ B We also introduce the notation D H(ˇ D) for the Hardy space – the image of ker(Dmax) under C ⊆ − 1 m the full trace mapping. Note that = P H(ˇ D)= P H 2 (Σ; E C ) by Theorem 1. CD C C ⊗ Theorem 3. Let D be an elliptic diﬀerential operator of order m > 0 as above. A boundary condition B H(ˇ D) is Fredholm if and only if (B, D) is a Fredholm pair in H(ˇ D). In this case, it holds that ⊆ C ind(D ) = ind(B, ) + dim ker(D ) dim ker(D† ). B CD min − min Moreover, ind(DB) is a homotopy invariant of (D,B) in the sense of Theorem 8.1 on page 59 (see also Theorem 8.5 on page 61).

The characterisation of Fredholm boundary conditions and the index formula is proven by abstract principles and can be found in Theorem 2.12 (see page 13) in the body of the text. Results similar to Theorem 3 for operators of order m = 1 can be found in [13, 15]. Theorem 4. Let D be an elliptic diﬀerential operator of order m > 0. Assume that P is a m− 1 m projection on H 2 (Σ; E C ) and set A := P (1 P ). Consider the following statements: ⊗ C − −

m− 1 m (1) The operator A is Fredholm on H 2 (Σ; E C ). ⊗ − 1 m (2) The operator P extends by continuity to H(ˇ D) and H 2 (Σ; E C ), and A deﬁnes a ⊗ Fredholm operator on H(ˇ D). − 1 m (3) The operator P extends by continuity to H 2 (Σ; E C ) and A deﬁnes a Fredholm operator − 1 m ⊗ on H 2 (Σ; E C ). ⊗ The following holds:

(i) If 1 and 2 holds, then P is boundary decomposing (see Deﬁnition 3.26 on page 30) and in particular

u ˇ (1 P )u m− 1 + Pu − 1 . k kH(D) ≃k − kH 2 (Σ;E⊗Cm) k kH 2 (Σ;E⊗Cm) (ii) If 1 and 3 holds, then the space m− 1 m B := (1 P )H 2 (Σ; E C ), P − ⊗ deﬁnes a regular boundary condition for D.

Moreover, if P is a zeroth order pseudodiﬀerential projection in the Douglis-Nirenberg calculus (see more details in Subsection 3.1), then the following are equivalent:

a) P is Shapiro-Lopatinskii elliptic with respect to D (in the sense of Deﬁnition 4.16). 0 m b) The operator PC (1 P ) Ψcl(Σ; E C ) is elliptic in the Douglis-Nirenberg calculus. c) The space − − ∈ ⊗ m− 1 m B := (1 P )H 2 (Σ; E C ), P − ⊗ is a regular boundary condition for D, so in particular, the realisation DB deﬁned from dom(D ) := u dom(D ): Pγu =0 = u Hm(M; E): Pγu =0 , B { ∈ max } { ∈ } 6 LASHIBANDARA, MAGNUS GOFFENG, ANDHEMANTHSARATCHANDRAN

is regular. d) The space m− 1 m B := (1 P )H 2 (Σ; E C ), P − ⊗ is a Fredholm boundary condition for D, so in particular, the realisation DB deﬁned from dom(D ) := u dom(D ): Pγu =0 , B { ∈ max } is a Fredholm operator.

If any of the equivalent conditions a)-d) holds, and moreover [P, PC ] is order m in the Douglis- Nirenberg calculus, then P is also boundary decomposing. −

The reader can find item i) stated as Theorem 3.30 (see page 30) and item ii) stated as Theorem 4.14 (see page 37) in the body of the text. The equivalence of item a) and b) is found in Proposition 4.18 (see page 38), and the equivalence of item b), c) and d) is found in Theorem 4.15 (see page 37). The final conclusion of the theorem appears as Corollary 3.32 (see page 31). Remark 5. The reader should be wary of the fact that the structure of the Cauchy data space H(ˇ D) is quite different from that of a Sobolev space. We give three instances of how this manifests:

If P is a zeroth order pseudodifferential projection in the Douglis-Nirenberg calculus and • m− 1 m PC (1 P ) is elliptic then BP = (1 P )H 2 (Σ; E C ) is a regular boundary condition but− if (1− P )P P is not of order m− (in the Douglis-Nirenberg⊗ calculus) then P fails to be − C C − boundary decomposing and even fails to act boundedly on H(ˇ D). This follows from Lemma 3.31. The obvious projectors on the Cauchy data space defining Dirichlet or Neumann conditions • for a Laplacian are not bounded operators on the Cauchy data space by Example 3.34, but a more refined projection (projecting along the Hardy space) produces a bounded projector as in Subsection 6.3. The pseudodifferential operators in the Douglis-Nirenberg calculus that act boundedly on the Cauchy data space are characterised in Lemma 3.31. We give an example in Section 9 (see page 63) of a formally self-adjoint first order elliptic • operator on the unit disc, with associated adapted boundary operator A such that the classical pseudodifferential operator P χ+(A) on the boundary (i.e. the unit circle) is of order C − 1 but does not act compactly on H(ˇ D). Therefore, the contrast between the approach in −this paper to that in [6] (where spectral projectors χ±(A) topologise the Cauchy data space) goes beyond compact perturbations even in the first order case. The image of χ+(A) differs substantially from the image of PC – the Hardy space – as seen in Proposition 9.12 containing an example where the two images’ intersection is a finite-dimensional space of smooth functions.

Much of the work in [6] concerning regular boundary conditions relied on graphical decompositions. The following theorem extends [6, Theorem 2.9] to elliptic diﬀerential operators of any order > 0. The theorem can be found as Theorem 5.6 (see page 40) in the body of the text.

Theorem 6. Let D be an elliptic diﬀerential operator of order m> 0, + a boundary decomposing projection (see Deﬁnition 3.26 on page 30), and B a boundary conditionP for D. The following are equivalent:

(i) B H(ˇ D) is an regular boundary condition, ⊂ (ii) B is + graphically decomposable (see Definition 5.3 on page 39), P m− 1 (iii) B is Fredholm decomposable in H 2 (see Definition 5.4 on page 40), P+ (iv) B is Fredholm decomposable in Hˇ (see Definition 5.5 on page 40). P+

The ideas of Theorem 1 can also be applied to study higher boundary regularity. Inspired by [6], we say that a boundary condition B H(ˇ D) is s-semiregular, for s 0, if whenever ξ B ⊆ ≥ ∈ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 7

m−1 s+m− 1 m s+m− 1 m satisﬁes that (1 P )ξ H 2 (Σ; E C ) then ξ H 2 (Σ; E C ). For s = 0, 0- − C ∈ j=0 ⊗ ∈ ⊗ semiregularity is equivalent to semi-regularity. In [6], a notion of s-semiregular boundary conditions L were introduced to prove higher boundary regularity for operators of order m = 1 that implies our 1 1 notion of s 2 -semiregularity for s> 2 . The following theorem can be found as Theorem 6.6 (see page 52) in− the body of the text. Theorem 7. Let D be an elliptic operator of order m > 0, s 0 and B a boundary condition. ≥ Then B is s-semiregular if and only if DB is s-semiregular, i.e. that whenever u dom(DB) satisﬁes D u Hs(M; F ) it holds that u Hs+m(M; E). ∈ B ∈ ∈

Theorem 7 generalises results from [5,6] from first order to general order. The method of proof can be localised to the boundary, so in the case that M is noncompact with compact boundary and D is complete, it holds that B is s-semiregular if and only if whenever u dom(DB) satisfies that s+m ∈ D u Hs (M; F ) it holds that u H (M; E) (where we use the notation Hs from [5, 6]). B ∈ loc ∈ loc loc Theorem 8. Let D be a formally self-adjoint elliptic operator of order m> 0 acting on a vector bundle E M. Assume that D has positive interior principal symbol σD and define → m − n 1 n − m cD := n TrE(σD(x, ξ) )dxdξ , n(2π) ˆ ∗ S M where n is the dimension of M. Then any lower semi-bounded self-adjoint realisation DB with dom(D ) Hm(M; E) is bounded from below with discrete real spectrum λ (D ) λ (D ) B ⊆ 0 B ≤ 1 B ≤ λ2(DB) satisfying that ≤ · · · m m λ (D )= c k n + o(k n ), as k + . k B D → ∞

Theorem 8 appears as Theorem 7.4 (see page 57) in the body of the text. The result might not be surprising as Weyl laws with error estimates are known under mild assumptions, for a brief historical overview see Section 7. Our assumptions are even milder, and we include the result as it showcases yet another feature of regular realisations that only depend on abstract principles. The proof of Theorem 8 was suggested to us by Gerd Grubb and relies on a precise description of the resolvent of DB, see Theorem 7.1 on page 56, and previous work of Grubb [34, 38]. For historical note, we mention that for Laplace operators on manifolds with boundary (see for instance [52] for a historical overview) and for elliptic pseudodiﬀerential operators on closed manifolds (see for instance [51, Chapter XXIX] or [67, Chapter III]), the remainder terms in the Weyl law are known more precisely.

1.2. Overview of contents. Let us brieﬂy describe the contents of the paper.

In section 2 we set up the theory for elliptic differential operators, building heavily on an abstract viewpoint presented in Appendix B. The abstract theory allows us to set up boundary conditions (Subsection 2.2) as in [5, 6] and characterise when they define Fredholm realisations. It allow us to make some straightforward applications to wellposedness in Subsection 2.4. We also consider a number of examples: in Subsection 2.5 we describe some classes of boundary conditions, in Subsection 2.6 we consider first order elliptic operators reconciling with results obtained in [6]. In Subsection 2.7, we consider scalar properly elliptic operators reconciling with more classical theory, e.g. [34, 57, 61].

In Section 3 we recall Seeley’s work on Calderón projectors and study its implications for the Cauchy data space and its analytic structure. The required Douglis-Nirenberg calculus and Hörmander’s description of the symbolic structure of a Calderón projection is described in Subsection 3.1. The Calderón projection from [65] and further consequences for boundary conditions are given in Subsection 3.2. In Subsection 3.3 we introduce the notion of boundary decomposing projections, the projections that characterise the analytic structure of the Cauchy data space.

In Section 4 we further analyse the Cauchy data space. The results of Subsection 4.1 shows that the Cauchy data space can be locally deﬁned and that the Calder´on projection could equally well be 8 LASHIBANDARA, MAGNUS GOFFENG, ANDHEMANTHSARATCHANDRAN

replaced with a projection constructed from a ﬁnite number of terms in H¨ormander’s construction in Subsection 3.1. In Subsection 4.2 we give a precise description of the boundary pairing between the Cauchy data space of an elliptic operator and the Cauchy data space of its adjoint, allowing us to characterise adjoint boundary conditions and regularity of pseudolocal boundary conditions in Subsection 4.3.

In Section 5 we characterise regular boundary conditions via graphical decompositions, in Section 6 we characterise boundary conditions admitting higher regularity, in Section 7 we prove the Weyl law for elliptic differential operators with regular boundary conditions, and in Section 8 we prove a rigidity result for the index of Fredholm realisations of elliptic differential operators. Following this, Section 9 consists of a computational exercise comparing a Calderón projection to the spectral projectors used in [6], quantifying their qualitative differences.

1.3. Notation. Throughout the paper, we use the analyst’s inequality a . b to mean that a Cb for some constant C where the dependence is apparent from context or explicitly speciﬁed.≤ By a b, we mean that a . b and b . a. ≃ The notation + is used for internal sums of vector subspaces and is used for direct sums. For two subspaces V , V V we use the notation V V = V + V ⊕to indicate W := V V , note 1 2 ⊆ 1 ⊕W 2 1 2 1 ∩ 2 that V1 W V2 ∼= (V1 V2)/d(W ) for the diagonal embedding d. In general, does not imply that the sum⊕ is orthogonal.⊕ We use the notation ⊥ for orthogonal direct sums.⊕ The term projection will be used for an idempotent operator on a Hilbert⊕ space; in general, they will not be orthogonal.

All manifolds and their boundaries are assumed to be smooth. For a manifold X without boundary, ′ ∞ we write (X) for the Fréchet space of distributions, i.e. the topological dual of Cc (X). For a manifoldD with boundary M we use the convention that the boundary is included in M. We write Σ := ∂M and M˚ := M Σ. \ For an operator T : H H over a Hilbert space H , potentially unbounded, we denote its domain by dom(T ). The kernel→and range are then given by ker(T ) and ran(T ) respectively. The graph 2 2 norm of T is T = + T , and the operator T is said to be closed if (dom(T ), T ) is a Banach space.k·k Since thek·k graphk norm·k polarises, (dom(T ), ) is a Hilbert space if T isk·k closed. p T If S,T are two operators, then we write S T if dom(S)k·kdom(T ) and S = T on dom(S). If S T are two closed operators then dom(S)⊂ dom(T ) is a⊂ closed subspace with respect to . ⊆ ⊂ k·kT An operator S is said to be adjoint to T if Tu,v = u,Sv for u dom(T ) and v dom(S). A densely-defined T has a unique adjoint T ∗ withh domaini h i ∈ ∈ dom(T ∗)= u H : C u,Tv C v , v dom(T ) . { ∈ ∃ u,T | h i|≤ u,T k k ∀ ∈ } A closed operator T is Fredholm if it has closed range with finite dimensional ker(T ) and ker(T ∗).

The operator T is said to be invertible if it is injective with dense range and T −1 : ran(T ) H is a bounded map. An invertible T has a unique extension T −1 : H H . The spectrum →of the operator T are the points λ C for which (λ T ) is not invertible. The→ set of all such points are denoted by spec(T ). There∈ are a number of non-equivalent− deﬁnitions of essential spectrum, but for us, this means the points λ spec(T ) for which (λ T ) fails to be Fredholm. The resolvent set is the complement of spec(T ) in∈ C and it is denoted by− res(T ).

A signiﬁcation notion throughout this paper is that of a Formally Adjointed Pair (FAP) (cf. Appendix B). This notion was used, without being named, in [34, Chapter II]. Given Hilbert H H † spaces 1 and 2, these are a pair of densely-deﬁned and closed operators (Tmin,Tmin) with H H † H H † ∗ † ∗ Tmin : 1 2 and Tmin : 2 1 adjoint to each other (i.e. Tmin (Tmin) and Tmin Tmin). These yield→ the following important→ maximal extensions ⊆ ⊆

† ∗ † ∗ Tmax := (Tmin) and Tmax := Tmin. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 9

A realisation of a FAP is an extension Tmin Tˆ Tmax. We will see FAPs arising from geometry and elliptic diﬀerential operators. ⊆ ⊆

It is of fundamental importance is to understand extensions, closed or otherwise, of Tmin (and † respectively Tmin). For this, the space

dom(Tmax) HˇT := dom(Tmin) is of fundamental importance. A Cauchy data space is a pair (γ, H(ˇ T )), where γ : dom(T ) max → H(ˇ T ) is a bounded surjection with ker γ = dom(Tmin). It is clear from the open mapping theorem Hˇ ˇ that γ induces an isomorphism (T ) ∼= H(T ).

1.4. Acknowledgements. LB was supported by SPP2026 from the German Research Founda- tion (DFG). MG was supported by the Swedish Research Council Grant VR 2018-0350. HS was supported by the Australian Research Council, through the Australian Laureate Fellowship FL170100020 held by V. Mathai. The authors would also like to thank Christian B¨ar and Andreas Ros´en for useful discussions. The authors are most grateful to Gerd Grubb, who motivated and inspired us through a multitude of comments and helpful suggestions on earlier versions of the paper and providing us with the method of proof used in Section 7.

2. Elliptic differential operators and boundary conditions

2.1. Setup. Let M be a compact manifold with boundary ∂M = Σ carrying a smooth measure µ. In our convention, Σ M and the interior of M is denoted by M˚ = M Σ. ⊂ \ Let (E,hE) M be a hermitian vector bundle and let C∞(M; E) denote the smooth sections over E. In particular,→ the support of such sections are allowed to touch the boundary. The subspace ∞ ˚ ∞ Cc (M; E) consists of u C (M; E) such that spt u Σ= ∅. By an abuse of notation, we write C∞(Σ; E) for the space of∈ smooth sections of E , and similarly∩ for other function spaces of sections. |Σ We shall also fix a smooth interior vectorfield T~ transversal to ∂M = Σ and let ν denote the smooth volume measure on Σ, induced by µ and T~. In what is to follow, the coordinate and derivative, s respectively, in this transversal direction will be denoted xn and ∂xn . For s R, by H (M; E) and Hs(Σ; E), we denote the L2 Sobolev spaces of s derivatives on M and Σ, respectively.∈ We let s ∞ ˚ s ˆ ˆ H0(M; E) denote the closure of Cc (M; E) in H (M; E) for some closed manifold M containing M as an open subset with smooth boundary. Note that Hs(M; E) can be identified with the quotient s ˆ H (M; E) Hs (Mˆ M; E). 0 \ For (F,hF ) M another hermitian vector bundle, we will be concerned with elliptic differential operators D→:C∞(M; E) C∞(M; F ) of order m> 0. Further, by D† :C∞(M; F ) C∞(M; E), → → let us denote the formal adjoint, obtained via integration by parts (cf. [5]). By Dc, we denote D ∞ ˚ † † ∞ ˚ with domain dom(Dc)=Cc (M; E), and analogously Dc is defined from dom(Dc)=Cc (M; F ). The maximal and minimal domains of D are then obtained as follows: † ∗ Dmax = (Dc) and Dmin = Dc.

It is clear that both these operators are closed and that Dmin Dmax. Moreover, it holds that ∗ † ∗ † ⊂ † Dmin = Dmax and Dmax = Dmin. In the language of Appendix B, (Dmin,Dmin) is a formally adjointed pair (FAP), see Definition B.1. The next result shows that the FAP associated with an elliptic operator of order m> 0 is a Fredholm FAP (see Definition B.14). Proposition 2.1. If D : C∞(M; E) C∞(M; F ) is an elliptic differential operator of order m> 0 then the following holds: →

m (1) dom(Dmin) = H0 (M, E); 10 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

∗ − 1 (2) Dmin has “compact resolvent” in the sense that (1 + DminDmin) 2 is a compact operator on L2(M, E); (3) Dmin has ﬁnite-dimensional kernel, closed range and Dmax has closed range. Moreover, L2(M; E) = ker(D ) ⊥ ran(D† ) and L2(M; F ) = ker(D† ) ⊥ ran(D ). min ⊕ max min ⊕ max

Proof. Item 1 follows from the G˚arding inequality showing that the graph norm of D is equivalent m ∞ ˚ ∗ − 1 to the H (M; E)-norm on Cc (M; E). Item 2 follows from the fact that (1 + DminDmin) 2 factors 2 over the domain inclusion dom(Dmin) ֒ L (M, E) which is compact by item 1 as m> 0. Item 3 follows from item 2 by Proposition B.15→.

Using the transversal vectorfield T~ at the boundary, for j = 0, 1, 2,..., we can define a trace mapping ∞ ∞ j γj :C (M, E) C (Σ, E), u ∂ u Σ, → 7→ xn | where xn denotes the variable transversal to the boundary defined from the vector field T~. Theorem 2.2. Let D :C∞(M; E) C∞(M; F ) be an elliptic differential operator of order m> 0. The trace mapping → γ :C∞(M; E) C∞(Σ; E Cm), u (γ u)m−1, → ⊗ 7→ j j=0 extends to a continuous mapping

m−1 − 1 −j γ : dom(D ) H 2 (Σ; E), max → j=0 M with dense range and m ker γ = H0 (M; E).

This result is classical and can be found in [56, 57]. The result is also proven in [65].

Following the notation of Appendix B we use the notation

Hˇ Hˇ dom(Dmax) D := (D ,D† ) . min min ≡ dom(Dmin) m We note that since dom(Dmin) = H0 (M; E) = ker γ, setting

H(ˇ D) := γdom(Dmax), the map γ induces a bounded inclusion

m−1 − 1 −j ,(Hˇ ֒ H 2 (Σ, E D → j=0 M with range H(ˇ D). The reader should be aware that this inclusion is not a Banach space isomorphism onto its image. However, by giving H(ˇ D) the induced topology from this inclusion, we obtain that

m−1 − 1 −j Hˇ = H(ˇ D) H 2 (Σ, E). D ∼ ⊆ j=0 M Since H(ˇ D) is isomorphic to the Hilbert space quotient dom(Dmax) , the Banach space dom(Dmin) H(ˇ D) is in fact a Hilbert space (although we shall not make use of a speciﬁc choice of inner product). We also note that since Hm(M; E) dom(D ) we have continuous inclusions with dense ranges ⊆ max m−1 m−1 m− 1 −j − 1 −j H 2 (Σ, E) H(ˇ D) H 2 (Σ, E). ⊆ ⊆ j=0 j=0 M M REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 11

Hˇ ˇ † Similarly, we obtain D† and H(D ) and continuous inclusions with dense ranges m−1 m−1 m− 1 −j † − 1 −j H 2 (Σ, F ) H(ˇ D ) H 2 (Σ, F ). ⊆ ⊆ j=0 j=0 M M

2.2. Boundary conditions. Any elliptic operator D :C∞(M; E) C∞(M; F ) as above defines † → the FAP (Dmin,Dmin) in the terminology of Appendix B. Therefore, realisations of elliptic operators are readily described from abstract principles as subspaces of the Cauchy data space H(ˇ D). We reiterate that realisations and boundary conditions of elliptic operators form, to say the least, a well understood topic. Our ambition is solely to provide a new perspective consistent with the perspective due to Bär-Ballmann in the first order case.

A generalised boundary condition B for D is a subspace of H(ˇ D). It is simply called a boundary condition if B is further required to be closed. We note that by Proposition B.5, there is a one- to-one correspondence between realisations of an elliptic operator and its generalised boundary conditions. This correspondence is set up by associating the realisation DB defined from dom(D ) := f dom(D ): γ(f) B , B { ∈ max ∈ } with B, and conversely to associate the subspace γ(dom(De)) H(ˇ D) with a realisation De of D. We also note that by Proposition B.7, for any generalised boundary⊆ condition B of D, it holds that ∗ † (DB) = DB∗ , where B∗ H(ˇ D†) is the adjoint boundary condition defined by ⊂ B∗ := η H(ˇ D†): D†f,g = f,Dg f γ−1( η ), g dom(D ) . { ∈ h i h i ∀ ∈ { } ∈ B } In Subsection 4.3 (see page 36) we further describe the adjoint boundary condition as an annihilator of B with respect to an explicit continuous sesquilinear form H(ˇ D†) H(ˇ D) C, compare also to Appendix B.2, on page 74. Note, B∗ is necessarily closed. For convenience,× → we shall simply refer to a realisation or boundary condition for D when speaking of a realisation or boundary condition † for (Dmin,Dmin). This terminology is consistent with the terminology for boundary conditions for first order elliptic differential operators from [5, 6].

Deﬁnition 2.3. Let D be an elliptic diﬀerential operator of order m> 0 and DB a closed realisation. We say that that DB is:

m semi-regular if dom(DB) H (M; E), • regular if D and (D )∗ ⊆are semi-regular. • B B

In light of Proposition B.5, we say that B is (semi-) regular if DB is (semi-) regular. Remark 2.4. The literature on boundary value problems contains a broad range of refined notions for ellipticity and regularity. Historically, the term ellipticity has been used for a local condition at the leading symbol level (including possible boundary symbols) that implies regularity properties. For the boundary conditions consider in this paper and in [5, 6], there is in general no notion of m boundary symbols. In [5, 6], the regularity property dom(DB) H (M; E) was therefore termed semi-ellipticity. To better reflect the terminology in bulk of the literature⊆ and to capture the fact that m dom(DB) H (M; E) is a regularity property, we use the term semi-regular boundary condition in this paper.⊆ Proposition 2.5. Let D be an elliptic differential operator of order m > 0 and B a boundary m−1 m− 1 −j condition. Then D is semi-regular if and only if B H 2 (Σ; E). B ⊆ j=0 L m−1 1 m− 2 −j Proof. If DB is semi-regular, then γ(dom(DB)) j=0 H (Σ; E) by the trace theorem. We m−1 m− 1⊆−j conclude that B = γ(dom(D )) H 2 (Σ; E). B ⊆ j=0 L L 12 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

m−1 1 m− 2 −j If B j=0 H (Σ; E), then any x dom(DB) can be written as x = x0 + xB where ⊆ m ∈ x0 dom(Dmin) = H0 (M; E) and xB dom(Dmax) is a pre-image of γ(x) B under γ. By the ∈ L m ∈ m−1 m− 1 −j ∈ trace theorem, we can take x H (M; E) since γ(x) H 2 (Σ; E). We conclude that B ∈ ∈ j=0 dom(D ) dom(D ) + Hm(M; E) = Hm(M; E). B ⊆ min L Proposition 2.6. Let D be an elliptic diﬀerential operator of order m> 0 and B a semi-regular ∗ boundary condition. Then, ker(DB) is ﬁnite dimensional and ran(DB) and ran(DB) are closed. Moreover, L2(M; E) = ker(D ) ⊥ ran(D∗ ) and L2(M; F ) = ker(D∗ ) ⊥ ran(D ). B ⊕ B B ⊕ B In particular, if B is regular then DB is Fredholm.

† † ∗ Proof. If B is semi-regular, the pair B := (DB,Dmin) is a FAP (indeed DB Dmax = (Dmin) ) † † ∗ T 2 ⊆ and Dmin DB∗ = (DB) ) with compact domain inclusion dom(DB) ֒ L (M; E). The claimed result now⊆ follows from Proposition B.15. → Remark 2.7. We return to the study of regular boundary conditions below in Subsection 4.3 and Subsection 5.

For an elliptic operator D, we use the notation m−1 − 1 −j := γ ker D H(ˇ D) H 2 (Σ, E). CD max ⊆ ⊆ j=0 M m−1 − 1 −j The space is called the Hardy space of D. We shall later see that H 2 (Σ, E) is, CD CD ⊆ j=0 in fact, closed in H(ˇ D). In the setting of Appendix B, more precisely Deﬁnition B.11, the space L D is denoted by (D ,D† ). C C min min Proposition 2.8. Let D be an elliptic diﬀerential operator of order m> 0 and B B′ be boundary conditions. ⊆

(i) It holds that D D ′ and there is a short exact sequence of Hilbert spaces B ⊆ B B′ 0 dom(D ) dom(D ′ ) 0. → B → B → B → In particular, dom(DB) = B, and dom(Dmin) ∼ ˇ dom(Dmax) = H(D) dom(DB) ∼ B hold for a boundary condition B. (ii) The inclusion and restriction mappings ﬁt into a short exact sequence of Hilbert spaces 0 ker(D ) ker(D ) B 0. → min → B →CD ∩ → In particular, we have the short exact sequence γ 0 ker(D ) ker(D ) 0. → min → max −→CD →

Proof. The result follows from the Propositions B.10 and B.12. Theorem 2.9. Let D be an elliptic diﬀerential operator of order m > 0 and B be a generalised boundary condition. Then the following holds:

(i) ran(D ) = ran(D ) and it is closed if and only if B + is a boundary condition (i.e. B B+CD CD closed in H(ˇ D)). (ii) ker(D ) is finite-dimensional if and only if B is finite-dimensional. B ∩CD (iii) ran(DB) has finite algebraic codimension if and only if B + D has finite algebraic codi- C ⊥ mension in H(ˇ D) if and only if ran(DB) is closed and ran(DB) is finite-dimensional. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 13

Proof. Item (i) follows from Theorem B.23, Proposition 2.1, and Lemma B.20. Item (ii) follows from Proposition 2.8. Item (iii) follows from Proposition B.26. Corollary 2.10. Let D be an elliptic diﬀerential operator of order m > 0 and B m−1 m− 1 −j ⊆ H 2 (Σ, E) be closed in H(ˇ D). Then B + H(ˇ D) is closed. j=0 CD ⊆ L Proof. The space B deﬁnes a semi-regular boundary condition by Proposition 2.5. By Proposition 2.6, D has closed range so B + H(ˇ D) is closed by Theorem 2.9. B CD ⊆

A signiﬁcant consequence of Theorem 2.9 is that it allows us to relate the closedness of the range of a realisation of a boundary condition to the closedness of an associated boundary condition. It may seem that this implicitly asserts that an operator needs to be closed in order to have closed range. This is not the case, as highlighted by the following quintessential example. Example 2.11. Consider the generalised boundary condition m−1 m− 1 −j BSob = H 2 (Σ; E), j=0 M for an elliptic operator D of order m> 0. The space BSob is not a closed subspace of H(ˇ D), so the associated operator DBSob is not a closed operator. One can in fact show that DBSob = DBSob+C = ˇ Dmax. We can prove that DBSob has closed range and as such, BSob + D is closed in H(D). We return to this example below in in Example 3.21 below and prove, amongstC other things, that B + = H(ˇ D). Sob CD

Let us prove that DBSob has closed range. By a similar argument as in Proposition 2.5, we see that m dom(DBSob ) = H (M; E). We can assume that M is a domain with smooth boundary in a compact manifold Mˆ , that E, F extend to Hermitian vector bundles E,ˆ Fˆ Mˆ and that there exists an elliptic differential operator → Dˆ of order m extending D to Mˆ . Let Tˆ Ψ−m(Mˆ ; F,ˆ Eˆ) denote a parametrix of Dˆ such that DˆTˆ 1 ∈ cl − is a projection onto the finite-dimensional space ker(Dˆ †) and TˆDˆ 1 is a projection onto the finite- − dimensional space ker(Dˆ). We define the continuous operator T := Tˆ : L2(M; F ) Hm(M; E) which by construction inverts D : Hm(M; E) L2(M; F ) (viewed as a| continuous operator)→ from the right up to smoothing operators. From the→ compactness of (1 DT ) and Fredholm’s theorem, we conclude that − ran(D ) = ran(D : Hm(M; E) L2(M; F )) ran(DT :L2(M; F ) L2(M; F )), BSob → ⊇ → has finite algebraic codimension and is closed in L2(M; F ).

By understanding closedness of the ranges of an operator, we are able to understand which boundary conditions yield Fredholm operators. In applications to index theory, it is essential to understand which boundary conditions yield such realisations. This is described in the following theorem. Theorem 2.12. Let D be an elliptic operator of order m > 0 and B be a boundary condition. Then the following are equivalent:

(i) (B, ) is a Fredholm pair in H(ˇ D); CD (ii) DB is a Fredholm operator.

If either of these equivalent conditions hold, we have that

∗ H(ˇ D) B † = . D ∼ (B + D) ∩C C Moreover, ind(D ) = ind(B, ) + dim ker(D ) dim ker(D† ). B CD min − min 14 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Proof. Follows from Theorem B.28. Remark 2.13. A direct corollary of Theorem 2.12 is that whenever B and B′ are two Fredholm boundary condition that are norm-continuously homotopic via Fredholm boundary conditions2 then it holds that ind(DB) = ind(DB′ ). However, Theorem 8.1 below provides a more general result concerning homotopy invariance so we omit the details of this argument. Corollary 2.14. Let D be an elliptic operator of order m > 0. Assume that B m−1 m− 1 −j ∗ m−1 m− 1 −j ⊆ H 2 (Σ; E) is closed in H(ˇ D) and satisﬁes that B H 2 (Σ; F ). Then j=0 ⊆ j=0 (B, ) is a Fredholm pair in H(ˇ D) and L CD L ∗ H(ˇ D) † B D ∼= (B + ). ∩C CD

Proof. Follows from Theorem 2.12 by noting that our assumptions are equivalent to B being regular which implies that it is Fredholm by Proposition 2.6.

2.3. The boundary pairing. Consider Hˇ with the quotient map dom(D ) D max → dom(Dmax) = HˇD as a Cauchy data space. Then, by abstract nonsense, the boundary dom(Dmin) pairing ω : Hˇ † Hˇ C, D D × D → defined for f dom(D† ), g dom(D ) as ∈ max ∈ max † ω ([f], [g]) := D f,g 2 f,Dg 2 , D h iL (M;E) − h iL (M;F ) is well-defined and continuous. By an abuse of notation, we consider the boundary pairing as a sesquilinear mapping ω : H(ˇ D†) H(ˇ D) C. D × → To describe ω on H(ˇ D) and H(ˇ D†) in greater detail, we introduce the following m m-matrix D × 0 0 0 0 1 0 0 ··· 0 1 0  ··· . 0 0 1 0 . τ =  ···  (1) . . . .  . . .. 0 .    0 1 0 0 0   1 0 ··· 0 0 0  ···    ∞ We introduce the notation Al = (Al(xn))xn∈[0,1] for the family of differential operators C (Σ, E) C∞(Σ, F ) of order l such that → m m−l D = AlDxn , Xl=0 near Σ. The following proposition follows from the computations on [65, page 794]. See also [42, Proposition 1.3.2]. Proposition 2.15. Let D be an elliptic differential operator of order m > 0. There exists an m m matrix of differential operators × A 0 0 0 0 0 ··· A1,m−1 A0 0 0 0  ···. . .  A A .. 0 . . a =  2,m−2 1,m−2  ,  . . .   . . .. A 0 0   0  e A A A A 0   m−2,2 m−3,2 1,2 0  A A ··· A A A   m−1,1 m−2,1 ··· 2,1 1,1 0   2Norm-continuously homotopic in the sense that B and B′ are images of norm-continuously homotopic projectors where all projections along the homotopy define Fredholm boundary conditions. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 15

where Aj,k is a diﬀerential operator of order j with the same principal symbol as Aj xn=0, such that | A A A A A m−1,1 m−2,1 ··· 2,1 1,1 0 Am−2,2 Am−3,2 A1,2 A0 0  . ··· .  A .. A 0 . a := iτa = i  m−3,3 0  , − −  . . .   . .. 0 .   .  e A A ..   1,m−2 0   A 0 0 0 0   0 ···  satisﬁes   † D u, v L2(M;E) u,Dv L2(M,F ) = γu, aγv L2(Σ;F ⊗Cm), h i − h i h i for all u C∞(M; F ) and v C∞(M; E). ∈ ∈

The fact that the order of the diﬀerential operators appearing in a remains constant along diagonals in the matrix shows that, in fact, a Ψ0 (Σ; E Cm, F Cm) via the Douglis-Nirenberg calculus ∈ cl ⊗ ⊗ we review in Subsection 3.1. The reader should note that we abusee notation and identify A0 with ∞ A0 xm=0 which is of order 0, i.e. eA0 C (Σ; Hom(E, F )). Since A0 is the restriction of the | ′ ∈ principal symbol to xn = 0, ξ = 0 and ξn = 1, ellipticity implies that A0 : E Σ F Σ is a vector bundle isomorphism. | → |

Lemma 2.16. Let D be an elliptic diﬀerential operator of order m> 0. Let a and a† denote the matrices of diﬀerential operators constructed from D and D†, respectively, as in Proposition 2.15. Recalling that a = iτa, it holds that e e − ∗ a + a† =0. e

Proof. The lemma follows from that a deﬁnes the boundary pairing as in Proposition 2.15 and the symmetry condition on the boundary pairing in Proposition B.4. Lemma 2.17. The operators a (from Proposition 2.15) and τ satisfy the following.

(i) For any s R, τ definese a unitary isomorphism ∈ m−1 m−1 Hs+j(Σ; F ) Hm−1+s−j(Σ; F ) → j=0 j=0 M M (ii) The matrix of differential operators a is invertible as a matrix of differential operators and −1 A0 0 0 0 0 e−1 ··· B1,m−1 A0 0 0 0  ···. . .  −1 B B .. 0 . . a =  2,m−2 1,m−2  ,  . . .   . . .. A−1 0 0   0  e B B B A−1 0   m−2,2 m−3,2 ··· 1,2 0  B B B B A−1  m−1,1 m−2,1 2,1 1,1 0   ···  where Bj,k is a differential operator of order j. (iii) For any s R, a defines a Banach space isomorphism ∈ m−1 m−1 e Hs−j(Σ; E) Hs−j(Σ; F ), → j=0 j=0 M M and a = iτa defines a Banach space isomorphism − m−1 m−1 e Hs+j(Σ; E) Hm−1+s−j(Σ; F ), → j=0 j=0 M M 16 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

−1 Proof. Item (i) is immediate. Item (ii) follows from that A0 a is lower triangular with the identity −1 operator on the diagonal, and as such, the inverse of A0 a is computed from polynomial operations on its entries through Gaussian elimination. The fact that thee diﬀerential operator Bj,k is of order j follows from a short inspection of the Gaussian elimination.e To prove item (iii), we note that any m−1 matrix (Cj,k)j,k=0 of diﬀerential operators with the order of Cj,k being j k, acts continuously on m−1 s−j − j=0 H (Σ; F ). Now item (iii) follows from item (ii). L Following Appendix B, we use the notation ω : H(ˇ D†) H(ˇ D) C, D × → for the sesquilinear boundary pairing

† ω (η, ξ) := D u, v 2 u,Dv 2 , D h iL (M;E) − h iL (M,F ) for u γ−1( η ) and v γ−1( ξ ). This pairing is well deﬁned and non-degenerate by Proposition B.4.∈ Below{ in} Theorem∈4.12,{ we} shall see that it is perfect. For now we settle with the pairing being perfect on the Sobolev spaces.

† Proposition 2.18. The boundary pairing ωD : H(ˇ D ) H(ˇ D) C extends by continuity to a perfect pairing × → m−1 m−1 ω : H−s−j(Σ; F ) Hm−1+s−j(Σ; E) C, D,s × → j=0 j=0 M M for any s R with ∈ ωD,s(η, ξ)= η, aξ L2(Σ;F ⊗Cm), h i for η m−1 H−s−j(Σ; F ) and ξ m−1 Hm−1+s−j(Σ; E). ∈ j=0 ∈ j=0 L L ′ ′ ∞ ′ ∞ Proof. By Proposition 2.15, ωD([ξ], [ξ ]) = ξ, aξ L2(Σ;F ⊗Cm) for ξ C (Σ; F ) and ξ C (Σ; E). The statement now follows from Lemma 2.17h . i ∈ ∈

2.4. Well-posedness in the absence of kernels. In the situation of boundary conditions without kernels, we are able to understand some properties of associated PDEs. Proposition 2.19. Let D be an elliptic operator of order m> 0. Suppose that B is a semi-regular ∗ boundary condition such that ker(DB)=0. Then, the spectrum of DBDB consists solely of discrete spectrum in (0, ) and letting λ1 denote its smallest nonzero eigenvalue, we have the Poincar´e inequality ∞

λ u 2 D u 2 1k kL (M;E) ≤k B kL (M;F ) for all u dom(D ). p ∈ B

Under an additional hypothesis on B, we give a precise description of the spectral asymptotics of ∗ DBDB in Corollary 7.7 below.

∗ ∗ m Proof. The operator DBDB satisﬁes dom(DBDB) H (M; E), and moreover, it is easy to see that ⊂ ∗ it is non-negative self-adjoint. By the Rellich embedding theorem, we obtain that spec(DBDB) is ∗ ∗ discrete, real, non-negative. Also, ker(DBDB) = ker(DB) = 0 and therefore, spec(DBDB) = 0 < λ λ ... . Then, via numerical range considerations, { 1 ≤ 2 ≤ } D∗ D u,u λ u 2. h B B i≥ 1k k ∗ ∗ ∗ for u dom(DBDB). However, dom(DBDB) is dense in dom( DBDB) = dom(DB) and therefore, we obtain∈ the desired inequality. p REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 17

Proposition 2.20. Let D be an elliptic operator of order m> 0. Suppose that B is a semi-regular boundary condition and that ker(DB)=0. Then,

DBu = f γ(u) B ( ∈ for f ran(D ) is well-posed. ∈ B

∗ 2 Proof. By Proposition 2.6, we have that DB and DB has closed range. But L (M; E) = ker(DB) ∗ ∗ ⊕ ran(DB) = ran(DB), and so on applying [72, Corollary 1, Chapter VII, Section 5], we obtain that D : dom(D ) L2(Σ; E) ran(D ) B B ⊂ → B has a bounded inverse. Therefore, u u + Du = D−1f + f . f . k kDB ≃k k k k k B k k k k k This is exactly that the problem in the statement of the corollary is well-posed.

We apply this to understand the well-posedness of the Dirichlet problem. An important and well known notion is weak UCP (unique continuation property) for an operator D. This means that for smooth u, when Du = 0 and u = 0 on a nonempty open subset Ω M, then u = 0. Similarly, weak inner UCP is satisﬁed by D if for smooth u, when Du = 0 and⊂u = 0, this implies u = 0. | Σ Corollary 2.21. Let D be an elliptic diﬀerential operator satisfying weak inner UCP. Then,

Du = f (γ(u)=0 is well-posed on ran(DCD ) = ran(Dmin).

Proof. We obtain well-posedness on ran(Dmin) from Proposition 2.20 with the semi-regular boundary condition B = 0 upon showing that ker(Dmin) = 0. By elliptic regularity, ker(Dmin) ∞ ⊂ C (M; E) so the weak inner UCP property yields that ker(Dmin) = 0. Lastly, we have that ran(Dmin) = ran(D0+CD ) = ran(DCD ) from Theorem 2.9 item (i).

Now, let us focus on the first order case. In the first order case, Dmin corresponds to the Dirichlet problem for D. As discussed in [14, Section 1.2], weak UCP implies weak inner UCP for elliptic first order differential operators. However, focusing on Dirac-type operators, the following holds. Corollary 2.22. Let D be first order and Dirac-type. Then,

Du = f (γ(u)=0 is well-posed on ran(DCD ) = ran(Dmin).

Proof. As mentioned in [14, Remark 2.2], it is well known that all ﬁrst order Dirac-type operators satisfy weak UCP. This, in turn, implies weak inner UCP, and therefore, the conclusion follows.

Remark 2.23. It is a well known fact that well-posedness fails for the Dirichlet problem for first order Dirac-type operators on L2. However, the corollary shows that on a very large space, well-posedness actually holds. In fact, by Proposition 2.6, it is easily seen that the obstruction to † ∗ well-posedness is indeed large. It is precisely ker(Dmax) = ker(Dmin). On a manifold of dimension exceeding 1, we see from Theorem 3.11 that this is, indeed, an infinite dimensional space. That is, in general, the obstruction to well-posedness for the Dirichlet problem is infinite dimensional. 18 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

2.5. Local, pseudo-local and bundle-like boundary conditions. Let us give some examples of boundary conditions. Deﬁnition 2.24. Let D be an elliptic operator of order m> 0 and B a boundary condition.

We say that B is pseudo-local if there exists an m m matrix P of pseudo-differential • operators on E with P 2 = P and × |Σ B = H(ˇ D) ker(P : ′(Σ; E Cm) ′(Σ; E Cm)). ∩ D ⊗ → D ⊗ We say that the pseudo-local boundary condition B is defined from P , and to indicate the dependence on P we sometimes use the notation BP for B. m ′ We say that B is local if there exists a differential operator b on E Σ C E (for some • vector bundle E′ Σ) such that | ⊗ → → B = H(ˇ D) ker(b : ′(Σ; E Cm) ′(Σ; E′)). ∩ D ⊗ → D We say that the local boundary condition B is defined from b. m ′ ′ m If B is a local boundary condition defined from b on E Σ C E where E E Σ C • is a subbundle and b is a projection onto a complement| ⊗ of E′,→ we say that the⊆ associated| ⊗ boundary condition is bundle-like and defined from E′.

In [6], the bundle-like boundary conditions were called local. However, in the higher order case more general local boundary conditions naturally arise and are well studied, e.g. in [34, 57, 69].

The reader should note that a bundle-like boundary condition is both local and pseudo-local. If B is a local boundary condition defined from b, then dom(D )= u dom(D ): bγu =0 . B { ∈ max } Here bγu is interpreted in a distributional sense. In particular, B is indeed a boundary condition because the topology on H(ˇ D) induced from the space of distributions is weaker than the norm topology of H(ˇ D). A similar argument shows that a pseudo-local boundary condition indeed is a boundary condition. Example 2.25. For 0 k m, we consider the order zero operator b that projects onto the first k coordinates of E C≤m.≤ Let B be the associated bundle-like boundary condition. We have that |Σ ⊗ k B = ξ H(ˇ D): ξ = ξ = ... = ξ =0 , and k { ∈ 0 1 k−1 } dom(D )= u dom(D ): γ u = γ u = ...γ u =0 = dom(D ) Hk(M). Bk { ∈ max 0 1 k−1 } max ∩ 0 Proposition 2.26. Let D be an elliptic operator of order m> 0 and BP a pseudo-local boundary ∗ † condition defined from P . Then (BP ) is the pseudo-local boundary condition BP† for D defined ∗ −1 ∗ ∗ from P† := (a ) (1 P )a where a is the differential operator from Proposition 2.15. −

Proof. We remark that (a∗)−1(1 P ∗)a∗ is again an idempotent matrix of pseudodiﬀerential operators because (a∗)−1 is a matrix− of diﬀerential operators by Lemma 2.17. The space B∗ is ∗ characterised by the property that η B if and only if ωD(η, ξ) = 0 for all ξ B. We need to prove that ∈ ∈ B∗ = H(ˇ D†) ker((1 P ∗)a∗ : ′(Σ; F Cm) ′(Σ; E Cm)). ∩ − D ⊗ → D ⊗

Since B = H(ˇ D) ker(P : ′(Σ; E Cm) ′(Σ; E Cm)), any ξ B satisﬁes ξ = (1 P )ξ. In ∩ D ⊗ → D ⊗ ∈ − particular, if η H(ˇ D†) ker((1 P ∗)a∗ : ′(Σ; F Cm) ′(Σ; E Cm)) and ξ B then ∈ ∩ − D ⊗ → D ⊗ ∈ ∗ ∗ ωD(η, ξ)= η, a(1 P )ξ L2(Σ;F ⊗Cm) = (1 P )a η, ξ L2(Σ;F ⊗Cm) =0. h − i h − i We conclude that H(ˇ D†) ker((1 P ∗)a∗ : ′(Σ; F Cm) ′(Σ; E Cm)) B∗. ∩ − D ⊗ → D ⊗ ⊆ To prove the converse inclusion, if η B∗ then for any ξ C∞(Σ; E Cm) we have that ∈ 0 ∈ ⊗ ∗ ∗ ωD(η, (1 P )ξ0)= (1 P )a η, ξ0 L2(Σ;F ⊗Cm). − h − i REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 19

∗ ∗ ∞ m Since ξ := (1 P )ξ0 B, we have that (1 P )a η, ξ0 L2(Σ;F ⊗Cm) = 0 forany ξ0 C (Σ; E C ). − ∈ h − i ∈ ⊗ In particular, (1 P ∗)a∗η = 0 in distributional sense if η B∗. Therefore H(ˇ D†) ker((1 P ∗)a∗ : ′(Σ; F Cm) − ′(Σ; E Cm)) B∗. ∈ ∩ − D ⊗ → D ⊗ ⊇ Example 2.27. Returning to Example 2.25, assume for simplicity that E = F and that the elliptic m operator D takes the form D = Dxn + A near the boundary for a diﬀerential operator A on Σ (with ∗ no dependence on xn). A short computation with integration by parts show that a = 0 σ(A) τ where 0:Σ T ∗Σ denotes the zero section and τ is as in Equation (1). Using this it is readily⊗ → ∗ veriﬁed via Proposition 2.26 that Bk = Bm−k−1 for k =0,...,m with the convention that B−1 =0. If the order of D is even, then Bm/2 should be interpreted as a Dirichlet condition. We return to this condition below in Subsection 2.7 for scalar properly elliptic operators.

2.6. Elliptic first order operators. The situation on which we have modelled our description of boundary conditions is that of first order operators described in [5, 6]. For simplicity, we continue to focus on compact manifolds with boundary, while [5,6] allowed for non-compact manifolds with compact boundary. The theory of boundary value problems for first order elliptic operators is further described in [43] and the special case of Dirac operators is studied in [15].

We follow the setup of Subsection 2.1 and, throughout this subsection, consider an elliptic differential operator D order m = 1. Associated to such an operator D, there are adapted boundary operators A on the bundle E Σ which are first order elliptic differential operators on Σ whose principal symbols satisfy | σ(A)(x, ξ′)= σ(D)(x, ξ′ =0, ξ = 1)−1 σ(D)(x, ξ′). n ◦ Alternatively, an adapted boundary operator A is an operator on Σ such that near Σ we can write D = σ (∂ + A + R ), · xn 0 for an xn-dependent first order differential operator R0 on Σ such that R0 xn=0 is of order zero. ′ | Here we have shortened the notation σ = σ(D)(x, ξ =0, ξn = 1). The reader should note that in the notation of Proposition 2.15,

σ = A0 = a and A1 = σ (A + R0). ·

In [6] by B¨ar and Bandara, the authors show that there exists an admissible cut, which is an r R so that Ar := A r is invertible ωr-bisectorial. This means that spec(Ar) is contained in a closed∈ bisector S = −ζ C : arg ζ ω of angle ω < π/2 in the complex plane, and that for ωr { ∈ ± ≤ r} r µ (ωr,π/2), there exists a constant Cµ > 0 so that whenever ζ is outside of the closed bisector ∈ −1 Sµ of angle µ, we have that ζ (ζ Ar) Cµ. The framework in [6] extends the work of B¨ar and Ballmann in [5] where they| |k make− the additionalk≤ assumption that A can be chosen self-adjoint.

Given that the spectrum of Ar is contained in the open left and right half-planes, a consequence ± ± of the bisectoriality of Ar is that we can consider spectral projectors χ (Ar), where χ (ζ)=1 for Re ζ > 0 and 0 otherwise. Not only do these spectral projectors exist, they are pseudo-differential ±operators of order zero (see [46]). This means that they act boundedly across all Sobolev scales and we define the space − 1 + − 1 Hˇ (D) := χ (A )H 2 (Σ; E) χ (A )H 2 (Σ; E). A r ⊕ r The following is an important property of this space that we will use later. We refer its proof to Appendix A. Lemma 2.28. Let A be an adapted boundary operator for a first order elliptic operator acting on a manifold with boundary of dimension > 1. For any admissible cut r R, the spaces ± 2 ∈ χ (Ar)L (Σ; E) are infinite dimensional. Moreover, the space Hˇ A(D) = 0 is an infinite- dimensional space with 6 { } 1 − 1 H 2 (Σ; E) Hˇ (D) H 2 (Σ; E), ⊆ A ⊆ 20 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

and each inclusion is dense.

∞ In [6, Theorem 2.3], the trace map γ : u u Σ, initially deﬁned on C (Σ; E), is extended to a bounded surjection 7→ | u u : dom(D ) Hˇ (D) 7→ | Σ max → A with kernel ker(u u ) = dom(D ). That is, H(ˇ D) = Hˇ (D) with γ is a Cauchy data 7→ | Σ min A space. The topology of H(ˇ D) is given purely in terms of an associated diﬀerential operator on the boundary, namely, the adapted boundary operator A. We give a direct argument for the equality H(ˇ D)= Hˇ A(D) below in Example 3.33.

+ A pseudo-differential projection P of order zero, for which 1 P χ (Ar) is a Fredholm operator 1 − − induces a regular boundary condition via BP = (1 P )H 2 (Σ; E). In particular, for any admissible + − − 1 cut r R, P = χ (A ) is such a projection. That is, B = χ (A )H 2 (Σ; E) is a regular boundary ∈ r r r r condition. In particular, the (generalised) APS-realisation DAPS, defined from dom(D ) := u H1(M; E): χ+(A )γ(u)=0 , APS { ∈ r } is regular. For notational convenience, we assume r = 0 and drop it from the notation for the remainder of the paper. The reader can find further examples of boundary conditions for first order elliptic operators in [5, 6, 30] and [43, Section 4].

2.7. Scalar properly elliptic operators. Let us consider an example that historically has played an important role and has long been well understood, see for instance [34, 57, 69]. We consider an elliptic differential operator D of order 2m acting between sections of E and F . We say that the differential operator is scalar if E and F are line bundles; the terminology is justified by the fact ∞ that Hom(E, F ) is a line bundle (trivialisable at Σ due to the existence of A0 C (Σ; Aut(E, F ))), and as such the symbol calculus is completely scalar. We recall the notion∈ of a properly elliptic scalar operator, see [61]. Definition 2.29. Let D be a scalar elliptic differential operator of order 2m. We say that D is properly elliptic if for each (x′, ξ′) S∗Σ, the polynomial equation ∈ σ(D) (x′, ξ′,z)=0, (2) |Σ has exactly m complex solutions with positive imaginary part. Remark 2.30. Let us consider some special cases where scalar elliptic differential operators are automatically properly elliptic. If σ(D) Σ is real-valued, then z is a solution of Equation (2) if and only if z¯ is, so real valued principal symbol| ensures that a scalar elliptic differential operator is properly elliptic.

If dim(M) > 2, the ﬁbres of the cosphere bundle of Σ are connected. Since D is elliptic, Equation (2) has no real solutions and as such the number of solutions with positive imaginary part is locally constant. We note that if z is a solution to Equation (2) at (x′, ξ′), then z is a solution to ′ ′ ′ ′ − ′ ′ Equation (2) at (x , ξ ) because of the symmetry condition σ(D) Σ(x , ξ ,z)= σ(D) Σ(x , ξ , z). This proves that all− scalar elliptic diﬀerential operators are properly| elliptic if dim(|M) >−2. −

For a scalar properly elliptic differential operator D of order 2m, we define the bundle-like boundary condition B from the subbundle E′ := E Cm, viewed as a subbundle of E C2m by embedding Dir ⊗ ⊗ it in the last m coordinates. Compare to Example 2.25. We write DDir := DBDir and call it the Dirichlet realisation of D. More explicitly, we have that dom(D )= u dom(D ): γ u = γ u = = γ u =0 . Dir { ∈ max 0 1 ··· m−1 } The following theorem reformulates several results from [34] to our setting, the reader is referred to [34] for proofs. Theorem 2.31. Let D be a scalar properly elliptic differential operator of order 2m. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 21

(i) BDir is a regular boundary condition, and so the realisation DDir of D with domain dom(D ) =H2m(M; E) Hm(M; E)= Dir ∩ 0 = u H2m(M; E): γ u = γ u = = γ u =0 , { ∈ 0 1 ··· m−1 } is regular. (ii) Also D† is properly elliptic and

∗ † DDir = DDir, † 2m m is the realisation of D with domain H (M; F ) H0 (M; F ). (iii) The map ∩ m−1 −j− 1 H 2 (M; E), CD → j=0 M 2m−1 1 m−1 1 −j− 2 −j− 2 induced from the projection j=0 H (M; E) j=0 H (M; E), is a Fredholm operator. → (iv) We can write L L dom(D ) = H2m(M; E) Hm(M; E) ker(D ), max ∩ 0 ⊕ker DDir max and 2m−1 2m−j− 1 H(ˇ D)= H 2 (M; E) . ⊕γ ker(DDir) CD j=m M

The reader should note that the theorem above holds for more general local (regular) boundary conditions than Dirichlet conditions, and are studied using diﬀerent methods in [34]. We remark that from these type of results, several Fredholm type results follow readily for properly elliptic boundary value problems with regular boundary conditions.

3. Calderon´ projectors and the Douglis-Nirenberg calculus

In this section we will review results of Hörmander and Seeley concerning Calderón projectors. See more in [50, Chapter XX], [65] and also in [45, Chapter 11]. Calderón projections are projections on function spaces on the boundary Σ onto the space of boundary values of homogeneous solutions Du = 0 in the interior, i.e. the Hardy space D. For higher order operators, it is necessary to keep track of different order traces, so evenC though the Calderón projection is a matrix of pseudodifferential operators, its entries will have different order. However, the orders remain constant along the diagonals in the Calderón projection which makes it amenable for study via the calculus of Douglis-Nirenberg.

3.1. Approximate Calderón projectors. First, we will in this subsection recall some constructions found in Hörmander’s book [50]. It will provide a way of constructing the full symbol of the Calderón projection.

As above, we write xn for the coordinate near the boundary defined from the transversal vector field T~. The coordinate x identifies a neighbourhood of the boundary with a collar Σ [0, 1] with n × the boundary defined by the equation xn = 0. Consider the differential operator Dxn = i∂xn defined in the collar neighbourhood of the boundary. Near the boundary, we can write − m m−j D = Aj Dxn . j=0 X For each j, A = (A (x )) is a family of differential operators C∞(Σ, E) C∞(Σ, F ) of j j n xn∈[0,1] → order j. Let a denote the principal symbol of D and aj the principal symbol of Aj . In coordinates 22 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

′ ′ ∗ x = (x , xn) on Σ [0, 1] with associated cotangent coordinates ξ = (ξ , ξn) on T M Σ, near Σ we can write × | m ′ ′ ′ ′ m−j a(x , xn, ξ , ξn)= aj (x , xn, ξ )ξn . j=0 X ∗ We deﬁne the boundary symbol σ∂ (A) as the ordinary diﬀerential operator valued function on T Σ given by m ′ ′ ′ ′ m−j σ∂ (D)(x , ξ ) := aj (x , 0, ξ )Dt . (3) j=0 ∞ ∗ X ∞ We consider σ∂ (D) as an element of C (T Σ, Hom(E Σ, F Σ) (C [0, ))). The conormal symbol of D is the symbol of the boundary symbol | | ⊗ L ∞ m ′ ′ ′ ′ m−j σcn(D)(x , ξ , ξn) := aj (x , 0, ξ )ξn , (4) j=0 X ′ ′ ′ ′ so that σ∂ (D)(x , ξ ) = σcn(D)(x , ξ ,Dt) and σcn(D) = σ(D) Σ. For more details on the yoga of boundary symbols, see [42, 60, 62] or [59] for related notions. |

The following result follows from [50, Theorem XX.1.3], which we state below as Theorem 3.8. Proposition 3.1. Let D be elliptic of order m > 0. The following construction produces a well- defined vector bundle ∗ E+(D) S Σ. ∗ → m ′ ′ ∗ Define E+(D) as the subset E+(D) π (E) S∗Σ C whose fibre over (x , ξ ) S Σ consists of m ⊆ | ⊗ ∞ ∈ (v0, v1,...,vm−1) Ex C such that the solution v C ([0, ), Ex) of the ordinary differential equation ∈ ⊗ ∈ ∞ σ (D)(x′, ξ′)v(t)=0, t> 0, ∂ (5) Dj v(0) = v , j =0,...,m 1, ( t j − is exponentially decaying as t + . → ∞ ∗ The subset E−(D) π (E) S∗Σ given as the set of vectors whose solution to (5) is exponentially ⊆ | ∗ decaying as t is a well-defined vector bundle defining a complement to E+(D) in π (E) S∗Σ Cm. → −∞ | ⊗ ∞ ∗ ∗ m Definition 3.2. Let p+(D) C (S Σ, Hom(π (E) S∗Σ C )) denote the projection onto E+(D) along E (D) and p (D) :=∈ 1 p (D) its complementary| ⊗ projection. − − − + Proposition 3.3. Let D be elliptic of order m> 0. It holds that −1 † ∗ p+(D)= σ0 (a )p−(D ) σ0(a), where a is the matrix of differential operators from Proposition 2.15 and σ0(a) is its principal symbol (in the Douglis-Nirenberg calculus we review below) σ (A ) σ (A ) σ (A ) σ (A ) A m−1 m−1 m−2 m−2 ··· 2 2 1 1 0 σm−2(Am−2) σm−3(Am−3) σ1(A1) A0 0  ···  .. . σm−3(Am−3) . A0 0 . σ0(a)= i   . −  . . .   . .. 0 .   .   σ (A ) A ..   1 1 0   A 0 0 0 0   0 ···    In particular, σ0 (a) induces isomorphisms † E±(D) ∼= E∓(D ).

We postpone the proof of Proposition 3.3 until after the statement of Theorem 3.8 below. Proposition 3.4. Let D be an elliptic diﬀerential operator of order m > 0 acting on a manifold ∗ of dim(M) > 1. The vector bundles E+(D), E−(D) S Σ are non-trivial over each component of S∗Σ. → REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 23

′ ′ m ′ ′ Proof. We note that σcn(D)(x , ξ ,z) = ( 1) σcn(D)(x , ξ , z), so z C solves ′ ′ −′ ′ − − m ∈ det σcn(D)(x , ξ ,z) = 0 if and only if det σcn(D)(x , ξ , z) = 0. Since E C = E+(D) E−(D) ′ ′ −∗ − ⊗ ⊕ by Proposition 3.1, we conclude that given (x , ξ ) S Σ, either E+(D)(x′,ξ′) =0or E−(D)(x′,ξ′) = ′ ′ ∗ ∈ 6 6 0. Therefore, given (x , ξ ) S Σ then at least one of the vector spaces E (D) ′ ′ and ∈ + (x ,ξ ) E+(D)(x′,−ξ′) is non-zero and at least one of the vector spaces E−(D)(x′,ξ′) and E−(D)(x′,−ξ′) is non-zero.

∗ Remark 3.5. We remark that if dim(M) > 2, then π0(S Σ) = π0(Σ) and in this case ∗ ∗ E+(D), E−(D) S Σ are non-trivial over each component of S Σ. If dim(M)=2, the “odd-to- → c even” part of a spin Dirac operator twisted by a line bundle provides an example where E+(D) and E−(D) satisfy rk(E+(D))+rk(E−(D))=1 but are non-vanishing over each connected component of the boundary; their ﬁbrewise supports are on complementary components of S∗Σ= S1 Σ ˙ S1 Σ. × ∪ ×

To construct the Calderón projection, we use the Douglis-Nirenberg calculus. The reader is referred to [1, 37], see also [50, Chapter XIX.5], for details. We introduce some notation. If E Σ is a Hermitian vector bundle orthogonally graded by R (viewed as a discrete group), for s R →we define the graded Sobolev space ∈ s s′ H (Σ,EE) := ′ H (Σ,EE[t]), ⊕s +t=s where E = tE[t] is the grading of E. If EE,FF Σ are two Hermitian vector bundles or- ⊕ → m thogonally graded by R (viewed as a discrete group), we can define the space Ψcl (Σ;EE,FF ) of Douglis-Nirenberg operators from E to F of order m R as the subspace of classical pseudo- ∗ ∈ differential operators Ψcl(Σ;EE,FF ) that act homogeneously jointly in the Sobolev degree and the nE nF gradings of the vector bundles. More precisely, writing E = k=1E[sk] and F = j=1F [rj ] ⊕ ∗ ⊕ for some s1,...,snE , r1,...,rnF R, a matrix (Ajk)j=1,...,nE ,k=1,...,nF Ψcl(Σ;EE,FF ) belongs to α ∈ ∈ Ψcl(Σ;EE,FF ) if and only if the order αj,k of Aj,k satisfies αj,k = α rk + sj for all j and k. −∞∞ −s −∞ − By construction, Ψ (Σ;EE,FF ) := s∈RΨ (Σ;EE,FF )=Ψ (Σ;EE,FF ) – the ordinary smoothing α ∩α operators. We write Ψcl(Σ;E) := Ψcl(Σ;EE,EE).

We define the principal symbol σα(A) to be the matrix of principal symbols (σαj,k (Aj,k))j,k which ∞ ∗ ∗ ∗ α defines an element of C (S Σ, Hom(π EE, π F )). We say that A Ψcl(Σ;EE,FF ) is elliptic if σα (A) ∞ ∗ ∗ ∗ ∈ is invertible in all points, i.e. that σα (A) C (S Σ, Iso(π EE, π F )). By the standard construction, A Ψα (Σ;EE,FF ) is elliptic if and only if∈ there exists a parametrix B Ψ−α(Σ;F ,EE). ∈ cl ∈ cl By construction, any A Ψα (Σ;EE,FF ) defines a continuous operator ∈ cl ′ A : Hs(Σ;E) Hs (Σ;F ), → ′ ′ ′ for any s,s R with s s α. If σα (A) = 0, this operator is compact if and only if s

The graded vector bundles we shall be concerned with take the form E Cm for a trivially graded Hermitian vector bundle E. We shall tacitly grade this vector bundle as⊗ follows. We view Cm as a graded bundle by C, if s =0, 1,...,m 1, Cm[s]= − . (0, otherwise. We grade E Cm as a tensor product, i.e. ⊗ E, if s =0, 1,...,m 1, (E Cm)[s]= − ⊗ (0, otherwise. In particular, m−1 Hs(Σ; E Cm)= Hs−j(Σ; E). ⊗ j=0 M 24 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~m We also introduce the graded bundle Cop which is the trivial vector bundle of rank m, graded by~~

m m C, if s =0, 1,..., m +1, Cop[s]= C [ s]= − − . − (0, otherwise. m m The reader should note that Cop can be identified with the graded dual bundle of C , and the m m Hermitian structure of E allow us to identify E Cop with the graded dual bundle of E C . In particular, ⊗ ⊗ m−1 Hs(Σ; E Cm )= Hs+j(Σ; E). ⊗ op j=0 M We can conclude the following proposition. Proposition 3.6. The L2-pairing induces a perfect pairing Hs(Σ; E Cm) H−s(Σ; E Cm ) C, ⊗ × ⊗ op → for any s R. ∈ Definition 3.7. Let D be an elliptic differential operator of order m > 0 on a manifold M with boundary Σ acting between sections of the Hermitian vector bundles E and F . An approximate 0 m Calderón projection for D is a classical pseudo-differential operator Q Ψcl(Σ; E C ) which is order zero in the Douglis-Nirenberg sense and satisfies the following ∈ ⊗

Q2 Q is smoothing. • − Q is an approximate projection onto the Hardy space D in the following sense: • − 1 mC ∞ (1) If u ker(Dmax) then v := γu H 2 (Σ; E C ) satisfies that v Qv C (Σ; E Cm).∈ ∈ ⊗ − ∈ ⊗ − 1 m ∞ (2) For any v H 2 (Σ; E C ), there is a u ker(Dmax)+C (M, E) (smooth up to the boundary)∈ such that⊗Qv = γu. ∈ Theorem 3.8 ([50], Theorem XX.1.3). Let D be an elliptic differential operator of order m> 0. 0 m There exists an approximate Calderón projection Q Ψcl(Σ; E C ) for D with principal symbol given by ∈ ⊗

~~σ0 (Q)= p+(D), m−1 (as deﬁned in Deﬁntion 3.2) and the full symbol of Q = (Qj,k)j,k=0 computed from the formula [50, Equation XX.1.7]:~~

~~m−1−k Q f = ( i)j+l+1γ T [A f δ(l) ], (6) j,k − j ◦ m−l−k−1 ⊗ t=0 Xl=0 where T is a pseudodiﬀerential parametrix to D computed in a neighbourhood of Σ.~~

Remark 3.9. We remark that the formula for Qj,k might seem difficult to parse at first. But the fact that the trace mappings are from inside of M makes it possible to compute the full symbol by means of residue calculus and the Paley-Wiener theorem. For instance, [50, Equation XX.1.8] computes the principal symbol of Q by residue calculus as ′ ′ ′ ′ σ0(Q) (x , ξ ) σ (Q )(x , ξ )= (7) 0 j,k ≡ k−j j,k m−k−1 j+l ′ −1 ′ ′ = Resξn ξn a(x , 0,ξ,ξn) am−k−l−1(x , 0, ξ ) " # Im(Xξn)>0 Xl=0 ′ ′ ∗ ′ ′ ∗ where (x , ξ ) are coordinates on T Σ and ((x ,t), (ξ , ξn)) are coordinates on T M near Σ. The proof of [50, Theorem XX.1.3] identifies this expression with p+(D). The lower order terms in Q are, in general, hard to compute. We shall give some examples of such computations below for order 1 operators. We can condense the formula (7) into

~~′ ′ ′ ′ −1 ′ ′ σ0 (Q)(x , ξ )= Resξn a(x , 0, ξ , ξn) e(ξn)σ0 (a)(x , ξ ) , (8) Im(ξ )>0 Xn REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 25~~

∗ j m−1 −1 ∗ where e(ξn) := v(ξn) v(ξn) and v(ξn) = (ξn)j=0 . Here, we abuse notation and view a T M|Σ\Σ ⊗ m −1 | ∗ and e as endomorphisms of E Σ C . Note that a T M|Σ\Σ and e commute because they act on diﬀerent factors in the tensor| product⊗ E Cm. | |Σ ⊗

Proof of Proposition 3.3. In light of Theorem 3.8 and Remark 3.9, the proof consists of an exercise with boundary symbols and residue calculus. We compute that ∗ † ∗ ′ ′ ∗ ′ ′ −1 ′ ′ p−(D ) (x , ξ )= Resξn a (x , 0, ξ , ξn) e(ξn)σ0 (a†)(x , ξ ) = −  Im(ξ )<0 Xn  ∗ ′ ′ −1 ′ ′ ∗ = Resξn a (x , 0, ξ , ξn) e(ξn)σ0 (a†)(x , ξ ) = − Im(ξ )>0 Xn h i ∗ ′ ′ ′ ′ −1 = Resξn σ0 (a )(x , ξ )a(x , 0, ξ , ξn) e(ξn) = − † Im(ξ )>0 Xn ′ ′ ′ ′ −1 = Resξn σ0 (a)(x , ξ )a(x , 0, ξ , ξn) e(ξn) . Im(ξ )>0 Xn † † In the first equality we used that p−(D ) is the complementary projection to p+(D ) which is the sum of all residues in the lower half plane, with a minus sign due to the orientation. In the second equality we used that the adjoint is antilinear, and interchanges the upper and lower half plane. In the last equality we used Lemma 2.16. We arrive at ′ ′ −1 † ∗ ′ ′ ′ ′ σ0(a)(x , ξ ) p−(D ) (x , ξ )σ0 (a)(x , ξ )= ′ ′ −1 ′ ′ ′ ′ −1 ′ ′ =σ0 (a)(x , ξ ) Resξn σ0(a)(x , ξ )a(x , 0, ξ , ξn) e(ξn) σ0(a)(x , ξ )= Im(ξ )>0 Xn ′ ′ −1 ′ ′ = Resξn a(x , 0, ξ , ξn) e(ξn)σ0 (a)(x , ξ ) = p+(D). Im(ξ )>0 Xn Corollary 3.10. Let D be an elliptic differential operator of order m> 0 acting on a manifold of dim(M) > 1 and let Q be the approximate Calderón projection from Theorem 3.8. For any s R, the operator ∈ Q : Hs(Σ; E Cm) Hs(Σ; E Cm), (9) ⊗ → ⊗ is continuous but neither Q nor 1 Q is compact. −

~~This result should be compared to Lemma A.2 in the appendix.~~

Proof. Continuity of (9) follows from that Q is order zero in the Douglis-Nirenberg sense. By standard arguments in pseudodifferential calculus, (9) is compact for some s if and only if it is compact for all s if and only if the principal symbol vanishes. The principal symbol of Q is p+(D) which is non-trivial by Proposition 3.4, so Q is never compact on Hs(Σ; E Cm). By the same ⊗ argument, the principal symbol of 1 Q is p−(D) which is non-trivial by Proposition 3.4, so1 Q is never compact on Hs(Σ; E Cm).− − ⊗ Theorem 3.11. Let D be an elliptic differential operator of order m> 0 on a manifold of dimension > 1. Then ker(Dmax) is infinite-dimensional.

Despite this result being well known and an immediate consequence of results of Seeley that we recall in the next subsection, let us provide a short proof that remains at a symbolic level for an approximate Calder´on projection.

Proof. By Proposition 2.1, ker(Dmax) is inﬁnite-dimensional if and only if the Hardy space D is inﬁnite-dimensional. We will argue by contradiction. C 26 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Assume that is finite-dimensional. Since the space CD − 1 m H 2 (Σ; E C ), CD ⊆ ⊗ − 1 m − 1 m is finite-dimensional it is closed. Consider the operator Q : H 2 (Σ; E C ) H 2 (Σ; E C ) − 1 m ⊗ → ⊗ as in Equation (9). By Theorem 3.8, for any v H 2 (Σ; E C ) it holds that ∈ ⊗ Qv +C∞(Σ; E Cm). ∈CD ⊗ − 1 m − 1 m We conclude that Q : H 2 (Σ; E C ) H 2 (Σ; E C ) factors over the inclusion ⊗ → ⊗ ∞ m − 1 m ,( C (Σ; E C ) ֒ H 2 (Σ; E C+ CD ⊗ → ⊗ which is compact if D is finite-dimensional. Therefore, if D is finite-dimensional then Q is − 1 C m C compact on H 2 (Σ; E C ) which contradicts Corollary 3.10. ⊗ Corollary 3.12. Assume that (F,hF ) = (E,hE ) and let D : C∞(M; E) C∞(M; E) be an elliptic differential operator of order m > 0 acting on a manifold of dim(M)→> 1. We then have that spec(Dmax) = specpt(Dmax)= C. Consequently, spec(Dmin)= C.

Proof. Fix λ C, and note that D λ is again an elliptic diﬀerential operator of order m> 0. Now, Theorem 3.11∈implies that dim ker(−D λ) = . But that is exactly that λ spec (D ) max − ∞ ∈ pt max and so we conclude that spec(Dmax) = specpt(Dmax)= C. For the statement concerning Dmin, we † † ∗ note that by mirroring this argument, we obtain that spec(Dmax)= C and since Dmin = (Dmax) , † conj we obtain that spec(Dmin) = spec(Dmax) = C.

Remark 3.13. For D with ker(Dmin) = 0 (i.e., ﬁrst order Dirac-type operators), we have that 0 specpt(Dmin). However, from Corollary 3.12, we know that 0 spec(Dmin). In fact, 6∈ ∈ 2 0 specres(Dmin), i.e., it is in the residue spectrum. This is because we have that L (M; E) = ∈ † 2 ker(Dmax) ran(Dmin) from Proposition 2.6 which shows that Dmin : dom(Dmin) L (M; E) does not have⊕ dense range. In fact, the range fails to be dense on the inﬁnite dimensional→ subspace † ker(Dmax). However, despite these shortcomings, and a complete lack of spectral theory, Proposition

2.20 shows that the Dirichlet problem is indeed well-posed when restricted to ran(Dmin) = ran(DCD ) (see more in Subsection 2.4). Corollary 3.14. Assume that (F,hF ) = (E,hE ) and that D : C∞(M; E) C∞(M; E) is ﬁrst → order elliptic and Dirac-type. Then, spec(Dmin) = specres(Dmin)= C.

Proof. For λ C, the operator Dλ = D λ is again ﬁrst order Dirac-type. By Remark 3.13, we conclude that∈ 0 spec ((D ) ). But (−D ) = D λ and so the conclusion holds. ∈ res λ min λ min min −

3.2. Seeley’s results on Calderón projectors. Definition 3.15. Let D be an elliptic differential operator of order m > 0 on a manifold M with boundary Σ acting between sections of the Hermitian vector bundles E and F . A Calderón 0 m projection for D is an approximate Calderón projection PC Ψcl(Σ; E C ) such that it is a projection onto ∈ ⊗ − 1 m H 2 (Σ; E C ). CD ⊆ ⊗ Remark 3.16. Calderón projectors are not uniquely determined. For instance, if PC is a Calderón 3 projection, then for any self-adjoint smoothing operator R which is “small” and R CD =0 we have −1 | that PC(1+R) also satisfies the conditions of the theorem. We shall tacitly use only the Calderón projection constructed in the next theorem of Seeley.

~~Let us recall the following theorem of Seeley. We remark that Seeley proved this result in the larger generality of higher regularity (cf. Theorem 6.1 below) on Lp-spaces, 1~~

~~3How small can be made precise by pointwise estimates on the kernel, but it is irrelevant for the sake of this argument. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 27~~

Theorem 3.17. [Seeley [65]] Let D be an elliptic differential operator of order m> 0 on a manifold M with boundary Σ acting between sections of the Hermitian vector bundles E and F . Then there 0 m exists a Calderón projection PC Ψcl(Σ; E C ) whose full symbol is uniquely determined as that of the approximate Calderón projection∈ in Theorem⊗ 3.8. In particular,

~~σ0 (PC)= p+(D).~~

In fact, Seeley obtained even stronger results in [65] that we partly summarise in the following theorem. Theorem 3.18. Let D be an elliptic diﬀerential operator of order m > 0 on a manifold M with boundary Σ acting between sections of the Hermitian vector bundles E and F . There exists a Calder´on projection PC such that

(i) There exists a continuous Poisson operator − 1 m : H 2 (Σ; E C ) ker(D ), K ⊗ → max with PC = γ (and in particular, = PC ). (ii) The image of◦ K K K◦ − 1 m γ : dom(D ) H 2 (Σ; E C ), max → ⊗ is precisely the subspace − 1 m m− 1 m P H 2 (Σ; E C ) (1 P )H 2 (Σ; E C ). C ⊗ ⊕ − C ⊗ In particular, the identity map induces an isomorphism of Banach spaces − 1 m m− 1 m H(ˇ D) = P H 2 (Σ; E C ) (1 P )H 2 (Σ; E C ). ∼ C ⊗ ⊕ − C ⊗

Proof. The ﬁrst item is proven in [65]. The second item is a reformulation of a statement on [65, top of page 782] which was stated but not proven there. We include its proof for the convenience of the reader.

To prove the second item, it suffices to prove that m− 1 m (1 P )γdom(D ) H 2 (Σ; E C ). − C max ⊆ ⊗ We can assume that M is a domain with smooth boundary in a compact manifold Mˆ , that E and F extend to Hermitian vector bundles E,ˆ Fˆ Mˆ and that there exists an elliptic differential → operator Dˆ of order m extending D to Mˆ . Write D′ for the elliptic differential operator of order m defined from restricting Dˆ to M c. Take a g (1 P )γdom(D ) γdom(D ) and pick ∈ − C max ⊆ max f dom(Dmax) lifting g. Since g (1 PC)γdom(Dmax), [65, Lemma 5] implies that there ∈ ′ ′ 2 c ∈ − 2 exists an f ker(D ) L (M , Eˆ M c ) lifting g. We define F L (M,ˆ E) as f on M and as ′ c ∈ ⊆ ′ | ∈ f on M . Since γ(f) = g = γ(f ), we have that F dom(Dˆmax). By elliptic regularity on the m ∈ m closed manifold Mˆ , it holds that dom(Dˆmax) = H (M,ˆ E). We conclude that f H (M, E) and m− 1 m ∈ therefore g = γ(f) H 2 (Σ; E C ). ∈ ⊗ Remark 3.19. It is also proven in [65] that the complementary projection (1 PC) is (up to a finite rank projection) a projection onto the Hardy space of the complementary− manifold with boundary Mˆ M˚. Note that Seeley’s construction depends on the choice of the closed manifold Mˆ \ and the extension of D to an elliptic operator Dˆ on Mˆ . In other words, the Calderon projection is a projection onto the Hardy space along the Hardy space of the complement. For this reason, the + + Calderon projection was denoted by P in [65] and C in [45]. Our notation PC is chosen to avoid confusion with the notation for spectral projectors from [5, 6].

We remark that a Calderon projection PC such that (1 PC ) is (up to a ﬁnite rank projection) a projection onto the Hardy space of the complementary− manifold is uniquely determined modulo smoothing operator. We therefore implicitly will use the Calderon projection PC from Theorem 3.18 throughout the paper. 28 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Corollary 3.20. Let D be an elliptic diﬀerential operator of order m> 0. The short exact sequence of Hilbert spaces γ 0 dom(D ) dom(D ) H(ˇ D) 0, → min → max −→ → is split by the continuous mapping E : H(ˇ D) dom(D ), E = + E (1 P ), → max K 0 ◦ − C m− 1 m m where E0 : H 2 (Σ; E C ) H (M; E) is any continuous splitting of the trace mapping m m− 1 ⊗ m → H (M; E) H 2 (Σ; E C ). → ⊗

Proof. We have that E = EP +E(1 P ). The operators EP = and E(1 P )= E (1 P ) are C − C C K − C 0 − C continuous by Theorem 3.17, so E is continuous. Moreover, we have that γE = γ +γE0(1 PC)= P + (1 P )=1so E is a splitting. K − C − C Example 3.21. Let us provide some further context for Example 2.11. We again consider the m−1 1 m− 2 −j generalised boundary condition BSob = j=0 H (Σ; E) for an elliptic diﬀerential operator D of order m> 0. Letting P c = (I P ), the complementary projection to P , C − C L C m− 1 m c B + =P c H 2 (Σ; E C ) = = H(ˇ D), Sob CD C ⊗ ⊕CD C ⊕C by Theorem 3.18. We can now use Theorem B.23 to give an alternate proof of the fact that ran(DBSob ) is closed. We can also use Lemma B.20 to conclude ran(DBSob ) = ran(Dmax) which we formulate in the following corollary. Corollary 3.22. Let D be an elliptic diﬀerential operator of order m> 0 on a manifold M with boundary Σ acting between sections of the Hermitian vector bundles E and F . Then it holds that m ran(Dmax)= DH (M; E), 2 † as closed subspaces of L (M; F ) whose codimension is dim ker(Dmin).

The next result is another corollary of Theorem 3.18. Corollary 3.23. Let D be an elliptic diﬀerential operator of order m> 0. The boundary condition c m− 1 m := (1 P )H 2 (Σ; E C ) is regular and satisﬁes: C − C ⊗

~~c m− 1 m c is closed in H 2 (Σ; E C ) and in H(ˇ D) and for x , • C ⊗ ∈C~~

x m− 1 x ˇ . k kH 2 (Σ;E⊗Cm) ≃k kH(D) c c D =0 and + D = H(ˇ D). •The C ∩C inclusion mapsC deﬁnesC a Banach space isomorphism • c = H(ˇ D), C ⊕CD ∼ − 1 m where D is equipped with either the norm from H(ˇ D) or the norm from H 2 (Σ; E C ) (the twoC are equivalent on ). ⊗ CD Remark 3.24. It follows from the proof Theorem 2.9 and Corollary 3.23 that an elliptic diﬀerential † operator D admits an invertible realisation if and only if ker(Dmin)=0 and ker(Dmin)=0. Indeed, † ∗ for any realisation DB we have the inclusions ker(Dmin) ker(DB) and ker(Dmin) = ker(DB) † ⊆ so ker(Dmin)=0 and ker(Dmin)=0 is a necessary condition. The converse follows from that † ker(Dmin)=0 and ker(Dmin)=0 if and only if DCc is invertible by the proof Theorem 2.9 and Corollary 3.23.

For general regular boundary conditions, we have a statement similar to Corollary 3.23: Proposition 3.25. Let D be an elliptic diﬀerential operator of order m > 0 and B be a regular boundary condition. Then the following holds: REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 29

(i) (B, ) is a Fredholm pair in H(ˇ D) and CD ∗ H(ˇ D) B † = . D ∼ (B + D) ∩C C m− 1 m (ii) There is a finite dimensional space F H 2 (Σ; E C ) and a subspace B B + F ⊆ ⊗ 0 ⊆ of finite codimension defining a regular boundary condition with ker DB0 = ker(Dmin) for which we can write dom(D ) = dom(D ) ker D , max B0 ⊕ker Dmin max and H(ˇ D)= B . 0 ⊕CD

The contents of item ii) in Theorem 3.25 can be interpreted as the statement that any regular boundary condition is up to ﬁnite dimensional perturbations a complement to the Hardy space in the Cauchy data space. In light of Theorem 2.12, or rather Corollary B.29, there are constraints on the dimension of F and the codimension of B B + F given by the identity: 0 ⊆ dim(F ) dim B + F = dim ker(D ) dim ker(D† ) ind(D ). − B0 min − min − B

∗ † Proof. Item (i) follows from Theorem 2.12. If B satisfies that ker(DB) = ker(Dmin) and ker DB = ker(Dmin), then with B0 = B and F = 0, item (ii) follows from Corollary B.27 applied to the FAP with closed range := (D ,D† ). TB B min To prove item (ii), we first prove that for any regular boundary condition B there is a finite m− 1 m dimensional space F H 2 (Σ; E C ) with (B + F ) D = B D and B + F + D = H(ˇ D). Indeed, B +F is clearly⊆ semi-regular⊗ and will be regular since∩C (B +F∩C)∗ B∗. To constructC such an ˇ ⊆ F , we note that item (i) implies that the inclusion B H(D) has finite codimension. B D → D Indeed, we have that ∩C C (H(ˇ D)/ D) H(ˇ D) C (B/B ) = (B + ). ∩CD CD As such, there are plenty of finite-dimensional subspaces F H(ˇ D) with (B + F ) D = B D m⊆− 1 m ∩C ∩C and B + F + = H(ˇ D). That we can take F H(ˇ D) H 2 (Σ; E C ) follows from Example CD ⊆ ∩ ⊗ m− 1 m H(ˇ D) induces an isomorphism which implies that the inclusion H 2 (Σ; E C ) ֒ 3.21 ⊗ → CD m− 1 m H 2 (Σ; E C ) H(ˇ D) m− 1 m = . ⊗ H 2 (Σ; E C ) ∼ D ⊗ ∩CD C

To finish the proof, we need to construct B0. Upon replacing B with B + F , the argument in the preceding paragraph allow us to assume that the regular boundary condition B satisfies ∗ † ker(DB) = ker(Dmin). We need to construct a regular boundary condition B0 B of finite m−⊆1 m codimension such that B0 + D = B + D and B0 D = 0. Consider any B0 H 2 (Σ; E C ) C C ∩C m− 1⊆ m ⊗ which forms a complement of B in B. Since B is regular, B H 2 (Σ; E C ) is finite- ∩CD ∩CD ⊆ ⊗ dimensional so B B has finite codimension. Moreover, B is closed in H(ˇ D) and by construction 0 ⊆ 0 B0 + D = B + D and B0 D = 0. It remains to prove that we can take the complement B0 C C∗ m− 1 ∩C m ∗ ∗ to satisfy that B0 H 2 (Σ; F C ). We have that B B0 has finite codimension and using ⊆ ⊗ ⊆ m− 1 m B∗ H 2 (Σ; F C ) Proposition 2.18 we deduce that we can choose B0 such that 0B∗ ⊗ B∗ which proves the theorem. ⊆

3.3. Boundary decomposing projectors. With Theorem 3.17, we are able to prove results of use when studying graphical decompositions of regular boundary conditions, see more in [6] and Subsection 5 below. We make the following deﬁnition that we shall make use of in the ﬁrst order case below in Subsection 5 30 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~Definition 3.26. Let D be an elliptic differential operator of order m> 0. We say that a projection is boundary decomposing for D if the following conditions are satisfied P+~~

α m α m 1 1 (P1) + : H (Σ; E C ) H (Σ; E C ) is a bounded projection for α 2 ,m 2 , P ⊗ → ⊗ m− 1 m∈ − − (P2) : H(ˇ D) H(ˇ D), and := (I ): H(ˇ D) H 2 (Σ; E C ), and P+ → P− −P+ → ⊗ (P3) u ˇ −u m− 1 + +u − 1 . k kH(D) ≃ kP kH 2 (Σ;E⊗Cm) kP kH 2 (Σ;E⊗Cm)

~~The next proposition follows from the open mapping theorem.~~

Proposition 3.27. Let D be an elliptic diﬀerential operator of order m > 0 and + a boundary decomposing projection for D. Then P − 1 m− 1 H(ˇ D)= H 2 (Σ; E) and H(ˇ D)= H 2 (Σ; E), P+ P+ P− P− with equivalent norms.

~~Let us state yet another corollary of Theorem 3.17.~~

Corollary 3.28. The Calderón projection PC of an elliptic differential operator D is bound- c ary decomposing for D. Moreover, the complementary subspace := (I PC)H(ˇ D) = (1 m− 1 m C − − P )H 2 (Σ; E C ) defines a regular boundary condition with ker(D c ) = ker(D ) and char- C ⊗ C min acterises dom(Dmax) via the following continuously split short exact sequence of Hilbert spaces

~~PC ◦γ 0 dom(D c ) dom(D ) 0. → C → max −−−→C →~~

Proof. By the fact that PC is a pseudo-differential operator of order zero in the Douglis-Nirenberg calculus, it automatically satisfies (P1). The properties (P2) and (P3) follow from Theorem 3.18. It follows that c is a regular boundary condition. Since the restricted trace map is a surjection c C ker(D c ) = 0 with kernel ker(D ), we conclude that ker(D c ) = ker(D ). C →C ∩C min C min Example 3.29. The Hardy space itself is a boundary condition, so we may consider the closed operator DCD . The realisation DCD is called the soft extension, or the Krein extension, and is self-adjoint if D is formally self-adjoint. Its properties were further studied in [39]. It is clear that ker(DCD ) = ker(Dmax), which we have noted is infinite dimensional if dim(M) > 1. By Proposition B.10, setting B = H(ˇ D) and B = , we obtain that 2 1 CD ˇ dom(Dmax) = H(D) = c, dom(D ) ∼ D ∼ CD C C where c is the kernel of P . By Corollary 3.28, we have that c is infinite dimensional. That is, C C C DC is an operator which differs from Dmax on an infinite dimensional subspace but still has infinite dimensional kernel.

This can be repeated on replacing by any infinite dimensional closed subspace of . Therefore, there’s an abundance of boundaryC conditions with infinite dimensional kernel whichC is different from Dmax on an infinite dimensional subspace.

We now state a sufficient condition for a projection to be boundary decomposing. Theorem 3.30. Let D be an elliptic differential operator of order m > 0 and P a continuous m− 1 m projection on H 2 (Σ; E C ). Assume that P satisfies the following: ⊗

− 1 m P is continuous also in the H(ˇ D)-norm and the H 2 (Σ; E C )-norm. • ⊗ m− 1 m The operator A := P (1 P ) deﬁnes a Fredholm operator on H 2 (Σ; E C ) and by • C − − ⊗ continuity extends to a Fredholm operator on H(ˇ D).

~~Then P is boundary decomposing, and in particular~~

~~u ˇ (1 P )u m− 1 + Pu − 1 . k kH(D) ≃k − kH 2 (Σ;E⊗Cm) k kH 2 (Σ;E⊗Cm) REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 31~~

~~Proof. By the open mapping theorem and density it suﬃces to prove that the assumptions on P ensure that~~

(1 P )u ˇ (1 P )u m− 1 and Pu ˇ Pu − 1 , k − kH(D) ≃k − kH 2 (Σ;E⊗Cm) k kH(D) ≃k kH 2 (Σ;E⊗Cm) m− 1 m for u H 2 (Σ; E C ). ∈ ⊗ m− 1 m Since we have a continuous embedding H 2 (Σ; E C ) ֒ H(ˇ D), we conclude that there is an ⊗ → 1 m− 2 m estimate (1 P )u ˇ . (1 P )u m− 1 for u H (Σ; E C ). We now prove the k − kH(D) k − kH 2 (Σ;E⊗Cm) ∈ ⊗ m− 1 m converse estimate. Since A is Fredholm on H 2 (Σ; E C ), we conclude that there is a ﬁnite m− 1 m ⊗ rank operator F on H 2 (Σ; E C ) such that ⊗

~~u m− 1 Au m− 1 + Fu m− 1 . k kH 2 (Σ;E⊗Cm) ≃k kH 2 (Σ;E⊗Cm) k kH 2 (Σ;E⊗Cm) By density, we can assume that F extends to a ﬁnite rank operator on H(ˇ D). We then have that~~

(1 P )u m− 1 . A(1 P )u m− 1 + F (1 P )u m− 1 . (10) k − kH 2 (Σ;E⊗Cm) k − kH 2 (Σ;E⊗Cm) k − kH 2 (Σ;E⊗Cm) But A(1 P ) = (P (1 P ))(1 P ) = (P 1)(1 P ) so − C − − − C − −

~~A(1 P )u m− 1 + F (1 P )u m− 1 = (11) k − kH 2 (Σ;E⊗Cm) k − kH 2 (Σ;E⊗Cm)~~

~~= (1 PC)(1 P )u m− 1 + F (1 P )u m− 1 = k − − kH 2 (Σ;E⊗Cm) k − kH 2 (Σ;E⊗Cm)~~

= (1 PC)(1 P )u ˇ + F (1 P )u m− 1 . k − − kH(D) k − kH 2 (Σ;E⊗Cm) . (1 P )u ˇ + F (1 P )u ˇ . (1 P )u ˇ . k − kH(D) k − kH(D) k − kH(D) In the second last estimate we used that all norms on finite-dimensional spaces are equivalent and in the third estimate we used that F extends to a finite rank operator on H(ˇ D). Combining the estimates (10) and (11) we arrive at the estimate (1 P )u m− 1 . (1 P )u ˇ . k − kH 2 (Σ;E⊗Cm) k − kH(D) ˇ The proof that Pu ˇ Pu − 1 goes ad verbatim with H(D) instead of k kH(D) ≃ k kH 2 (Σ;E⊗Cm) m− 1 m − 1 m H 2 (Σ; E C ) and H 2 (Σ; E C ) instead of H(ˇ D) and using that A is Fredholm on H(ˇ D). ⊗ ⊗ Lemma 3.31. Let D be an elliptic differential operator of order m> 0 and T Ψ0 (Σ; E Cm). ∈ cl ⊗ Then T defines a bounded operator on H(ˇ D) if and only if (1 P )T P Ψ−m(Σ; E Cm). − C C ∈ cl ⊗

Proof. Assume first that (1 P )T P Ψ−m(Σ; E Cm) holds. The property (1 P )T P − C C ∈ cl ⊗ − C C ∈ Ψ−m(Σ; E Cm) implies that cl ⊗ T =P T P + (1 P )T (1 P )+ P T (1 P )+(1 P )T P C C − C − C C − C − C C =P T P + (1 P )T (1 P )+ P T (1 P )+Ψ−m(Σ; E Cm). C C − C − C C − C cl ⊗ ˇ −m m It now follows that T acts boundedly on H(D) from the facts that Ψcl (Σ; E C ) acts boundedly ˇ 0 m s m t m⊗ on H(D) and Ψcl(Σ; E C ) acts boundedly H (Σ; E C ) H (Σ; E C ) for all s,t R with t s. ⊗ ⊗ → ⊗ ∈ ≤ Conversely, assume that T acts boundedly on H(ˇ D). Since T Ψ0 (Σ; E Cm), the argument above ∈ cl ⊗ implies that the pseudodifferential operator (1 PC)T PC act boundedly on H(ˇ D). In particular, − − 1 m m− 1 m (1 PC )T PC will have to extend to a continuous operator H 2 (Σ; E C ) H 2 (Σ; E C ) − −m ⊗ → ⊗ and so it must be an operator of order m, i.e. (1 P )T P Ψ−m(Σ; E Cm). − − C C ∈ cl ⊗ 0 m Corollary 3.32. Let D be an elliptic differential operator of order m> 0 and P Ψcl(Σ; E C ) a projection. Assume that P satisfies the following: ∈ ⊗

~~[P , P ] Ψ−m(Σ; E Cm). • C ∈ cl ⊗ The operator A := P (1 P ) Ψ0 (Σ; E Cm) is elliptic. • C − − ∈ cl ⊗ 32 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN~~

~~Then P is boundary decomposing, and in particular~~

~~u ˇ (1 P )u m− 1 + Pu − 1 . k kH(D) ≃k − kH 2 (Σ;E⊗Cm) k kH 2 (Σ;E⊗Cm)~~

Proof. Since (1 P )P P = [P , P ]P Ψ−m(Σ; E Cm), the projection P extends to a con- − C C − C C ∈ cl ⊗ tinuous projection on H(ˇ D) by Lemma 3.31. Therefore P defines a continuous projection on m− 1 m − 1 m H 2 (Σ; E C ), H 2 (Σ; E C ) and H(ˇ D). Since A := PC (1 P ) is elliptic, it defines a ⊗ m− 1 ⊗ m − − Fredholm operator on H 2 (Σ; E C ). ⊗ Let us now prove that A extends to a Fredholm operator on H(ˇ D). Let R denote a parametrix of 0 m −∞ m A. We proceed by computing in the formal symbol algebra Ψcl(Σ; E C )/Ψ (Σ; E C ). For a zeroth order pseudodifferential operator T we write T for its associated⊗ full symbol.⊗ In this notation, R = A −1. We compute that ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ [P , R] =[ P , A −1]= A −1[ P , A ] A −1 = ⌊ C ⌋ ⌊ C ⌋ ⌊ ⌋ −⌊ ⌋ ⌊ C ⌋ ⌊ ⌋ ⌊ ⌋ = A −1[ P , P ] A −1 = R[P , P ]R . −⌊ ⌋ ⌊ C⌋ ⌊ ⌋ ⌊ ⌋ −⌊ C ⌋ Therefore it holds that [P , R]= R[P , P ]R+Ψ−∞(Σ; E Cm). Since [P , P ] Ψ−m(Σ; E Cm), C − C ⊗ C ∈ cl ⊗ we conclude that (1 P )RP = [P , R]P Ψ−m(Σ; E Cm) so R acts boundedly on H(ˇ D). − C C − C C ∈ cl ⊗ Therefore, density and the fact that smoothing operators act compactly on H(ˇ D) implies that A extends to a Fredholm operator on H(ˇ D).

~~The corollary now follows from Theorem 3.30.~~

Below in Theorem 4.15 we shall see that under assumptions on P similar to those in Corollary m− 1 m 3.32, BP := (1 P )H 2 (Σ; E C ) is a regular boundary condition. For the precise relations between the assumptions− in Theorem⊗ 4.15 and Corollary 3.32, see the formulation of Theorem 4. Example 3.33. Let us briefly comment on the case of a first order elliptic differential operator 0 D. In this case, Corollary 3.32 shows that any P Ψ (Σ; E) with P and PC commuting at ∞ ∗ ∈ ∗ principal symbol level (as elements of C (S M; End(π E))) and with p+(D) (1 σ0(P )) C∞(S∗M; Aut(π∗E)), defines a boundary decomposing projection. Below in Lemma− 9.1−, we prove∈ + that p+(D) coincides with the principal symbol of the positive spectral projection χ (A) for an adapted boundary operator. In particular, P := χ+(A) defines a boundary decomposing projection by Corollary 3.32. This proves the claimed equality H(ˇ D) = Hˇ A(D) from Subsection 2.6, thus reconciling the topological considerations of H(ˇ D) from Subsection 3.2 with that of [6] in the first order case. Example 3.34. Consider the case of a second order elliptic differential operator D acting on a vector bundle E with scalar positive principal symbol (e.g. a Laplace type operator on a vector bundle). In this case, it is readily verified from solving Equation (5) that ′ −1 ′ ′ 1 1E ξ 1E σ0 (PC )(x , ξ )= ′ | | , 2 ξ 1E 1E | | for the Riemannian metric on Σ induced from the Riemannian metric that σ2(D) defines on M. We can consider the projectors defining Dirichlet or Neumann conditions (from the rule BP := m− 1 2 (1 P )H 2 (Σ; E C ) or the explicit condition Pγu =0) are given by − ⊗ 1 0 0 0 P := and P := . D 0 0 N 0 1 We then have that ′ −1 ′ −1 1 0 ξ 1E 1 0 ξ 1E [PD, PC]= ′ | | and [PN , PC]= ′ −| | . 2 ξ 1E 0 2 ξ 1E 0 −| | | | Therefore, (1 P )P P , (1 P )P P Ψ0 (Σ; E C2) with non-vanishing principal symbol. By − C N C − C D C ∈ cl ⊗ Lemma 3.31, we conclude that neither PD nor PN act boundedly on H(ˇ D). In particular, these REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 33

~~projectors are not boundary decomposing. A more appropriate projector is considered in Subsection 6.3.~~

~~4. The Cauchy data space of an elliptic differential operator~~

~~In this section we will reﬁne our analysis of the Cauchy data space and boundary conditions further.~~

4.1. Constructing regular pseudodifferential boundary conditions. Theorem 3.18 implies that we can describe the Cauchy data space H(ˇ D) in terms of the Calderón projection. Unfortu- nately, the construction of the Calderón projection requires knowledge from the interior of M and not just symbolic information of D at the boundary. In this subsection, we give a construction of projectors from the Douglis-Nirenberg calculus that rely only on the germ of D at the boundary yet still describes the Cauchy data space as in Corollary 3.23. The considerations of this subsection implies that many structural properties of H(ˇ D) extends to the case of noncompact M with compact boundary. Recall the definition of a boundary decomposing projection from Definition 3.26. Lemma 4.1. Let D be an elliptic differential operator of order m> 0. Then the following scheme produces a boundary decomposing pseudodifferential projection P Ψ0 (Σ; E Cm) with P P ∈ cl ⊗ − C ∈ Ψ−m(Σ; E Cm). cl ⊗

0 m (1) Choose P1 Ψcl(Σ; E C ) with σ0 (P1)= p+(D). ∈ 0 ⊗ m −1 m (2) Construct P Ψ (Σ; E C ) by choosing a T Ψ (Σ; E C ) with σ 1(T ) = 2 ∈ cl ⊗ 2 ∈ cl ⊗ −1 2 σ−1(P1 PC ) and set P2 := P1 T2. − − 0 m (3) Proceed iteratively and deﬁne Pk+1 Ψcl(Σ; E C ) from Pk by choosing a Tk+1 −k m ∈ ⊗ ∈ Ψcl (Σ; E C ) with σ−k (Tk+1) = σ−k (Pk PC) and set Pk+1 := Pk Tk+1. Stop at k = m. ⊗ − − (4) Choose a compact complex contour Γ in the open right half-plane containing 1 in its interior − 1 m − 1 m domain and not intersecting the spectrum of Pm : H 2 (Σ; E C ) H 2 (Σ; E C ), and deﬁne P as the Riesz projection ⊗ → ⊗

~~1 −1 P := (λ Pm) dλ. 2πi ˆΓ −~~

~~m− 1 m Moreover, B := (1 P )H 2 (Σ; E C ) is a regular boundary condition for D that characterises P − ⊗ dom(Dmax) via the following continuously split short exact sequence of Hilbert spaces~~

~~P ◦γ − 1 m 0 dom(D ) dom(D ) P H 2 (Σ; E C ) 0. → B → max −−−→ ⊗ →~~

Proof. We need to verify that the stated scheme is possible to carry out and that it produces a 0 m boundary decomposing pseudodifferential projection P Ψcl(Σ; E C ) that defines a regular boundary condition. For the proof, it is beneficial to bear∈ in mind tha⊗ t if e is an idempotent in a unital algebra, then the spectrum of e is 0, 1 and for λ =0, 1 we have that { } 6 (λ e)−1 = λ−1(1 e) + (λ 1)−1e. − − − In particular, if e is an idempotent in a unital algebra its Riesz projectors are well defined and that onto the component 1 coincides with e. { } Surjectivity of the symbol mappings ensures that the construction of P Ψ0 (Σ; E Cm) is m ∈ cl ⊗ possible. Indeed, Pm will in local charts have a full symbol that up to degree m is determined by the formula (6). We note that the spectrum of P : Hs(Σ; E Cm) Hs(Σ;− E Cm) is by m ⊗ → ⊗ elliptic regularity independent of s R. Since σ0 (Pm) by construction is a projection, the essential s m ∈ s m spectrum of Pm : H (Σ; E C ) H (Σ; E C ) is precisely the set 0, 1 C. This ensures the existence of the complex⊗ contour→ Γ. By⊗ construction, P will be a projection{ }⊆ and belong to Ψ0 (Σ; E Cm) (since Γ is compact). cl ⊗ 34 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

−m m −1 −1 To prove that P PC Ψcl (Σ; E C ) it suffices to prove that (λ Pm) (λ PC) −m m − ∈ ⊗ − −−m − m∈ Ψ (Σ; E C ) for λ Γ. By construction, σ m−1(P P )=0,so P P Ψ (Σ; E C ) cl ⊗ ∈ −m−1 m − C m − C ∈ cl ⊗ and (λ P )−1 (λ P )−1 Ψ−m(Σ; E Cm) follows from the identity − m − − C ∈ cl ⊗ (λ P )−1 (λ P )−1 = (λ P )−1(P P )(λ P )−1. − m − − C − m C − m − C −m m Since P PC Ψcl (Σ; E C ), P is boundary decomposing because of Corollary 3.32. The proof that− P defines∈ a regular⊗ boundary condition is postponed until Theorem 4.14 below. Remark 4.2. The reader should be aware that the condition P P Ψ−m(Σ; E Cm) is far − C ∈ cl ⊗ from sufficient to guarantee that P P is compact on H(ˇ D). This happens despite P P being − C − C compact on Hs(Σ; E Cm) for all s R (it even has singular values behaving like O(k−m/ dim(M)) as k ). We give⊗ an example thereof∈ in Proposition 9.12 below in order m =1. → ∞ −m m We also note that the condition P PC Ψcl (Σ; E C ) is far from necessary for P to be − ∈ ⊗ m− 1 m boundary decomposing (cf. Corollary 3.32) or even for BP := (1 P )H 2 (Σ; E C ) to be a regular boundary condition for D (cf. Theorem 4.14 below). − ⊗ Proposition 4.3 (Localizability of H(ˇ D)). Let D be an elliptic differential operator of order m> 0 on a manifold with boundary M. Pick a point x0 Σ and coordinates κ : U U0 M around ∈ → ∗ ⊆ x0. Let D0 be an elliptic differential operator of order m on U such that D0 and κ D coincide on −1 n 0 ∗ κ (U0 Σ) = U ∂R+ to order m. Let QU Ψcl(U; κ E) differ from an approximate Calderón ∩ ∩ −m ∗ ∈ projection for D0 by a term in Ψcl (U; κ E).

′ ˇ ∞ Then for any u (Σ; E), it holds that χ0u H(D) for all χ0 Cc (U0) if and only if (1 m∈− 1 D ∈ − 1 ∈ − Q )[χκ∗u] H 2 (U; κ∗E Cm) and Q [χκ∗u] H 2 (U; κ∗E Cm) for all χ C∞(U). U ∈ loc ⊗ U ∈ loc ⊗ ∈ c

′ ∞ ∗ ′ −m ∗ Proof. It follows from Theorem 3.8 that for any χ,χ Cc (U), χ(κ PC QU )χ Ψcl (U; κ E). m− 1 m ∈ ∞ − ∈ ∗ Therefore, (1 PC )χ0u H 2 (Σ; E C ) for all χ0 Cc (U0) if and only if (1 QU )[χκ u] m− 1 − ∈ ⊗ ∈ 1 − ∈ 2 ∗ m ∞ − 2 m ∞ Hloc (U; κ E C ) for all χ Cc (U) and PCχ0u H (Σ; E C ) for all χ0 Cc (U0) if and ⊗ − 1 ∈ ∈ ⊗ ∈ only if Q [χκ∗u] H 2 (U; κ∗E Cm) for all χ C∞(U). The proposition follows. U ∈ loc ⊗ ∈ c

4.2. The boundary pairing and Calderón projectors of the formal adjoint. In this subsection, we shall study the relation between a Calderón projection of D and a Calderón projection of its formal adjoint D†. We apply this to the boundary pairing. First, we shall revisit the operators τ and a from Subsection 2.3. The next proposition follows from Lemma 2.17 and the following observations.

m−1 ∗ For a matrix T = (Ti,j )i,j=0 of pseudo-diﬀerential operators on a closed manifold, we write T = ∗ 2 (Tj,i)i,j for its L -adjoint and # ∗ ∗ T := τT τ = (Tm−i−1,m−j−1)i,j , where τ is deﬁned in (1). A short computation shows that if T Ψα (Σ; E Cm, F Cm) then ∈ cl ⊗ ⊗ T # Ψα (Σ; F Cm, E Cm) and T ∗ Ψα (Σ; F Cm , E Cm ). ∈ cl ⊗ ⊗ ∈ cl ⊗ op ⊗ op Proposition 4.4. The operators a, a, and τ from Subsection 2.3 satisfy the following.

(i) For any s R, τ defines a unitarye isomorphism ∈ Hs(Σ; F Cm ) Hs+m−1(Σ; F Cm), ⊗ op → ⊗ and for any α R, conjugation by τ defines a product preserving isomorphism ∈ Ψα (Σ; E Cm, F Cm) Ψα (Σ; E Cm , F Cm ). cl ⊗ ⊗ → cl ⊗ op ⊗ op (ii) For any s R, a defines a Banach space isomorphism ∈ Hs(Σ; E Cm) Hs(Σ; F Cm), e ⊗ → ⊗ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 35

and a defines a Banach space isomorphism Hs(Σ; E Cm) Hs−m+1(Σ; F Cm ), ⊗ → ⊗ op and for any α R, conjugation by a (i.e. the mapping T aT a−1) defines a product preserving isomorphism∈ 7→ Ψα (Σ; E Cm) Ψα (Σ; F Cm ). cl ⊗ → cl ⊗ op Proposition 4.5. Let D be an elliptic differential operator of order m> 0. Let a and a† denote the matrices of differential operators constructed from D and D†, respectively, as in Proposition 0 m m 0 m m 2.15. It holds that a Ψcl(Σ; E C , F C ) and a† Ψcl(Σ; F C , E C e) are elliptice and invertible in the Douglis-Nirenberg∈ ⊗ calculus⊗ and satisfy ∈ ⊗ ⊗ e # e a = a†.

~~e e Proof. The proposition is a consequence of Proposition 2.15 and Lemma 2.16.~~

~~The following theorem reﬁnes Proposition 3.3 to the operator level.~~

Theorem 4.6. Let D be an elliptic differential operator of order m > 0. We assume that PC 0 m 0 m ∈ Ψ (Σ; E C ) is a Calderón projection for D and P † Ψ (Σ; F C ) is a Calderón projection cl ⊗ C ∈ cl ⊗ for D†. Then it holds that −1 ∗ −∞ m P † =a (1 P )a† +Ψ (Σ; F C )= C † − C cl ⊗ −1 ∗ −∞ m =[a(1 PC)a ] +Ψ (Σ; F C ). − cl ⊗

−1 ∗ Proof. Before proving the main identity of the theorem, we remark that a† (1 PC )a† = [a(1 −1 ∗ −1 0 m − − PC)a ] by Lemma 2.16 and a(1 PC)a Ψ (Σ; F C ) by Proposition 4.4. − ∈ cl ⊗ op Let us proceed with the proof. It follows from [34, Chapter II, Lemma 1.2] that, up to a smoothing operator, we have ω (ξ, η)= ω (P † ξ, η) ω (ξ, P η), D D C − D C for ξ H(ˇ D†) and η H(ˇ D). By Proposition 2.15 and Lemma 2.16, this is equivalent to the equality∈ (valid up to a∈ smoothing operator)

~~a†ξ, η L2(Σ;E⊗Cm) = a†ξ, PC η L2(Σ;E⊗Cm) P † ξ, aη L2(Σ;F ⊗Cm). h i h i − h C i In particular, we have that~~

a†ξ, (1 PC)η L2(Σ;E⊗Cm) = P † ξ, aη L2(Σ;F ⊗Cm), h − i −h C i ∗ up to a smoothing operator. We deduce from Lemma 2.16 that (1 PC )a† = a†PC† up to a smoothing operator. − Remark 4.7. The reader should compare Theorem 4.6 to Proposition 2.26 describing adjoints of boundary conditions deﬁned from projectors.

Corollary 4.8. Let D be an elliptic differential operator of order m > 0. We assume that PC 0 m 0 m ∈ Ψ (Σ; E C ) is a Calderón projection for D and P † Ψ (Σ; F C ) is a Calderón projection cl ⊗ C ∈ cl ⊗ for D†. Then a extends to a continuous and invertible operator m− 1 m ∗ 1 −m m (1 P )H 2 (Σ; E C ) (1 P † )H 2 (Σ; F C ) C − C ⊗ op a : − ⊗ . − 1 m −→ ∗ 1 m P H 2 (Σ; E C ) P H 2 (Σ; F C ) C L ⊗ C† L ⊗ op Moreover, in this decomposition + 0 a −∞ m m a = − +Ψ (Σ; E C , F C ), a 0 ⊗ ⊗ where + ∗ − ∗ a = (1 P † )aPC and a = P † a(1 PC ). − C C − 36 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Proof. This follows directly from Proposition 4.4 and the relation between PC and PC† from The- orem 4.6. Definition 4.9. Let D be an elliptic differential operator of order m> 0. We define the space ∗ 1 −m m (1 P )H 2 (Σ; E C ) − C ⊗ op H(ˆ D)= ∗ 1 m P H 2 (Σ; E C ) C L ⊗ op Lemma 4.10. The L2-pairing induces a perfect pairing , : H(ˆ D) H(ˇ D)∗ and hence a Banach space isomorphism h· ·i × ˆ ˇ ∗ H(D) ∼= H(D) . Moreover, the operator a defines an isomorphism a : H(ˇ D) H(ˆ D†). →

Proof. By Theorem 3.18 we have that m− 1 m (1 PC)H 2 (Σ; E C ) H(ˇ D)= − ⊗ . − 1 m P H 2 (Σ; E C ) C L ⊗ 2 ˆ ˇ ∗ Proposition 3.6 implies that the L -pairing induces a Banach space isomorphism H(D) ∼= H(D) . The fact that a : H(ˇ D) H(ˆ D†) is an isomorphism follows from Corollary 4.8. → Remark 4.11. If D is an elliptic diﬀerential operator of order m = 1 with adapted boundary operator A, the results of [6] shows that − ∗ − 1 + ∗ 1 H(ˆ D)= χ (A )H 2 (Σ; E) χ (A )H 2 (Σ; E). ⊕ We use the notation Hˆ A(D) for the right hand side of this equation. The reader should be aware that in [6], this space was denoted by H(ˆ A∗). Theorem 4.12. Let D be an elliptic diﬀerential operator of order m> 0. The boundary pairing ω : H(ˇ D†) H(ˇ D) C. D × → takes the form ′ ′ ˇ † ′ ˇ ωD(ξ, ξ )= ξ, aξ L2(Σ;F ⊗Cm), ξ H(D ), ξ H(D), h i ∈ ∈ and is a perfect pairing.

~~Proof. The formula for the boundary pairing for smooth sections follows from Proposition 2.15 and that it can be written as the stated L2-pairing follows from Lemma 4.10.~~

4.3. Adjoint boundary conditions and a sufficient condition for regularity. In light of the results of last subsection, we can further describe adjoint boundary conditions. In particular, we can derive a sufficient condition for certain semi-regular boundary conditions to be regular. Proposition 4.13. Let D be an elliptic differential operator of order m > 0 and B H(ˇ D) a ⊆ generalised boundary condition. The adjoint boundary condition B∗ H(ˇ D†) takes the form ⊆ ∗ ⊥ −1 ⊥ B = (aB) = a† B , where denotes the induced annihilators with respect to L2-pairing. ⊥

Proof. Using the description in Lemma 4.10, the space (aB)⊥ H(ˇ D†) is the annihilator of ⊆ aB H(ˆ D†) in the perfect L2-pairing H(ˆ D†) H(ˇ D†) C. By Theorem 4.12, (aB)⊥ is the ⊆ × ∗ → ⊥ annihilator of B in the boundary pairing ωD, so B = (aB) follows. By a similar argument, ∗ ˆ ˇ 2 ∗ ⊥ a†B H(D) is the annihilator of B H(D) in the L -pairing, so a†B = B which implies ∗ ⊆−1 ⊥ ⊆ B = a† B . REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 37

The reader should compare Proposition 4.13 to Proposition 2.26 describing the adjoint boundary condition of a pseudo-local boundary condition. We shall now state a theorem providing a rather general sufficient condition for regularity. Theorem 4.14. Let D be an elliptic differential operator of order m > 0 and P a continuous m− 1 m − 1 m projection on H 2 (Σ; E C ) that extends by continuity to a projection on H 2 (Σ; E C ). ⊗ m− 1 ⊗ m Assume that the operator A := PC (1 P ) defines a Fredholm operator on H 2 (Σ; E C ) − − − 1 m ⊗ that extends by continuity to a Fredholm operator on H 2 (Σ; E C ). Then P defines an regular boundary condition ⊗ m− 1 m B := (1 P )H 2 (Σ; E C ), P − ⊗ for D. In other words, we have that u dom(D ): Pγu =0 = u Hm(M; E): Pγu =0 , { ∈ max } { ∈ } and this space defines the domain of an regular realisation of D.

Proof. By using the same argument as in the proof of Theorem 3.30, we see that if A := PC (1 P ) m− 1 m − − is Fredholm on H 2 (Σ; E C ), then ⊗ (1 P )u ˇ (1 P )u m− 1 , k − kH(D) ≃k − kH 2 (Σ;E⊗Cm) m− 1 m for u H 2 (Σ; E C ). In particular, B is closed in H(ˇ D). This proves that B is a semi-regular boundary∈ condition.⊗ It remains to prove that B∗ is a semi-regular boundary condition.

∗ m− 1 m We claim that B = (1 P†)H 2 (Σ; F C ) and that this is a semi-regular boundary condition. − −1 ⊗ 1 1 ∗ ∗ 2 m 2 m Here we use the notation P† := a† (1 P )a†, where P : H (Σ; E Cop) H (Σ; E Cop) is the − 2 − 1 m ⊗ 1 → m ⊗ continuous projection deﬁned from the L -pairing H 2 (Σ; E C ) H 2 (Σ; E C ) C. By the ⊗ × ⊗ op → arguments in the preceding paragraph and Proposition 2.26, this claim follows if A† := PC† (1 P†) m− 1 m − − is Fredholm on H 2 (Σ; F C ). However, using Theorem 4.6 we have that ⊗ −1 ∗ A† = a† A a†, − 1 m ∗ where we by an abuse of notation write A for its Fredholm extension to H 2 (Σ; E C ) and A 2 1 m ⊗ for its L -adjoint on H 2 (Σ; E Cop). The fact that A† is Fredholm now follows from that A is − 1 m ⊗ Fredholm on H 2 (Σ; E C ). ⊗

The next result specialises Theorem 4.14 to pseudolocal boundary conditions and characterises those that are regular. We encourage the reader to compare the assumptions on P in the next corollary to those in Corollary 3.32. 0 m Theorem 4.15. Let D be an elliptic diﬀerential operator of order m> 0 and P Ψcl(Σ; E C ) a projection. Deﬁne the generalised boundary condition ∈ ⊗ m− 1 m B := (1 P )H 2 (Σ; E C ). P − ⊗ Then the following are equivalent:

~~i) The operator P (1 P ) Ψ0 (Σ; E Cm) is elliptic. C − − ∈ cl ⊗ ii) BP is an regular boundary condition for D. iii) BP is a Fredholm boundary condition for D.~~

We remark that the space BP defined in the theorem above is a priori different from the pseudo- local boundary condition defined from P as in Definition 2.24. A posteriori, they produce the same boundary condition under any of the assumptions i)-iii) in Theorem 4.15.

Proof. Item i) implies item ii) by Theorem 4.14. Item ii) implies item iii) by Proposition 2.6. Using [50, Theorem XIX.5.1 and XIX.5.2], to prove that iii) implies i) it suﬃces to prove that P (1 P ) C − − is Fredholm as soon as BP is Fredholm boundary condition. We note that BP is Fredholm if and 38 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

only if B∗ is Fredholm if and only if (P H(ˇ D),B ) and ((1 P ∗)H(ˆ D),B⊥) are Fredholm pairs. P C P − C P As such, PC (1 P ) is Fredholm whenever BP is a Fredholm boundary condition by combining [6, Lemma A.9]− with− Atkinson’s theorem.

To add some more flavor to the regularity condition on a pseudo-local boundary condition as in Theorem 4.15, let us compare this to the well studied notion of Shapiro-Lopatinskii ellipticity. Definition 4.16. Let D be an elliptic differential operator of order m > 0 and B a pseudo-local 0 m boundary condition defined from a pseudo-differential projection P Ψcl(Σ; E C ). We say that P is Shapiro-Lopatinskii elliptic if for any (x′, ξ′) T ∗Σ, when the∈ differential⊗ equation ∈ ′ ′ σ∂ (D)(x , ξ )v(t)=0, t> 0, m−1 ′ ′ j (12) σ0 (P )(x , ξ ) Dt v(0) = h,  j=0 has a solution it is uniquely determined and exponentially decaying.

Compare the definition of Shapiro-Lopatinskii elliptic pseudo-differential projectors the the construction of E+(D) in Proposition 3.1. The reader can find more on Shapiro-Lopatinskii ellipticity of pseudo-local boundary conditions in [63]. Similarly, for more general boundary conditions, if for any (x′, ξ′) T ∗Σ, the differential equation ∈ ′ ′ σ∂ (D)(x , ξ )v(t)=0, t> 0, m−1 ′ ′ j σ(b)(x , ξ ) Dt v(0) = h,  j=0 has a unique exponentially decaying solution (for a suitably defined symbol σ(b)) for any right hand side h, the problem is said to be Shapiro-Lopatinskii elliptic. The reader can find more on Shapiro-Lopatinskii ellipticity of a local boundary condition in [3]. The constructions in [3] show how local considerations at the boundary prove to be a highly efficient tool for determining regularity of a local boundary condition. Indeed for the purpose of verifying regularity of local boundary conditions, local semiclassical methods are more direct than the global perspective on a boundary condition as a subspace B H(ˇ D) that dominates this paper. ⊆ Example 4.17. We return to the study of Dirichlet and Neumann conditions on a second order elliptic differential operator D acting on a vector bundle E with scalar positive principal symbol as in Example 3.34. We have that ′−1 ′ ′ 1 1E ξ 1E σ0 (PC (1 PD))(x , ξ )= ′ | | and − − 2 ξ 1E 1E | | − ′−1 ′ ′ 1 1E ξ 1E σ0 (PC (1 PN ))(x , ξ )= −′ | | . − − 2 ξ 1E 1E | | It follows that PC (1 PD) and PC (1 PN ) are elliptic. Therefore Theorem 4.15 reproves the well known fact− that− Dirichlet and− Neumann− conditions are regular. A short computation (or using Proposition 4.18 below) reproves the well known fact that Dirichlet and Neumann conditions are Shapiro-Lopatinskii elliptic. By Example 3.34, the projectors PD and PN are not boundary decomposing which shows that regularity of a boundary condition defined from a projection is not equivalent to the boundary decomposing property of a projector.

We now return to the general case of pseudo-local boundary conditions deﬁned from a projection in the Douglis-Nirenberg calculus. Proposition 4.18. Let D be an elliptic diﬀerential operator of order m> 0 and P Ψ0 (Σ; E ∈ cl ⊗ Cm). Then P is Shapiro-Lopatinskii elliptic if and only if P (1 P ) Ψ0 (Σ; E Cm) is elliptic. C − − ∈ cl ⊗ In particular, a Shapiro-Lopatinskii elliptic pseudo-local boundary condition is regular in the sense of Subsection 2.2. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 39

Proof. The condition of Shapiro-Lopatinskii ellipticity is equivalent to σ0(P ) deﬁning an isomor- m phism between E (D) and σ0(P )(E C ). By [6, Proof of Theorem 2.15], this is equivalent to + 0 ⊗ p+(D) σ0 (1 P ) being an isomorphism in all points, or in other words that PC (1 P ) is elliptic.− − − −

~~5. Characterisations of regularity via graphical decompositions~~

A graphical characterisation for regular boundary conditions was first given in [5], motivated by the desire to capture regular boundary conditions beyond those that are given by pseudodifferential projectors. The main application in mind was a remarkably conceptual proof of the relative index theorem, originally due to Gromov-Lawson [33]. In [5], the boundary decomposing projectors were assumed to be self-adjoint spectral projectors, with the assumption that the adapted operator on the boundary was self-adjoint. A particular luxury in this setting was that the obtained decomposition could be equated with orthogonal complements in L2. This was subsequently generalised [6, Theorem 2.9] for general first order operators, where orthogonal complements were no longer available and the decomposition had to be obtained separately for the boundary condition and its adjoint. In particular, this meant that there were double the number of spaces in the decomposition and we consider the most significant step towards our presentation here for the general order case.

5.1. Preliminaries. As above, we assume that D is an elliptic diﬀerential operator and + is a boundary decomposing projection for D (in the sense of Deﬁnition 3.26 on page 30). WeP use the notation =1 and ∗ for the appropriate dual operator to with respect to the duality P− −P+ P+ P+ induced by L2 inner product. Lemma 5.1. The pairing , : ∗ H−α(Σ; E Cn ) Hα(Σ; E Cm) C, h· ·i P± ⊗ op ×P± ⊗ → induced from the L2(Σ; E) inner product is a perfect pairing for α 1 ,m 1 . ∈ {− 2 − 2 }

Proof. The L2-inner product , induces a perfect pairing between H−α(Σ; E Cn ) and Hα(Σ; E h· ·i ⊗ op ⊗ Cm) by Proposition 3.6. The desired conclusion follows simply on making the right identiﬁcations to the spaces , ∗, X, Y in Lemma A.1. B B Proposition 5.2. For any boundary decomposing projection + for the elliptic diﬀerential operator D, we have that: P

~~− 1 m m− 1 m (i) +H(ˇ D)= +H 2 (Σ; E C ) and −H(ˇ D)= −H 2 (Σ; E C ), and P ∗ P 1 m⊗ ∗ 1 −Pm Pm ⊗ (ii) H(ˆ D)= H 2 (Σ; E C ) H 2 (Σ; E C ). P+ ⊗ op ⊕P− ⊗ op~~

Proof. Item (i) follows from Proposition 3.27. The proof of (ii) follows from (i) since H(ˇ D) = m− 1 m − 1 m −H 2 (Σ; E C ) +H 2 (Σ; E C ), Lemma 5.1, and [55, Theorem 4.8 (4.12) in Chapter 4,P Section 4]. ⊗ ⊕P ⊗

Deﬁnition 5.3 (Graphical decomposition). Let B be a boundary condition and + a boundary decomposing projection for an elliptic diﬀerential operator D of order m> 0. FollowingP [6], we say that a graphical decomposition of B (with respect to ) is a decomposition of the following form: P+

m− 1 m (i) there exist mutually complementary subspaces W and V of H 2 (Σ; E C ) satisfying: ± ± ⊗ m− 1 m W V = H 2 (Σ; E C ), ± ⊕ ± P± ⊗ ∗ 1 m ∗ 1 −m m (ii) it holds that W− H 2 (Σ; E Cop), and W± H 2 (Σ; E Cop) and the subspaces m− 1 ⊂ m ⊗ ⊂ ⊗ W H 2 (Σ; E C ) are ﬁnite dimensional, ± ⊂ ⊗ 40 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

(iii) there exists a continuous map g : V V such that − → + ∗ ∗ 1 m ∗ 1 m g (V H 2 (Σ; E C )) V H 2 (Σ; E C ), + ∩ ⊗ op ⊂ − ∩ ⊗ op ∗ 2 m− 1 m where g denotes the adjoint in the induced L -pairing between H 2 (Σ; E C ) and 1 −m m ⊗ H 2 (Σ; E C ) ⊗ op B = v + gv : v V W . { ∈ −}⊕ +

~~If a boundary condition admits a graphical decomposition (with respect to +), we say it is +- graphically decomposable. P P~~

∗ ∗ 1 −m m We remark that spaces V , W , denotes subspaces of H 2 (Σ; E C ) obtained via the in- ± ± ⊗ op duced pairing from the L2-inner product, formed with respect to the direct sum decomposition m− 1 m H 2 (Σ; E C )= V W V W . ⊗ + ⊕ + ⊕ − ⊕ − m− 1 m− 1 m Deﬁnition 5.4 ( Fredholm pair decomposed in H 2 ). Let B H 2 (Σ; E C ). Suppose: P+ ⊂ ⊗

m− 1 m (i) B is closed subspace of H 2 (Σ; E C ), m− 1 m ∗ ⊗1 m ⊥ 1 m (ii) ( +H 2 (Σ; E C ),B) and ( −H 2 (Σ; E Cop)),B H 2 (Σ; E Cop)) are Fredholm P m− 1 ⊗ m P 1 ⊗m ∩ ⊗ pairs in H 2 (Σ; E C ) and H 2 (Σ; E Cop)), respectively, and m− 1 ⊗ ∗ 1 ⊗ m ⊥ 1 m (iii) ind( H 2 (Σ; E),B)= ind( H 2 (Σ; E C )),B H 2 (Σ; E C )). P+ − P− ⊗ op ∩ ⊗ op

~~m− 1 Then, we say that B is Fredholm-pair decomposed with respect to in H 2 . P+ m− 1 m Deﬁnition 5.5 ( Fredholm pair decomposed in H(ˇ D)). Let B H 2 (Σ; E C ). Suppose: P+ ⊂ ⊗~~

ˇ ⊥ 1 m ˆ (i) B is closed subspace of H(D) and B H 2 (Σ; E Cop) is a closed subspace of H(D), 1 ∩ ⊗ ˇ ∗ ˆ ⊥ 2 m ˇ ˆ (ii) ( +H(D),B) and ( −H(D),B H (Σ; E Cop)) are Fredholm pairs in H(D) and H(D) respectively,P P ∩ ⊗ ∗ ⊥ 1 m (iii) ind( H(ˇ D),B)= ind( H(ˆ D),B H 2 (Σ; E C )). P+ − P− ∩ ⊗ op

~~Then, we say that B is Fredholm-pair decomposed with respect to in H(ˇ D). P+~~

~~Note that the appearing in these deﬁnitions taken with respect to the induced pairing from L2 1 −m⊥ m m− 1 m between H 2 (Σ; E C ) and H 2 (Σ; E C ). ⊗ op ⊗~~

~~5.2. Decomposability. The main theorem we prove in this section is the following.~~

~~Theorem 5.6. Let D be an elliptic diﬀerential operator of order m> 0. For + being a boundary decomposing projection for D, the following are equivalent: P~~

(i) B H(ˇ D) is a regular boundary condition, ⊂ (ii) B is + graphically decomposable, P m− 1 (iii) B is Fredholm decomposable in H 2 P+ (iv) B is Fredholm decomposable in H(ˇ D). P+ If one of these equivalent conditions are satisﬁed, then

∗ ⊥ ∗ ∗ 1 m ∗ a†B = B = u g u : u V H 2 (Σ; E C ) W . (13) − ∈ + ∩ ⊗ op ⊕ − and n o m− 1 m ind( +H 2 (Σ; E C ),B) = ind( +H(ˇ D),B), P ⊗ P (14) ∗ 1 m ⊥ 1 m ∗ ⊥ 1 m ind( H 2 (Σ; E C )),B H 2 (Σ; E C )) = ind( H(ˆ D),B H 2 (Σ; E C )). P− ⊗ op ∩ ⊗ op P− ∩ ⊗ op REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 41

m− 1 In particular, the properties of being graphically decomposable, Fredholm decomposable in H 2 and Fredholm decomposable in H(ˇ D) are independent of the choice of boundary decomposing projection.

The proof of this theorem is aided by the structure of [6, Theorem 2.9] which deals with the ﬁrst order case. In a sense, the essential ingredients are contained in that proof, although the subtleties of this decomposition requires signiﬁcant care. For that purpose, we give full details of the argument for the general order case. Before entering the proof of Theorem 5.6, let us give an index theoretical consequence. The following corollary follows directly from Theorem 2.12 and Equation (14)

m− 1 m Corollary 5.7. Let D be an elliptic diﬀerential operator of order m> 0 and B H 2 (Σ; E C ) 1 1 ⊆ 1 ⊗ m− 2 m− m m− m a regular boundary condition for D. Write := P H 2 (Σ; E C )= H 2 (Σ; E C ). CD C ⊗ CD∩ ⊗ The index of DB is given by m− 1 ind(D ) = ind(B, 2 ) + dim ker(D ) dim ker(D† ). B CD min − min 1 1 m− 2 m− m in terms of an index of the Fredholm pair (B, ) in H 2 (Σ; E C ). CD ⊗

~~Proof that (ii) = (i) in Theorem 5.6 . We shall prove that graphical decomposability implies regularity. First,⇒ note that a simple calculation yields~~

⊥,H(ˆ D) ∗ ∗ ∗ ∗ 1 m v + gv : v V = W W u g u : u V H 2 (Σ; E C ) . (15) { ∈ −} − ⊕ + ⊕ − ∈ + ∩ ⊗ op m− 1 m n o Moreover, we have that H 2 (Σ; E C )= W− V− W+ V+, and by Lemma 5.1, and using Lemma A.1, we have that ⊗ ⊕ ⊕ ⊕

~~1 −m ⊥,H 2 (Σ;E⊗Cm ) W op = V ∗ W ∗ V ∗. + − ⊕ − ⊕ + Therefore,~~

1 −m m ˆ 2 1 ⊥,H(D) ⊥,H (Σ;E⊗Cop) ∗ ∗ ∗ ∗ m W = W H(ˆ D)= V W (V H 2 (Σ; E C )). (16) + + ∩ − ⊕ − ⊕ + ∩P+ ⊗ op By [55, Theorem 4.8 (4.12) in Chapter 4, Section 4], we have that

ˆ B⊥ = (W v + gv : v V )⊥,H(D) + ⊕{ ∈ −} ˆ ˆ = W ⊥,H(D) v + gv : v V ⊥,H(D) + ∩{ ∈ −} ∗ ∗ ∗ 1 m = W u g u : u V H 2 (Σ; E C ) , − ⊕ − ∈ − ∩ ⊗ op n o where the ultimate equality follows from combining (15) and (16). Note that by Deﬁnition 5.3, we ⊥ 1 m have that B H 2 (Σ; E C ). Proposition 4.13 yields that ⊂ ⊗ op ∗ ⊥ ∗ ∗ 1 m ∗ 1 m a†B = B = u g u : u V H 2 (Σ; E C ) W H 2 (Σ; E C ), − ∈ + ∩ ⊗ op ⊕ − ⊂ ⊗ op n ∗ m− 1 m o and therefore, we have that B H 2 (Σ; F C ) so B is regular. ⊂ ⊗

The proof of (i) = (ii) is considerably more involved. In order to proceed with the proof (i) = (ii), the spaces in⇒ the Definition 5.3 need to be defined. Taking inspiration from [6], we define the⇒ following spaces: 1 −m ∗ ∗ 1 m ⊥,H 2 (Σ;E⊗Cm ) W := H 2 (Σ; E C ) B op − P− ⊗ op ∩ m− 1 m W := H 2 (Σ; E C ) B + P+ ⊗ ∩ − 1 (17) ∗ ∗ 1 −m m ∗ ⊥,H 2 (Σ;E⊗Cm) V := H 2 (Σ; E C ) (W ) − P− ⊗ op ∩ − 1 −m 2 m m− 1 m ⊥,H (Σ;E⊗Cop) V := H 2 (Σ; E C ) W + P+ ⊗ ∩ + 42 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

∗ ∗ 1 m Note that by definition W− −H 2 (Σ; E Cop). Therefore, we are able to consider its annihilator − 1 ⊂P ⊗ ∗ ⊥,H 2 (Σ;E⊗Cm) − 1 m (W ) in H 2 (Σ; E C ). Recalling the definition of the hybrid spaces − ⊗ − 1 m m−1 − 1 −j m−1 1 −m+j 1 −m m H 2 (Σ; E C )= H 2 (Σ; E) H 2 (Σ; E)= H 2 (Σ; E C ). ⊗ ⊕j=0 ⊃⊕j=0 ⊗ op ∗ Therefore, V− is well-defined. Similarly, it is readily verified that m− 1 m 1 −m m H 2 (Σ; E C ) H 2 (Σ; E C ), ⊗ ⊂ ⊗ op and therefore, we have that V+ is well defined.

~~This then allows us to deﬁne the remaining spaces as follows simply as the ranges of dual projectors or, as given by Lemma A.1, annihilators with respect to the corresponding duality.~~

1 −m 2 m ∗ ∗ 1 −m m ⊥,H (Σ;E⊗Cop) W := H 2 (Σ; E C ) V + P+ ⊗ op ∩ + m− 1 1 2 m m− 2 m ∗ ⊥,H (Σ;E⊗C ) W− := −H (Σ; E C ) (V−) P ⊗ ∩ (18) 1 −m 2 m ∗ ∗ 1 −m m ⊥,H (Σ;E⊗Cop) V := H 2 (Σ; E C ) W + P+ ⊗ op ∩ + m− 1 m− 1 m ∗ ⊥,H 2 (Σ;E⊗Cm) V := H 2 (Σ; E C ) (W ) − P− ⊗ ∩ − Lemma 5.8. Assume that B is regular and that V+, V−, W+ and W− are deﬁned as in the paragraphs above. The following hold:

m− 1 m ∗ 1 −m m (i) W± H 2 (Σ; E C ) and W± H 2 (Σ; E Cop) are ﬁnite dimensional. In partic- ⊆ ⊗ ⊂ 1 −m ⊗ 1 2 m ∗ ∗ −m m ⊥,H (Σ;E⊗Cop) ular, W− = −H 2 (Σ; E Cop) B . m−P1 m ⊗ ∗∩ ⊥ 1 m (ii) −B H 2 (Σ; E C ) and +B H 2 (Σ; E Cop) are closed subspaces. P ⊆ ⊗ P ⊆ ⊗ ∗ ∗ ∗ ∗ (iii) All of the subspaces V+, V−, W+ and W− as well as V+, V−, W+ and W− listed in (17) m− 1 m 1 −m m and (18) are closed in H 2 (Σ; E C ) and H 2 (Σ; E Cop) respectively. m− 1 m ⊗ ∗ 1 −m m⊗ ∗ ∗ (iv) H 2 (Σ; E C )= V W and H 2 (Σ; E C )= V W . P± ⊗ ± ⊕ ± P± ⊗ op ± ⊕ ±

1 ∗ ⊥ 2 m ⊥ Proof. Consider + B⊥ : B H (Σ; E Cop). By deﬁnition, we have that B is closed in P |⊥ → 1 ⊗ m 1 −m m H(ˆ D). Set X = (B , ), Y = H 2 (Σ; E C ), Z = H 2 (Σ; E C ) and the map k·kHˆ(D) ⊗ op ⊗ op K = ∗ : X Z. For for b⊥ B⊥ P−| B⊥ → ∈ ⊥ ∗ ⊥ ∗ ⊥ b ˆ ⊥ b 1 + ⊥ b 1 −m . H(D) + B H 2 Cm − B H 2 Cm k k ≃ kP | k (Σ;E⊗ op) kP | k (Σ;E⊗ op) Note that

∗ ⊥ 1 m 1 m compact 1 −m m ( B H 2 (Σ; E C ) H 2 (Σ; E C ) ֒ H 2 (Σ; E C : P−| B⊥ ⊂ ⊗ op → ⊗ op −−−−−→ ⊗ op ∗ and so on invoking [6, Lemma A.1] with this choice of X,Y,Z and K yields that ker + B⊥ is ﬁnite ∗ ⊥ 1 m P | dimensional and P B is closed in H 2 (Σ; E C ). Moreover, + ⊗ op ker ∗ = b⊥ B⊥ : ∗ b⊥ =0 = b⊥ B⊥ : b⊥ ∗ H(ˆ D)=0 P+| B⊥ ∈ P+ ∈ ∈P− ⊥ ∗ n1 −m m ⊥ ∗ o 1 m = B H 2 (Σ; E C )= B H 2 (Σ; E C ), ∩P− ⊗ op ∩P− ⊗ op where the penultimate equality follows from Proposition 5.2 (i) and the ultimate equality from the ⊥ 1 m fact that B H 2 (Σ; E C ). ⊂ ⊗ op

~~1 1 m− 2 m m− 2 m Similarly, consider the map − B : B H (Σ; E C ). Since B H (Σ; E C ), we have that whenever b B, P | → ⊗ ⊂ ⊗ ∈~~

~~b m− 1 b ˇ − b m− 1 + + b − 1 . k kH 2 (Σ;E⊗Cm) ≃k kH(D) ≃ kP | B kH 2 (Σ;E⊗Cm) kP | B kH 2 (Σ;E⊗Cm) REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 43~~

However, m− 1 m compact − 1 m ,( B H 2 (Σ; E C ) ֒ H 2 (Σ; E C : P+| B → ⊗ −−−−−→ ⊗ and therefore, on invoking [6, Lemma A.1] yields that W+ = ker( + B) is finite dimensional and m− 1 m P | B is closed in H 2 (Σ; E C ). P− ⊗ ∗ Therefore, we have proven the property (ii) as well as a part of the property (i), namely that W− ∗ and W+ are finite dimensional and that the asserted equality for W− holds. We prove that W− ∗ and W− are finite dimensional after proving (iii) and (iv).

∗ 1 −m m ∗ ∗ ∗ For that, we ﬁrst prove that −H 2 (Σ; E Cop) = V− W−. From the formula for W− P ⊗∗ ∗ 1 −m⊕ m that we have already proven, we have that W− −H 2 (Σ; E Cop) and it is closed since ∗ ⊂∗ P1 −m m⊗ it is ﬁnite dimensional. By construction V− −H 2 (Σ; E Cop). Therefore, we have that ∗ ∗ ∗ 1 −m m ⊂ P ⊗ V + W H 2 (Σ; E C ). − − ⊂P− ⊗ op

∗ ∗ ∗ ∗ ∗ 1 −m m Next, we show that V− W− = 0. Fix w V− W− = −H 2 (Σ; E Cop) − 1 ∩ ∈ ∩ P − 1 ⊗ ∩ ∗ 2 m ∗ 2 m ∗ ⊥,P−H (Σ;E⊗C ) ∗ ∗ ⊥,P−H (Σ;E⊗C ) ∗ (W−) W−. In particular, we have that w (W−) W−. ∗ ∩ 1 m ∈ ∩ Since w W− we also have that w H 2 (Σ; E Cop) which is in a perfect pairing with − 1 ∈ m ∈ ⊗ H 2 (Σ; E C ). Therefore, we have that w, w = 0. Moreover, it is readily verified that 1 ⊗ m m−1 2 h i H 2 (Σ; E C ) L (Σ; E) and therefore, ⊗ op ⊂⊕j=0 2 w 2 = w, w 2 = w, w 1 − 1 =0, L (Σ;E) L (Σ;E) H 2 Cm H 2 Cm k k h i h i (Σ;E⊗ op)× (Σ;E⊗ ) ⊥ ⊥ ∗ 1 −m m from which we conclude that w = 0. This proves that V W H 2 (Σ; E C ). − ⊕ − ⊂P− ⊗ op ∗ We now prove the reverse containment. For that, let k = dim W− and fix a basis e1,...,ek. By considering each e m−1L2(Σ; E), we can run the Gram-Schmidt process to obtain an L2- i ∈ ⊕j=0 orthonormal basise ˜i. Therefore, without the loss of generality, assume that ei are orthonormal 2 − 1 m { } in L . For u H 2 (Σ; E C ), define ∈ ⊗ k Pu = u,e e . h ii i i=1 X − 1 m − 1 It is readily checked that this is a projector and in fact, P : H 2 (Σ; E C ) H 2 (Σ; E − 1 ⊗ → ⊗ m − 1 m ∗ ∗ ⊥,H 2 (Σ;E⊗Cm) C ) with P H 2 (Σ; E C ) = W− and ker P = (W−) . Therefore, restricting ∗ 1 −m ⊗ m − 1 m to u −H 2 (Σ; E Cop) H 2 (Σ; E C ), we obtain that u = Pu (1 P )u and ∈ P ⊗ ⊂ ⊗ − 1 − − ∗ ∗ ⊥,H 2 (Σ;E⊗Cm) ∗ 1 −m m ∗ Pu W−. This means that (1 P )u (W−) −H 2 (Σ; E Cop) since W− ∗ ∈1 −m m − ∈ ∗ ∩P ⊗ ⊥ ⊥ ⊂ −H 2 (Σ; E Cop) and therefore, (1 P )u V−. Hence, we conclude that V− W− = P∗ 1 −m ⊗ m − ∈ ⊕ −H 2 (Σ; E Cop). By exploiting the duality yielded in Lemma 5.1, we conclude that V−, W− P ⊗ m− 1 m m− 1 m are closed in H 2 (Σ; E C ). Via Lemma A.1, we obtain H 2 (Σ; E C )= V W . P− ⊗ P− ⊗ − ⊕ − m− 1 m A similar argument then holds to show that +H 2 (Σ; E C )= V+ W+. Showing V+ +W+ m− 1 m P ⊗ ⊕ ⊂ +H 2 (Σ; E C ) and that V+ W+ = 0 is an easy calculation. The most important point to P ⊗ ∩ m− 1 m 1 −m m show the reverse containment is note that H 2 (Σ; E C ) H 2 (Σ; E Cop). This allows for 1 −m m ⊗ 1 −⊂m m⊗ 1 −m the construction of a projector Q : H 2 (Σ; E Cop) H 2 (Σ; E Cop) with QH 2 (Σ; E m− 1 m ⊗ → ⊗ ⊗ m ⊥,H 2 (Σ;E⊗C ) m− 1 m Cop) = W+ and ker Q = W+ and consequently, write +H 2 (Σ; E C ) ∗ ∗ P ∗ 1 −m ⊗ m⊂ V+ W+. As before, this yields that V+ and W+ are closed subspaces of +H 2 (Σ; E Cop) ⊕ ∗ 1 −m m ∗ ∗ P ⊗ and Lemma A.1 then yields H 2 (Σ; E C )= V W . P+ ⊗ op + ⊕ + ∗ This proves (iii) and (iv). It remains to show that W+ and W− are finite dimensional. For this, note that by using Lemma A.1, we have induced perfect pairings , : W W ∗ C and h· ·i + × + → , : W W ∗ C. But that means that W ′ = W ∗ and W ′ = W ∗ , where W ′ and W ′ are h· ·i − × − → + ∼ + − ∼ − + − 44 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

′ dual spaces of W+ and W− respectively. By ﬁnite dimensionality we have that W+ = W+ and ′ ∼ W− ∼= W− and this ﬁnishes the proof. Lemma 5.9. Under the same assumptions as in Lemma 5.8, the following hold:

~~∗ 1 m ∗ ⊥ (i) V H 2 (Σ; E C )= B , and + ∩ ⊗ op P+ (ii) V = B. − P−~~

~~Proof. First, we prove that~~

~~− 1 ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) ∗ − 1 m W = ( B ) H 2 (Σ; E C ). (19) + P+ ∩P+ ⊗ Then,~~

− 1 m x ( ∗ B⊥)⊥,H 2 (Σ;E⊗C ) 0= x, ∗ b⊥ b⊥ B⊥ ∈ P+ ⇐⇒ P+ ∀ ∈ ⊥ ⊥ ⊥ 0= +x, b b B ⇐⇒ P ∀ ∈ − 1 m = x (B⊥)⊥ ,H 2 (Σ;E⊗C ). ⇒ P+ ∈ − 1 ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) ∗ − 1 m Therefore, x ( +B ) +H 2 (Σ; E C ) implies that x = +x. Moreover, ∈ P 1 ∩P ⊗ 1 P ˇ − m ⊥ ⊥,H(ˆ D) m since by Proposition 5.2 H(D) H 2 (Σ; E C ) and B = B H 2 (Σ; E Cop), we obtain ⊂ − 1 ⊗ ⊂ ⊗ − 1 ⊥,H(ˆ D) H(ˇ D) ⊥ H 2 (Σ;E⊗Cm) ˇ ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) that B = (B ) = (B ) H(D). Therefore, x ( +B ) ∗ − 1 m ∩ ∈ P ∩ H 2 (Σ; E C ) then yields that x H(ˇ D) and hence, x B. This shows , P+ ⊗ ∈ ∈ − 1 ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) ∗ − 1 m − 1 m ( B ) H 2 (Σ; E C ) B H 2 (Σ; E C )= W . P+ ∩P+ ⊗ ⊂ ∩P+ ⊗ +

− 1 m ⊥ ⊥ For the reverse containment, ﬁx w W+ = B +H 2 (Σ; E C ). Therefore, for all b B , we have that ∈ ∩P ⊗ ∈ ⊥ ⊥ 0= w,b − 1 1 = w,b − 1 1 2 m 2 m + 2 m 2 m H (Σ;E⊗C )×H (Σ;E⊗Cop) P H (Σ;E⊗C )×H (Σ;E⊗Cop) ∗ ⊥ = w, b − 1 1 . + 2 m 2 m P H (Σ;E⊗C )×H (Σ;E⊗Cop)

− 1 ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) − 1 m Therefore, we have w ( +B ) and by assumption w +H 2 (Σ; E C ). That ∈ P − 1 ∈P ⊗ ∗ ⊥ ⊥,H 2 (Σ;E⊗Cm) ∗ − 1 m is precisely that w ( B ) H 2 (Σ; E C ). ∈ P+ ∩P+ ⊗ ∗ Now, by the deﬁnition of V+ in (18), we have that

1 −m 2 m ∗ 1 m ∗ 1 −m m ⊥,H (Σ;E⊗Cop) 1 m V H 2 (Σ; E C )= H 2 (Σ; E C ) W H 2 (Σ; E C ) + ∩ ⊗ op P+ ⊗ op ∩ + ∩ ⊗ op 1 2 m ∗ 1 m ⊥,H (Σ;E⊗Cop) = H 2 (Σ; E C ) W P+ ⊗ op ∩ + By the description of W+ in (19) and on using [55, Theorem 4.8 (4.12) in Chapter 4, Section 4], we obtain 1 2 m 1 m ⊥,H (Σ;E⊗Cop) ∗ ⊥ − 1 m ⊥,H 2 (Σ;E⊗C ) ∗ ⊥ ∗ 1 m W = B + ( H 2 (Σ; E C )) op = B + H 2 (Σ; E C ). + P+ P+ ⊗ P+ P− ⊗ op Therefore,

1 2 m ∗ 1 m ∗ 1 m ⊥,H (Σ;E⊗Cop) V H 2 (Σ; E C )= H 2 (Σ; E C ) W + ∩ ⊗ op P+ ⊗ op ∩ + ∗ 1 m ∗ ⊥ ∗ 1 m = H 2 (Σ; E C ) ( B + H 2 (Σ; E C ) P+ ⊗ op ∩ P+ P− ⊗ op ∗ 1 m ∗ ⊥ = H 2 (Σ; E C ) B P+ ⊗ op ∩P+ = ∗ B⊥. P+ This proves (i). REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 45

~~The proof of (ii) is similar. First, we prove that~~

1 −m ∗ ⊥,H 2 (Σ;E⊗Cm ) 1 −m m W = ( B) op H 2 (Σ; E C ). (20) − P− ∩P− ⊗ op 1 −m 1 2 m ∗ ∗ −m m ⊥,H (Σ;E⊗Cop) Recall that by Lemma 5.8 (i), W− = −H 2 (Σ; E Cop) B . and so for w W ∗ , we have that for all b B, P ⊗ ∩ ∈ − ∈

~~0= w,b 1 −m m− 1 2 m 2 m h iH (Σ;E⊗Cop)×H (Σ;E⊗C ) ∗ = w,b = w, b 1 −m m− 1 − − 2 m 2 m P h P iH (Σ;E⊗Cop)×H (Σ;E⊗C )~~

1 −m ⊥,H 2 (Σ;E⊗Cm ) This shows the containment “ ”. For the reverse containment, let x ( −B) op 1 −m m ⊂ ∈ P ∩ H 2 (Σ; E C ). Arguing similarly, for all b B, P− ⊗ op ∈ 0= x, b = ∗ x, b h P− i P− 1 −m ⊥,H 2 (Σ;E⊗Cm ) which yields that x = ∗ x B op . P− ∈ With this, we now note that

m− 1 m− 1 m ∗ ⊥,H 2 (Σ;E⊗Cm) V = H 2 (Σ; E C ) (W ) − P− ⊗ ∩ − m− 1 m m− 1 m = H 2 (Σ; E C ) ( B + H 2 (Σ; E C )) P− ⊗ ∩ P− P+ ⊗ m− 1 m = H 2 (Σ; E C ) B P− ⊗ ∩P− = B. P−

Proof that (i) = (ii) in Theorem 5.6 . We shall prove that regularity implies graphical decom- ⇒ posability. We use the spaces V+, V−, W+ and W− constructed on the preceding pages. Since 1 −m H 2 Cm m− 1 m 1 −m m ⊥, (Σ;E⊗ op) B H 2 (Σ; E C ) H 2 (Σ; E C ), we have that W B = 0. Deﬁne ⊂ ⊗ ⊂ ⊗ op + ∩ 6 ⊥ X− := − B∩W ⊥ : B W+ −B. P | + ∩ →P

We prove that this map is a bijection and hence, by the open mapping theorem, it is a Banach space isomorphism between (B, 1 −m ) and ( B, 1 −m ). First, we establish 2 m − 2 m k·kH (Σ;E⊗Cop) P k·kH (Σ;E⊗Cop) surjectivity of X−. For that, note that

1 −m 2 m 1 −m m ⊥,H (Σ;E⊗Cop) H 2 (Σ; E C )= W W , (21) ⊗ op + ⊕ + which follows from a construction of a projector as in the proof of (iii) in Lemma 5.8. Let P ⊥ W+,W+ 1 −m 2 m ⊥,H (Σ;E⊗Cop) denote the projection to W+ along W and the complementary projector P ⊥ . + W+ ,W+ Fix x = P b B and note − ∈P− x = P−(P ⊥ b + P ⊥ b)= P−P ⊥ b. W+ ,W+ W+,W+ W+ ,W+

1 −m 2 m ⊥,H (Σ;E⊗Cop) Since W+ B, PW ⊥,W b = b PW ⊥,W b B and hence PW ⊥,W b B W+ . This ⊂ + + − + + ∈ + + ∈ ∩ shows surjectivity. Injectivity follows easily by taking 0 = X−b = P−b which allows us to conclude

~~1 −m 1 −m 2 m 2 m m− 1 m ⊥,H (Σ;E⊗Cop) ⊥,H (Σ;E⊗Cop) b H 2 (Σ; E C ) B W = W W =0. ∈P+ ⊗ ∩ ∩ + + ∩ +~~

~~m− 1 m Deﬁne projectors H 2 (Σ; E C ) as: ⊗ 1 1 P+ := P m− m and Q+ := P m− m . V+,W+⊕P−H 2 (Σ;E⊗C ) W+,V+⊕P−H 2 (Σ;E⊗C )~~

Furthermore, note that since V− has an induced Hilbert structure and −B V− is closed, we obtain a closed complementary subspace V ′ such that V = B VP′ . Using⊂ the fact that − − P− ⊕ − X− : (B, m− 1 ) ( −B, m− 1 ) is an isomorphism, and the fact that k·kH 2 (Σ;E⊗Cm) → P k·kH 2 (Σ;E⊗Cm) 46 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~m− 1 m V− = −B from Lemma 5.9 (ii), deﬁne the H 2 (Σ; E C )-bounded map g0 : V− V+ by settingP ⊗ → g (x)= P (X )−1. 0 + ◦ −~~

First, we prove that W+ u + g0u : u −B B. By construction, we have that W+ B, so it remains to show u + g⊕{u : u V ∈PB. For}⊂ that, we compute ⊂ { 0 ∈ −}⊂ u + g u = u + P (X )−1u 0 + ◦ − = X−v + P+v = X v + P v + Q v Q v − + + − + = v + v Q v P− P+ − + 1 −m ⊥,H 2 (Σ;E⊗Cm ) = v Q v B W op + W B, − + ∈ ∩ + + ⊂ −1 where we have set v := (X−) u.

~~We now prove the reverse containment that B W+ u + g0u : u V− . Via the decomposition (21), we have a induced decomposition ⊂ ⊕{ ∈ }~~

1 −m 2 m 1 −m m ⊥,H (Σ;E⊗Cop) 1 −m m H 2 (Σ; E C )= W (W H 2 (Σ; E C )). ⊗ op + ⊕ + ∩ ⊗ op 1 −m m This decomposition holds because W+ H 2 (Σ; E Cop) and for the same reason, the projectors ⊂ ⊗ 1 −m m for this decomposition are the projectors PW ,W ⊥ and PW ⊥,W restricted to H 2 (Σ; E Cop). + + + + ⊗ 1 −m m ⊥ ⊥ For b B H 2 (Σ; E Cop), let b = PW ⊥,W b and b+ = PW ,W ⊥ b, so that b = b + b+. It is ∈ ⊂ ⊗ + + + + 1 −m ⊥,H 2 (Σ;E⊗Cm ) easy to see that, since W B, b⊥ B W op . Let v := X b⊥. Then, + ⊂ ∈ ∩ + − b⊥ = (X )−1v = (X )−1v + (X )−1v = v + P (X )−1v + Q (X )−1v − P− − P+ − + − + − = v + g v + Q (X )−1v u + g u : u B W . 0 + − ∈{ 0 ∈P− }⊕ +

Let − 1 ∗ ∗ ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) ∗ ⊥ X+ := + − 1 : B (W−) +B . ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) P | B ∩(W−) ∩ →P Note that, by using a similar projector construction as in the proof of Lemma 5.8 (ii), we obtain

− 1 − 1 m ∗ ∗ ⊥,H 2 (Σ;E⊗Cm) H 2 (Σ; E C )= W (W ) . (22) ⊗ − ⊕ − − 1 ∗ ∗ ⊥,H 2 (Σ;E⊗Cm) m−1 2 As before, w W− (W−) yields w = 0 since, in particular, w j=0 L (Σ; E) and ∈ ∩ 2 ∗ ⊥ ∗ ⊥ ∗ ∈⊕ ⊥ therefore, w, w = w 2 . For x = b B , we obtain x = PW ∗ ,(W ∗ )⊥ b . As before, h i k kL P+ ∈ P+ P+ − − − 1 ∗ ⊥ ⊥ ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) since W B , we have that P ∗ ∗ ⊥ b B (W ) which establishes the − W−,(W−) − ⊂ ∗ ∈ ∩ ∗ surjectivity of X+. Injectivity follows simply on noting that X+x = 0 implies

− 1 − 1 ∗ − 1 m ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) ∗ ∗ ⊥,H 2 (Σ;E⊗Cm) x H 2 (Σ; E C ) B (W ) = W (W ) =0. ∈P− ⊗ ∩ ∩ − − ∩ − ∗ This shows that X+ is a surjection and by the open mapping theorem, it is an isomorphism between − 1 ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) ∗ ⊥ (B (W ) , 1 ) and ( B , 1 ), where the former space − H 2 Cm + H 2 Cm ∩ k·k (Σ;E⊗ op) P k·k (Σ;E⊗ op) was established as a Banach space in Lemma 5.8 (ii).

1 −m m Next, deﬁne the following projectors on H 2 (Σ; E C ): ⊗ op ∗ ∗ P := P 1 −m and Q := P 1 −m . − ∗ ∗ ∗ 2 m − ∗ ∗ ∗ 2 m V−,W−⊕P+H (Σ;E⊗Cop) W−,V−⊕P+H (Σ;E⊗Cop)

∗ 1 m 1 m We have that P 1 is a bounded map from H 2 (Σ; E C ) to H 2 (Σ; E C ) since − H 2 Cm op op | (Σ;E⊗ op) ⊗ ⊗ P ∗ u = Q∗ u ∗ u − − −P+ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 47

∗ 1 m ∗ 1 −m m ∗ and 1 is a bounded map on H 2 (Σ; E C ) and Q H 2 (Σ; E C ) = W + 2 m op + op − P | H (Σ;E⊗Cop) ⊗ ⊗ ⊂ 1 m ∗ ⊥ ∗ 1 m H 2 (Σ; E C ). Therefore, using Lemma 5.9 (i) to equate B = V H 2 (Σ; E C ), ⊗ op P+ + ∩ ⊗ op ∗ ∗ −1 ∗ 1 m ∗ 1 m h := P (X ) : V H 2 (Σ; E C ) V H 2 (Σ; E C ). 0 − ◦ + + ∩ ⊗ op → − ∩ ⊗ op ∗ 1 m ∗ Since P− is H 2 (Σ; E Cop) bounded, the map X+ was established to be an isomorphism of the ⊗ 1 m respective Banach spaces it maps between, we have that h is bounded in the H 2 (Σ; E C )-norm. 0 ⊗ op To show W ∗ u + h u : u ∗ B⊥ B⊥, we note that W ∗ B⊥ by construction and that, − ⊕ 0 ∈P+ ⊂ − ⊂ setting w = (X∗ )−1u for u P ∗ B⊥, + ∈ + u + h u = w Q w B⊥ + W ∗ B⊥. 0 − − ∗ ∈ − ⊂ − 1 ⊥ ⊥ ∗ ⊥,H 2 (Σ;E⊗Cm) ∗ For the reverse containment, using (22), we write x = x + x− B (W−) W− ∗ ⊥ ∈ ∩ ⊕ and letting w = (X+)x , we deduce that x⊥ = w + h w + Q∗ (X∗ )−1w u + h u : v P ∗ B⊥ W ∗ . 0 − + ∈ 0 ∈ + ⊕ − ∗ ⊥ Lastly, we show that h0 = g 1 . For that, we note that B,B = 0 in the pairing 0 ∗ H 2 Cm − | V+ ∩ (Σ;E⊗ op) between H(ˇ D) and H(ˆ D) and so therefore, 0= u + g u, v + h v = u, v + u,h v + g u, v + g u,h v . h 0 0 i h i h 0 i h 0 i h 0 0 i However u V and v V ∗ and therefore, ∈ − ∈ + 0= u, v = g u,h v . h i h 0 0 i From this, we obtain that u,h v = u, g∗v h 0 i h − 0 i ∗ 1 m ∗ for all u V− and v V+ H 2 (Σ; E Cop). This shows us that h0v = g0v for all v ∗ 1 ∈ m ∈ ∩ ⊗ ∗ 1 m − ∈ V+ H 2 (Σ; E Cop) and moreover, since h0v V− H 2 (Σ; E Cop), by construction of h0, we obtain∩ that ⊗ ∈ ∩ ⊗ ∗ 1 m ∗ 1 m g (V H 2 (Σ; E C )) V H 2 (Σ; E C ). 0 + ∩ ⊗ op ⊂ − ∩ ⊗ op This ﬁnishes the proof.

~~Next, we move onto proving the assertions regarding Fredholm pairs characterisations of regular boundary conditions.~~

Proof that (ii) = (iii) in Theorem 5.6 . We shall prove that graphical decomposability implies ⇒ m− 1 Fredholm decomposability in H 2 . The argument is exactly as in the proof of [6, Theorem 2.9]. The idea is to use (ii) to write B = v + gv : v V W and { ∈ −}⊕ + ∗ ⊥ ∗ ∗ 1 m ∗ a†B = B = u g u : u V H 2 (Σ; E C ) W . − ∈ + ∩ ⊗ op ⊕ − This, along with Proposition 5.2 yieldsn o

m− 1 m ⊥ 1 m ∗ B H 2 (Σ; E C )= W , (B H 2 (Σ; E C )) H(ˆ D)= W , ∩P+ ⊗ + ∩ ⊗ op ∩ − m− 1 m H 2 (Σ; E C ) H(ˇ D) m− 1 m = = W−, and ⊗ B + H 2 (Σ; E C ) ∼ B + +H(ˇ D) ∼ P+ ⊗ P 1 m H 2 (Σ; E C ) H(ˆ D) ∗ op ⊥ ∗ = ⊥ ∗ 1 m = W . ⊗ B + H(ˆ D) ∼ B + H 2 (Σ; E C ) ∼ + P− P− ⊗ op ∗ Since W± ∼= W±, the conclusion follows.

~~Now, we prove the rest of the theorem in two parts. 48 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN~~

~~m− 1 Proof that (iii) = (iv) in Theorem 5.6 . We shall prove that Fredholm decomposability in H 2 ⇒ implies Fredholm decomposability in H(ˆ D). The argument is divided into three steps.~~

(a) B is closed in H(ˇ D). m− 1 m m− 1 m Set W+ := B +H 2 (Σ; E C ) = B +H(ˇ D), since B H 2 (Σ; E C ). Therefore, using Proposition∩P 5.2, ⊗ ∩P ⊂ ⊗ B = W (B W ⊥) and H(ˇ D)= W ( H(ˇ D) W ⊥). + ⊕ ∩ + P+ + ⊕ P+ ∩ + m− 1 m ⊥ ⊥ Also, B H 2 (Σ; E C ) W = W W = 0, and therefore, ∩P+ ⊗ ∩ + + ∩ + B + H(ˇ D)= B (B W ⊥). P+ ⊕ ∩ + Next, we claim that this space is closed. For that, we use [6, Lemma A.4] to see that m− 1 m B + H(ˇ D)= B H(ˇ D) and note that B H 2 (Σ; E C )= H(ˇ D) P+ P− ⊕P+ P− ⊂P− ⊗ P− is a closed subspace in H(ˇ D). ∗ 1 m ⊥ 1 m Now, using that ( −H 2 (Σ; E Cop)),B H 2 (Σ; E Cop)) is a Fredholm pair in 1 m P ⊗ − 1 ∩ m ⊗ ⊥ 1 m ⊥ H 2 (Σ; E Cop)), we obtain that ( +H 2 (Σ; E Cop)), (B H 2 (Σ; E Cop)) ) is a ⊗ − 1 m P ⊗ ∩ ⊗ Fredholm pair in H 2 (Σ; E C ) with opposite index. Using Proposition 5.2 and [6, ⊗ ⊥ Lemma A.5], we obtain that ( +H(ˇ D), Bˇ) is a Fredholm pair in H(ˇ D), where Bˇ = (B 1 m ⊥ P ⊥ ∩ H 2 (Σ; E C )) . This yields B (Bˇ H(ˇ D) W ) is closed in H(ˇ D). ⊗ op ⊕ ∩P+ ∩ + To prove the desired claim, we show that Bˇ H(ˇ D) W ⊥ = 0. For that, it suffices to ∩P+ ∩ + show that Bˇ H(ˇ D)= B H(ˇ D). Since B Bˇ, and the dimension of B H(ˇ D) ∩P+ ∩P+ ⊂ ∩P+ is finite, it suffices to show that dim(Bˇ H(ˇ D)) dim(B H(ˇ D)). For that, ∩P+ ≤ ∩P+ ˇ ˇ H(ˆ D) dim(B +H(D)) = dim ⊥ ∗ ˆ ∩P B + −H(D) P ∗ 1 m H 2 (Σ; E C ) = dim P+ ⊗ op ∗ ⊥ +B P ∗ 1 m H 2 (Σ; E C ) dim P+ ⊗ op ∗ ⊥ ≤ +B P m− 1 m = dim(B H 2 (Σ; E C )) = dim(B H(ˇ D)). ∩P+ ⊗ ∩P+ (b) ( H(ˇ D),B) is a Fredholm pair in H(ˇ D), with P+ m− 1 m ind( H(ˇ D),B) = ind( H 2 (Σ; E C ),B). P+ P+ ⊗

~~Contained in the proof of (a) was that B + H(ˇ D) is a closed subspace of H(ˇ D). It is P+ clear that B +H(ˇ D) is a closed subspace of H(ˇ D). Using [6, Lemma∩P A.4], we obtain that~~

m− 1 m ˇ H 2 (Σ; E C ) H(D) = − B + H(ˇ D) ∼ P ⊗ −B P+ P as well as m− 1 m H(ˇ D) −H 2 (Σ; E C ) m− 1 m = . B + H 2 (Σ; E C ) ∼ P ⊗ −B P+ ⊗ P m− 1 m This along with the fact that B +H(ˇ D) = B +H 2 (Σ; E C ) then yields the conclusion. ∩P ∩P ⊗ ⊥ 1 m ˆ ∗ ˆ ⊥ 1 m (c) B H 2 (Σ; E Cop) is closed in H(D) and ( −H(D),B H 2 (Σ; E Cop)) is a Fredholm pair∩ with ⊗ P ∩ ⊗

∗ ⊥ 1 m ∗ 1 m ⊥ 1 m ind( H(ˆ D),B H 2 (Σ; E C )) = ind( H 2 (Σ; E C )),B H 2 (Σ; E C )). P− ∩ ⊗ op P− ⊗ op ∩ ⊗ op ⊥ 1 m That B H 2 (Σ; E C ) is closed in H(ˆ D) is obtained via a repetition of the argument ∩ ⊗ op in (a) with the appropriate changes. Similarly, the remaining conclusion is obtained on mirroring the argument (b) with the necessary changes. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 49

~~The last remaining part to establish Theorem 5.6 is given below.~~

Proof that (iv) = (i) in Theorem 5.6 . We shall prove that Fredholm decomposability in H(ˆ D) ⇒ m− 1 m implies regularity. . Since B H 2 (Σ; E C ) and closed in H(ˇ D), we only need to show that ⊥ 1 m ⊂ ⊗ B H 2 (Σ; E C ). ⊆ ⊗ op ˇ ∗ ˆ ⊥ 1 m By hypothesis, we have that ind( +H(D),B)= ind( −H(D),B H 2 (Σ; E Cop)), and there- 1 P − P ∩ ⊗ ∗ ˆ ⊥ m ∗ ˆ ⊥ fore ind( −H(D),B H 2 (Σ; E Cop)) = ind( −H(D),B ). By [6, Lemma A.7], we obtain that ⊥ ⊥P 1 ∩ m ⊗ P B = B H 2 (Σ; E C ). ∩ ⊗ op

5.3. The first order case. For general boundary decomposing projectors in the first order P± case, a decomposition more in the spirit of [6] can be obtained. We remark that if + is boundary P s decomposing for a first order elliptic operator, then + extends by interpolation to H (Σ; E) for any s [ 1 , 1 ]. P ∈ − 2 2 2 Definition 5.10 (Graphical L -decomposition). Let B be a boundary condition and + a boundary decomposing projection for an elliptic differential operator D of order 1. We say thatP a graphical L2-decomposition (with respect to ) of B is the following decomposition: P+

2 (i) there exist mutually complementary subspaces W± and V± of L (Σ; E) satisfying: W V = L2(Σ; E), ± ⊕ ± P± ∗ 1 (ii) it holds that W± are ﬁnite dimensional with W±, W± H 2 (Σ; E), and (iii) there exists a bounded linear map g : V V such that⊂ − → + ∗ ∗ 1 ∗ 1 1 1 g (V H 2 (Σ; E)) V H 2 (Σ; E) and g(V H 2 (Σ; E)) V H 2 (Σ; E), + ∩ ⊂ − ∩ − ∩ ⊂ + ∩ where g∗ denotes the L2-adjoint and

~~1 B = v + gv : v V H 2 (Σ; E) W . ∈ − ∩ ⊕ + n o A boundary condition admitting a graphical L2-decomposition is said to be graphically L2- decomposable.~~

∗ As before, the spaces W± that appear in the definition, as before, are simply the ranges of the dual projectors. However, our computations are always in L2, and therefore, this is simply the ranges of 2 2 + the adjoint projectors in L with respect to the L inner product. The special case + = χ (A), for an adapted boundary operator A, of Definition (5.10) was stated in [6, DefinitionP 2.6].

We emphasise that in Definition 5.10, unlike for the graphical decomposition in Definition 5.3, the decomposition obtained is in L2. Indeed, the following theorem asserts that in the first order case, these decompositions are equivalent. The precise equivalence is set up as follows: a graphical 2 1 L -decomposition produces a graphical decomposition as in Definition 5.3 by intersecting with H 2 and the converse is constructed from L2-closures. We present this first order definition separately as it can be useful to compute in L2 in the first order case instead of the more complicated spaces that appear in the general order decomposition. Theorem 5.11. For a first order elliptic differential operator D, a boundary condition B is regular if and only if it is graphically L2-decomposable. In this case, we have that

⊥ ∗ ∗ 1 ∗ B = u g u : u V H 2 (Σ; E) W . − ∈ + ∩ ⊕ − n o ± The proof of this theorem runs as in the proof of [6, Theorem 2.9], with the projectors χ (Ar) replaced by . P± 50 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~6. Higher order boundary regularity~~

~~We will now prove results on higher order boundary regularity. These results rely on the fact that Theorem 3.18 due to Seeley was, in fact, stated in [65] also for higher order Sobolev spaces.~~

6.1. Preliminaries. For an elliptic differential operator D, we let Dmin,s and Dmax,s denote the minimal and maximal, respectively, closures of D acting densely defined on Hs(M; E) Hs(M; F ). s− 1 m → We let γ : dom(D ) H 2 (Σ; E C ) denote the trace mapping. s max,s → ⊗ Theorem 6.1. Let D be an elliptic differential operator of order m> 0. The Calderón projection P Ψ0 (Σ; E Cm) from Theorem 3.18 satisfies the following. C ∈ cl ⊗

(i) The operator from Theorem 3.18 deﬁnes a continuous Poisson operator K s− 1 m : H 2 (Σ; E C ) dom(D ), Ks ⊗ → max,s s− 1 m such that PC = γs s on H 2 (Σ; E C ). (ii) The image of ◦ K ⊗ s− 1 m γ : dom(D ) H 2 (Σ; E C ), s max,s → ⊗ is precisely the subspace

~~s− 1 m s+m− 1 m P H 2 (Σ; E C ) (1 P )H 2 (Σ; E C ). C ⊗ ⊕ − C ⊗~~

~~In particular, the trace map γs induces an isomorphism of Banach spaces~~

~~1 1 dom(Dmax,s) s− m s+m− m = PC H 2 (Σ; E C ) (1 PC)H 2 (Σ; E C ). dom(Dmin,s) ∼ ⊗ ⊕ − ⊗~~

While the Calderon projection is in general hard to compute, we shall formulate a condition on a boundary decomposing projection in order for it to implement higher semiregularity properties. For s 0, we introduce the notation ≥ 1 1 s s− m s+m− m dom(Dmax,s) Hˇ (D) := PCH 2 (Σ; E C ) (1 PC )H 2 (Σ; E C ) = , ⊗ ⊕ − ⊗ ∼ dom(Dmin,s) where the last isomorphism is implemented by γ by Theorem 6.1. Definition 6.2. Let D be an elliptic differential operator of order m> 0 and s 0 a positive num- ≥ ber. We say that a projection + is s-boundary decomposing for D if + is boundary decomposing for D and additionally satisfiesP the following conditions P

s+α m s+α m 1 1 (P1) + : H (Σ; E C ) H (Σ; E C ) is a bounded projection for α 2 ,m 2 , P s ⊗s → ⊗ s s+m− 1 m∈ − − (P2) : Hˇ (D) Hˇ (D), and := (I ): Hˇ (D) H 2 (Σ; E C ), and P+ → P− −P+ → ⊗ (P3) u ˇ s −u s+m− 1 + +u s− 1 . k kH (D) ≃ kP kH 2 (Σ;E⊗Cm) kP kH 2 (Σ;E⊗Cm)

Similarly to Theorem 3.30 and Corollary 3.32, we have the following result providing a sufficient condition for a projection being s-boundary decomposing. Theorem 6.3. Let D be an elliptic differential operator of order m> 0, s 0 a positive number s+m− 1 m ≥ and P a continuous projection on H 2 (Σ; E C ). Assume that P satisfies the following: ⊗

s − 1 m s− 1 m P is continuous also in the norms on H(ˇ D), Hˇ (D), H 2 (Σ; E C ), H 2 (Σ; E C ) • m− 1 m ⊗ ⊗ and H 2 (Σ; E C ). ⊗ s+m− 1 m The operator A := PC (1 P ) deﬁnes a Fredholm operator on H 2 (Σ; E C ) and • − − s m− 1 ⊗ m by continuity extends to Fredholm operators on H(ˇ D), Hˇ (D) and H 2 (Σ; E C ). ⊗ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 51

~~Then P is s-boundary decomposing, and in particular~~

u ˇ s (1 P )u s+m− 1 + Pu s− 1 . k kH (D) ≃k − kH 2 (Σ;E⊗Cm) k kH 2 (Σ;E⊗Cm) In the special case that P Ψ0 (Σ; E Cm) is a projection such that ∈ cl ⊗ [P , P ] Ψ−m(Σ; E Cm); and • C ∈ cl ⊗ the operator A := P (1 P ) Ψ0 (Σ; E Cm) is elliptic, • C − − ∈ cl ⊗ then P is s-boundary decomposing.

Let us consider a corollary of Theorem 6.1. Corollary 6.4. Let D be an elliptic diﬀerential operator of order m> 0 and s 0. Then it holds that ≥ dom(D ) Hs+m(M; E)= max ∩ s s+m− 1 m = u dom(D ): D u H (M; F ), P γu H 2 (Σ; E C ) . { ∈ max max ∈ C ∈ ⊗ }

Proof. The inclusion is clear, we need to prove that if u dom(Dmax) satisﬁes that Dmaxu s ⊆ s+m− 1 m s+m∈ ∈ H (M; F ) and P γu H 2 (Σ; E C ) then u H (M; E). After replacing u with u C ∈ ⊗ ∈ − s+mPC γu we can assume that PCγu = 0. In other words, we need to prove that if u dom(DCc ) K s s+m ∈ and D c u H (M; F ) then u H (M; E). C ∈ ∈ c s+m− 1 m H 2 C c Deﬁne s := (1 PC ) (Σ; E ) and consider the associated realisation DCs,s of D on s C − ⊗ s+m H (M; E). We note that DCc,s is semi-regular in the sense that dom(DCc,s) H (M; E). s s ⊆ ∞ c c Theorem 6.1 implies that C (M; E) dom(DC ) is a core for DCs,s for any s. Therefore, we have dense inclusions ∩ ∞ DCc (C (M; E) dom(DCc )) ran(DCc,s) and ∩ ⊂ s ∞ s D c (C (M; E) dom(D c )) ran(D c ) H (M; E). C ∩ C ⊂ C ∩ s+m s c c However, the spaces ran(DCs ,s) = ran(Dmax,s) = DH (M; E) and ran(DC ) H (M; E) are s s ∩ c c closed in H (M; E), and therefore, ran(DCs ,s) = ran(DC ) H (M; E). The condition that u s s ∩ ∈ dom(DCc ) with DCc u H (M; F ) and u H (M; E) therefore implies that there exists v s+m ∈ ∈ ∈ c c dom(DCs ,s) H (M; E) such that Dv = DC u. We then have that D(v u)=0, so γ(v u) c ⊆ ∞ − s+m − ∈ = 0, and therefore v u ker(Dmin) C (M; E). Hence, u = v + (u v) H (M; E)+ CC∩C∞(M; E) = Hs+m(M; E)− which∈ proves the⊂ required claim. − ∈

6.2. Higher order regularity of boundary conditions. In [5, 6], higher order regularity of boundary conditions was introduced under the name of regular boundary conditions. The regularity was stated in terms of graphical decompositions of regular boundary conditions. We shall formulate a slightly more general condition. Deﬁnition 6.5. Let D be an elliptic diﬀerential operator, B H(ˇ D) a semi-regular boundary condition (in L2-sense), s 0 and P an s-boundary decomposing⊆ projection. ≥

s+m− 1 We say that B is s-semiregular (w.r.t P ) if for any ξ B it holds that ξ H 2 (Σ; E • m s+m− 1 m ∈ ∈ ⊗ C ) if and only if (1 P )ξ H 2 (Σ; E C ). − ∈ ⊗ We say that the realisation DB is s-semiregular if whenever u dom(DB) satisﬁes that • D u Hs(M; F ) then u Hs+m(M; E). ∈ B ∈ ∈

The reader should note that the proof of Corollary 6.4 essentially boils down to proving that c is C s-semiregular for any s 0 (w.r.t PC ). Therefore, Corollary 6.4 partially proves the main result of this section (Theorem 6.6≥ below) in the special case of the boundary condition c. C 52 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~Theorem 6.6 (Higher order boundary regularity). Let D be an elliptic diﬀerential operator of order m> 0, s 0 and B H(ˇ D) a boundary condition. Then the following are equivalent: ≥ ⊆~~

~~(1) DB is s-semiregular. (2) B is s-semiregular with respect to PC (3) B is s-semiregular with respect to any s-boundary decomposing projection.~~

Proof. We start by showing that if P is an s-boundary decomposing projection and there is a s+m− 1 m s+m− 1 m ξ B with (1 P )ξ H 2 (M; E C ) but ξ / H 2 (M; E C ), then there is a ∈ − ∈ s ⊗ ∈ s+m ⊗ u dom(DB) satisfying that DBu H (M; F ) and u / H (M; E). Indeed, we take a preimage ∈ −1 ∈ ∈ u γ (ξ) written as u = u0 + u1 dom(Dmax,s) where γ(u0) = (1 P )ξ and γ(u1) = P ξ. ∈ s+m− 1 ∈m s+m− Since (1 P )ξ H 2 (M; E C ) we can assume that u0 H (M; E). However, if − s+m∈− 1 m ⊗ s+m− 1 m ∈ s+m− 1 m (1 P )ξ H 2 (M; E C ) but ξ / H 2 (M; E C ) then Pξ / H 2 (M; E C ) so −u / H∈s+m(M; E). This⊗ proves that 1)∈ implies 3). ⊗ ∈ ⊗ 1 ∈ It is clear that 3) implies 2). It remains to prove that 2) implies 1), i.e that if B is s-semiregular with s s+m respect to PC then whenever u dom(DB) satisﬁes that DBu H (M; F ) then u H (M; E). ∈ s ∈ ∈ Take a u dom(DB) satisfying that DBu H (M; F ). By applying the same argument as in the ∈ ∈ s+m c proof of Corollary 6.4, we have that w := u γu dom(DCs ,s) H (M; E). On the other − Ks+m∈− 1 m⊆ hand γ(w) = γ(u) PCγ(u)= (1 PC)γ(u) H 2 (M; E C ) so by weak s-semiregularity s+m− 1 − m − ∈ s+m− 1 ⊗ m γ(u) H 2 (M; E C ). Therefore PC γ(u) H 2 (M; E C ) and can conclude that ∈ s+m⊗ ∈ ⊗ s+m γu = s+mγu H (M; E) by Theorem 6.1. We conclude that u = w + γu H (M; E). WeK concludeK that∈ 1) and 2) are equivalent. K ∈

Based on the graphical decompositions of Section 5, let us provide a method of producing semiregular boundary conditions. Proposition 6.7. Let B be a boundary condition for an elliptic diﬀerential operator D of order m > 0, s 0 and P an s-boundary decomposing projection. Assume that B is graphically decomposable with≥ respect to P as in Deﬁnition 5.3. Then B is s-semiregular if and only if s+m− 1 m W H 2 (Σ; E C ) and + ⊆ ⊗ s+m− 1 m s+m− 1 m g(V H 2 (Σ; E C )) V H 2 (Σ; E C ). + ∩ ⊗ ⊆ − ∩ ⊗

The proof goes ad verbatim to [6, Lemma 8.9]. Corollary 6.8. Let D be an elliptic diﬀerential operator of order m> 0 and B H(ˇ D) a boundary ∗ † ⊆ condition. Then the realisation DBDB of D D is regular if B is regular and m-semiregular.

∗ ∗ Proof. Before getting down to brass tacks, we note that the operator DBDB is self-adjoint, so DBDB is a regular realisation if and only if it is a semi-regular realisation. If B is regular and m-semiregular ∗ ∗ m then any u dom(DBDB) satisﬁes DBu dom(DB) H (M; F ), because B is regular. Since B ∈ ∈ 2m ⊆ m is m-semiregular, Theorem 6.6 implies that u H (M; E) when DBu H (M; F ). We conclude that dom(D∗ D ) H2m(M; E) if B is regular∈ and m-semiregular. ∈ B B ⊆ Remark 6.9. We note that an interpolation argument shows that if D is an elliptic diﬀerential ˇ ∗ operator of order m> 0 and B H(D) a boundary condition such that DBDB is regular, then B is semi-regular. Since the proof of⊆ Corollary 6.8 only used semi-regularity of B∗ and m-semiregularity of B, we conclude that any m-semiregular boundary condition B with B∗ semi-regular is regular.

6.3. Higher regularity of the Dirichlet problem. As an example of our theory, we consider elliptic operators with a uniquely solvable Dirichlet problem. Such diﬀerential operators play a prominent role in [34–36]. The results of this subsection are well known. Along the way, we shall exemplify the results of this paper for operators with positive principal symbols. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 53

~~For an elliptic diﬀerential operator D of even order m, we deﬁne the Dirichlet condition as the space~~

~~m−1 1 −m m m B := ξ = (ξ ) H(ˇ D) H 2 (Σ; E C ): ξ =0, j =0,..., 1 . Dir { j j=0 ∈ ⊂ ⊗ j 2 − }~~

We say that D has a uniquely solvable Dirichlet problem if BDir is a regular boundary condition and DDir := DBDir is an invertible realisation. We note the following important example. Lemma 6.10. Let D = D† be a formally selfadjoint elliptic operator of order m acting on the sections of a hermitian vector bundle E on an n-dimensional manifold with boundary. Assume that D has positive interior principal symbol σD. Then DDir is the realisation with domain dom(DDir) := m m/2 H (M; E) H0 (M; E). Moreover, DDir is regular, self-adjoint and lower semibounded. In particular, for∩ large enough λ R, D + λ has a uniquely solvable Dirichlet problem. ∈

~~We remark that if D has positive interior symbol, the symmetry property σD(x, ξ) = ( 1)mσ (x, ξ) implies that m is even. − − D~~

Proof. By extending D to a self-adjoint elliptic diﬀerential operator with positive principal symbol on a closed manifold containing M as a smooth subdomain, we see that the G˚arding inequality implies that for λ> 0 large enough it holds that 2 (D + λ)ϕ, ϕ L2 (M;E) ϕ m/2 , h i ∼k kH0 (M;E) for ϕ C∞(M; E). Therefore D is lower semi-bounded and symmetric, and D is indeed the ∈ c c Dir Friedrichs extension of Dc. Therefore, DDir is self-adjoint and lower semibounded. Since DDir is semi-regular by construction, self-adjointness implies regularity.

We now describe a certain projection used in [34]. By Lemma 6.10 unique solvability of the Dirichlet problem is no serious restriction for operators with positive principal symbol. Under this assumption, γ restricts to an isomorphism ker(Dmax) ∼= D. Since DDir is invertible and regular, the operator C P := 1 D−1 D , ζ − Dir max is a continuous projection on dom(Dmax). The projection Pζ is the projection onto ker(Dmax) along dom(DDir). As such, Pζ deﬁnes a Banach space isomorphism (1 P ) P : dom(D ) ∼ dom(D ) ker(D ). − ζ ⊕ ζ max −→ Dir ⊕ max For more details, see [34, Chapter II]. Under γ, we arrive at a decomposition H(ˇ D)= B , Dir ⊕CD as Banach spaces when topologising B as a subspace of Hm−1/2(Σ; E Cm). We let P denote Dir ⊗ ζ,∂ the continuous projection in H(ˇ D) onto along B ; note that P is induced from P . CD Dir ζ,∂ ζ Lemma 6.11. Let D be an elliptic diﬀerential operator of even order with uniquely solvable Dirich- let problem. Let s 0. The projection P is s-boundary decomposing for D and is of order zero ≥ ζ,∂ in the Douglis-Nirenberg calculus P Ψ0 (Σ; E Cm). ζ,∂ ∈ cl ⊗

Proof. We write γ = (γD,γN ) where −1/2 m/2 k m/2−1 γD :dom(Dmax) H (Σ; E C ), u (∂ u Σ) , and → ⊗ 7→ xn | k=0 −m/2−1/2 m/2 k m−1 γN :dom(Dmax) H (Σ; E C ), u (∂ u Σ) . → ⊗ 7→ xn | k=m/2 m m/2 By [35], there is a Dirichlet-to-Neumann operator ΛDN Ψcl (Σ; E C ) deﬁned from ΛDN ξ := γ u where u ker(D ) satisﬁes γ u = ξ. A short computation∈ ⊗ shows that N ∈ max D 1 0 P = , (23) ζ,∂ Λ 0 DN 54 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

in the decomposition H(ˇ D) H−1/2(Σ; E Cm)= H−1/2(Σ; E Cm/2) H−m/2−1/2(Σ; E Cm/2). ⊆ ⊗ ⊗ ⊕ ⊗ It follows that P Ψ0 (Σ; E Cm). ζ,∂ ∈ cl ⊗ By construction, 1 P maps H(ˇ D) into B Hm−1/2(Σ; E Cm) and deﬁnes a decomposition − ζ,∂ Dir ⊆ ⊗ H(ˇ D) = BDir D+λ of Banach spaces. We conclude that Pζ,∂ is boundary decomposing for D. The fact that ⊕CP is s-boundary decomposing for all s 0 follows from Theorem 6.3. ζ,∂ ≥ Example 6.12. An immediate consequence of Lemma 6.11 and Theorem 6.6 is the well known fact that the Dirichlet problem is s-semiregular for all s 0 if it is uniquely solvable in L2. With ≥ respect to Pζ,∂ , we can graphically decompose the Dirichlet condition as B = v + gv : v V W , Dir { ∈ −}⊕ + for V =(1 P )H(ˇ D)=(1 P )Hm−1/2(Σ; E Cm), − − ζ,∂ − ζ,∂ ⊗ V =P H(ˇ D)= P Hm−1/2(Σ; E Cm), + ζ,∂ ζ,∂ ⊗ g =0 and W+ = W− =0. Example 6.13. Another consequence of Lemma 6.11 and Theorem 6.6 concerns Robin-type problems with claims concerning s-semiregularity for all s 0 if the Dirichlet problem is uniquely solvable in L2. This is studied in more detail and generality≥ in [36].

We give an example in the case that D is second order with positive principal symbol, e.g. a Laplace type operator. For an element a C∞(Σ; End(E)), consider the boundary condition ∈ B := (ξ , ξ ) H(ˇ D): aξ + ξ =0 . a { 0 1 ∈ 0 1 } This boundary condition corresponds to the Robin problem for D, i.e. for f L2(M; E), the ∈ equation DBa u = f is equivalent to the Robin problem

~~Du = f, in M au + ∂ u =0, on Σ. ( |Σ n |Σ~~

~~The boundary condition Ba is a pseudolocal boundary condition Ba = BPa where 0 0 P = . a a 1 A computation with Proposition 2.26 (cf. [36]) shows that Ba is regular.~~

We compute from Equation (23) that 0 (1 Pζ,∂ (λ))ξ = . (24) − ΛDN ξ0 + ξ1 − s+m−1/2 2 Take an s 0, and assume that ξ Ba satisfies (1 Pζ,∂ (λ))ξ H (Σ; E C ), or ≥ ∈ s+m−−3/2 ∈ ⊗ equivalently (by Equation (24)) that ΛDN ξ0 +ξ1 H (Σ; E). If (ξ0, ξ1) Ba, we therefore s+m−3/2 − ∈ 1 ∈ have that (ΛDN + a)ξ0 H (Σ; E). Since ΛDN Ψcl(Σ; E) is elliptic, we conclude from ∈ ∈ s+m−1/2 2 elliptic regularity that if ξ = (ξ0, ξ1) Ba satisfies (1 Pζ,∂ (λ))ξ H (Σ; E C ) then s+m−1/2 s+m−1/2∈ 2 − ∈ ⊗ ξ0 H (Σ; E) so ξ H (Σ; E C ). We conclude that Ba is s-semiregular for any s ∈0 when a C∞(Σ; End(∈E)). In particular,⊗ if f Hs(M; E), any solution u L2(M; E) to the Robin≥ problem∈ ∈ ∈ Du = f, in M au + ∂ u =0, on Σ, ( |Σ n |Σ satisfies u Hs+m(M; E). ∈ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 55

~~Let us graphically decompose the Robin condition Ba with respect to Pζ,∂ . For ξ = (ξ0, ξ1) = (ξ , aξ ) B , we can write 0 − 0 ∈ a ξ 0 ξ = 0 + . Λ ξ Λ ξ aξ DN 0 − DN 0 − 0~~

~~∈Pζ,∂ H(ˇ D) ∈(1−Pζ,∂ )H(ˇ D)~~

The operator ΛDN is first order elliptic,| {z and} | so is ΛDN{z + a.} So there exists an elliptic T Ψ−1(Σ; E) such that 1 (Λ + a)T and 1 T (Λ + a) are smoothing finite rank projectors,∈ cl − DN − DN ran((ΛDN + a)T ) = ran(ΛDN + a) and ker(T (ΛDN + a)) = ker(ΛDN + a). Define the space

0 3/2 2 V− := (1 Pζ,∂ )H (Σ; E C ):(ΛDN + a)T η1 = η1 . η1 ∈ − ⊗ The space 0 3/2 2 W− := (1 Pζ,∂ )H (Σ; E C ):(ΛDN + a)T η1 =0 , η1 ∈ − ⊗ is by the construction of T a ﬁnite-dimensional subspace of C∞(Σ; E C2) such that ⊗ V W = (1 P )H3/2(Σ; E C2). − ⊕ − − ζ,∂ ⊗ Deﬁne the space

η0 3/2 2 V+ := Pζ,∂ H (Σ; E C ): T (ΛDN + a)η0 = η0 . ΛDN η0 ∈ ⊗ The space η0 3/2 2 W+ := Pζ,∂ H (Σ; E C ): T (ΛDN + a)η0 =0 , ΛDN η0 ∈ ⊗ is by the construction of T a finite-dimensional subspace of C∞(Σ; E C2) such that ⊗ V W = P H3/2(Σ; E C2). + ⊕ + ζ,∂ ⊗ Define the map g : V− V+ by → 0 T η g := 1 . η Λ− T η 1 − DN 1 A short computation shows that ∗ ∗ 1 2 ∗ 1 2 g (V H 2 (Σ; E C )) V H 2 (Σ; E C ), + ∩ ⊗ op ⊂ − ∩ ⊗ op and that B = v + gv : v V W . a { ∈ −}⊕ + This gives a graphical decomposition of Ba. The existence of a graphical decomposition will by Theorem 5.6 give an alternative proof of the fact that Ba is elliptic. The construction of g in terms of a pseudodifferential operator, combined with Proposition 6.7, gives an alternative explanation for the fact that B is s-semiregular for any s 0. a ≥

~~7. Differential operators with positive principal symbols and their Weyl laws~~

In this section we shall prove a Weyl law for general lower semi-bounded self-adjoint regular realisations of an elliptic diﬀerential operator D of order m> 0 with positive interior principal symbol. Weyl laws with error estimates are known under mild assumptions from [2, 49], see also [38, The- orem 1]. The authors are grateful to Gerd Grubb who showed us how to prove the Weyl law for general boundary conditions by combining the spectral asymptotics of the Dirichlet realisation with [34].

The Weyl formula for the asymptotic behaviour of the eigenvalues of Dirichlet realisations of elliptic operators dates back to Weyl [71], where his focus was on the Laplacian. Weyl’s result was extended to more general operators in many contributions to the literature, as summed up e.g. in Agmon’s paper [2] (including the treatment by G˚arding [47] of variable-coeﬃcient operators). In [2] and subsequent treatments by H¨ormander [49], Duistermaat-Guillemin [24], Seeley [64] and Ivrii [53] 56 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

the focus has been on the remainder term. See more in the survey of Ivrii [52]. We shall here just study the principal asymptotics, however for realisations that are more general than those deﬁned by pseudodiﬀerential boundary conditions.

We now use results of [34,42] to describe the resolvent of DB under the assumptions stated above. The method of proof will be to reduce the problem to a well behaved boundary condition by studying the resolvent diﬀerence (D + λ)−1 (D + λ)−1, B − Dir for the Dirichlet extension DDir. The idea of studying this resolvent diﬀerence and its localisation to the boundary is due to Birman [11,12]. The precise description that we use follows Grubb [34]. Recall the results of Subsection 6.3 for the Dirichlet realisation. We write Pζ,∂ (λ) for the boundary decomposing projection on H(ˇ D) induced by the projection P (λ) := 1 (D + λ)−1(D + λ) ζ − Dir max on dom(Dmax) (cf. Lemma 6.11). Theorem 7.1. Let D = D† be formally selfadjoint elliptic operator of order m acting on the sections of a hermitian vector bundle E on an n-dimensional manifold with boundary. Assume that D has positive interior principal symbol σD and that B is a regular and self-adjoint boundary condition. Then DB has discrete spectrum and for some λ> 0 large enough then we can write −1 (DB + λ) = Q+ + G + GB, where

(i) Q is the truncation of a pseudo-diﬀerential operator Qˆ Ψ−m(Mˆ ; Eˆ) that acts on a + ∈ cl hermitian vector bundle Eˆ extending E to a closed manifold Mˆ containing M as a smooth domain, and satisﬁes that −1 σ (Qˆ) ∗ = σ . −m |T M\M D (ii) G is a singular Green operator of order m and class 0 (cf. [42]). − (iii) GB is an operator that factors over continuous mappings

P H(ˇ D) ˇ T−1 Pker(D +λ) γ Pζ,∂ (λ)B H(D) L2(M; E) max ker(D + λ) P (λ)B B −−−−−−−−→ max −→CD+λ −−−−−−−−−→ ζ,∂ −−−→ T−1 B m− 1 m − 1 m K 2 ,(P (λ)B ֒ H 2 (Σ; E C ) ֒ H 2 (Σ; E C ) L (M; E −−−→ ζ,∂ → ⊗ → ⊗ −→ ˇ T where Pker(Dmax+λ) and P H(D) denotes the orthogonal projectors and B : Pζ,∂ (λ)B H(ˇ D) Pζ,∂ (λ)B Pζ,∂ (λ)B is the operator constructed in [34, Chapter II] (and there denoted → H(ˇ D) by the densely deﬁned T : V W for V = W = P (λ)B with dom(T )= P (λ)B). → ζ,∂ ζ,∂

We remark that all three operators Q+, G and GB depend on the choice of λ. However, Q+ and G are independent of the boundary condition B (a priori there is an implicit dependence via the choice of λ, but it is irrelevant for the structural statements we are interested in).

Proof. It follows from [42, Section 3.3] that we, for λ > 0 outside a discrete set of real numbers, can write −1 (DDir + λ) = Q+ + G, where Q+ and G are as in the statement of the theorem. The operator DB is self-adjoint be- m cause D is formally self-adjoint and B is Lagrangian. Since dom(DB) H (M; E), the inclusion dom(D ) ֒ L2(M; E) is compact. Therefore D has only discrete spectrum.⊆ For λ outside a B → B discrete subset so that DB +λ and DDir +λ are invertible, it follows from [34, Chapter II, Theorem 1.4] that (D + λ)−1 (D + λ)−1 = G , B − Dir B where GB is as in the statement of the theorem. This proves the theorem. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 57

~~Lemma 7.2. The singular values of the operators appearing in Theorem 7.1 have the following asymptotics:~~

~~− m − m (i) µk(Q+)= cQk n + o(k n ) where~~

~~m n 1 n m cQ = n TrE (σm(Q)(x, ξ) )dxdξ . n(2π) ˆ ∗ S M − m (ii) µk(G)= O(k n−1 ) − m (iii) µk(GB)= O(k n−1 )~~

Proof. Item i) follows from [42, Proposition 4.5.3]. One can also prove item i) by hand by applying the Weyl law on a closed manifold, see e.g. [51, Chapter XXIX] or [67, Chapter III], to suitable perturbations of G+. The statement of item ii) follows from [40]. The statement of item iii) follows m− 1 m − 1 m from that G factors over the inclusion H 2 (Σ; E C ) ֒ H 2 (Σ; E C ) whose singular B ⊗ → ⊗ value sequence is O(k−m/(n−1)) as k . Indeed, using a Weyl law on the n 1-dimensional → ∞ − compact manifold Σ shows that the singular values of Hs+m(Σ; E) ֒ Hs(Σ; E) is O(k−m/(n−1)) as k for any s R. → → ∞ ∈ −1 m m − n − n Remark 7.3. The statement that µk(Q+ + G) = µk(DDir) = cQk + o(k ) is a direct consequence of the spectral asymptotics of the Dirichlet realisation, see [2]. Compare to [47] and − m − m − m [38, Theorem 1]. Therefore µk(Q+)= cQk n + o(k n ) follows from that µk(G)= o(k n ) by [40] and the Weyl-Fan inequality (see for instance [68, Theorem 1.7]). The proof above is included since it shows how the different components in the resolvent of DB contributes to the spectral asymptotics. Theorem 7.4. Let D = D† be formally selfadjoint elliptic operator of order m acting on the sections of a hermitian vector bundle E on an n-dimensional manifold with boundary. Assume that D has positive interior principal symbol σD and that B is a regular and self-adjoint boundary condition defining a lower semi-bounded realisation DB. Define the constant −m/n 1 −n/m cD = n TrE(σD(x, ξ) )dxdξ . n(2π) ˆ ∗ S M The operator DB has discrete real spectrum, and its spectrum λ0(DB) λ1(DB) λ2(DB) satisfies that ≤ ≤ ≤ · · · m/n m λ (D )= c k + o(k n ), as k + . k B D → ∞

Proof. Using Theorem 7.1, Lemma 7.2 and the Weyl-Fan inequality (see for instance [68, Theorem 1.7]) we conclude that m/n m µ (D )= c k + o(k n ), as k + . k B D → ∞ Here (µk(DB))k∈N denotes the singular value sequence of the unbounded operator DB, so by ∗ −1 −1 deﬁnition µk(DB)= µk((1 + DBDB) ) 1. Since DB is self-adjoint and lower semi-bounded, µ (D )= λ (D ) for k large enough and the− theorem follows. k B k B p

Remark 7.5. The operator DB is lower semi-bounded if and only if the operator TB from Theorem 7.1, item iii), is lower semi-bounded. For a certain class of elliptic boundary conditions, the operator TB is realised as a pseudodiﬀerential operator in [36, Section 8] and in this case the asymptotics of the negative eigenvalues was studied. Remark 7.6. Theorem 7.4 was proven in [2] under the additional hypothesis that there is a k > k k km n/m such that DB is regular, i.e. dom(DB) H (M; E). By complex interpolation, using ⊆ l lm that DB is self-adjoint, this condition implies dom(DB) H (M; E) for l = 1,...,k. Note k km ⊆ that dom(DB) H (M; E) if and only if B is m(k 1)-semiregular. Let us provide a simple example where the⊆ assumption appearing in [2] fails, but− Theorem 7.4 holds. Our example is quite similar to Example 6.13 and provided solely for the purpose of giving intuition for the diﬀerence in assumptions between Theorem 7.4 and [2]. 58 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Assume that M Rn is a compact domain with smooth boundary and consider a shift of the standard Laplacian⊆ n D =∆+1=1 ∂2 . − xj j=1 X In particular, both the Dirichlet problem and the Neumann problem for D are uniquely solvable. For a real-valued function a L∞(∂Ω, R), consider the Robin boundary condition ∈ B := (ξ , ξ ) H(ˇ D): aξ + ξ =0 . a { 0 1 ∈ 0 1 } A short computation with Proposition 2.26 (or in other words using Green’s formula) shows that

Ba is self-adjoint. Since the operator DBa is deﬁned from the closed lower semibounded quadratic form 2 2 2 1 qa(u) := ( u + u )dV (x)+ a u dS(x), u H (M), ˆM | | |∇ | ˆΣ | | ∈ the operator DBa is lower semi-bounded and so satisﬁes the assumptions of Theorem 7.4 as soon as dom(D ) Hm(M; E), i.e. B is semi-regular (or equivalently regular, by self-adjointness). Ba ⊆ a By the same argument as in Example 6.13, B is s-semiregular if and only if for any (ξ , ξ ) a 0 1 ∈ B , the property (Λ + a)ξ Hs+1/2(Σ) implies that ξ Hs+3/2(Σ). If a is such that the a DN 0 ∈ 0 ∈ multiplication operator a : Hs+3/2(Σ) Hs+1/2(Σ) is compact, then for a generic4 such a, the → boundary condition Ba is s-semiregular. Conversely, if Ba is s-semi-regular, then elliptic regularity of Λ and the fact that (Λ +a)ξ Hs+1/2(Σ) ξ Hs+3/2(Σ) implies that a : Hs+3/2(Σ) DN DN 0 ∈ ⇒ 0 ∈ → Hs+1/2(Σ) is continuous.

k km Let us now give a condition on a such that dom(DBa ) H (M; E) holds for k = 1 but not for ⊆ 2 k =2, and therefore for no k> 1. We note that u dom(DBa ) if and only if DBa u dom(DBa ). 2 4 ∈ ∈ Therefore, the inclusion dom(DBa ) H (M) holds if and only if for u dom(DBa ) such that 2 ⊆4 ∈ 2 4 DB u H (M) it holds that u H (M). From Theorem 6.6 we see that dom(D ) H (M) a ∈ ∈ Ba ⊆ if and only if Ba is 2-semiregular. By the argument above, this occurs if the function a makes the multiplication operator a : H3/2(Σ) H1/2(Σ) compact but the multiplication operator a : → H7/2(Σ) H5/2(Σ) is not even continuous. This occurs when a C1(Σ) but a / H5/2(Σ). Indeed, → ∈ ∈ continuity of a : H7/2(Σ) H5/2(Σ) fails if a = a1 / H5/2(Σ) and C1(Σ) acts continuously H1/2(Σ) → ∈ (by interpolation of the continuous actions H1(Σ) and L2(Σ)) and H3/2(Σ) H1/2(Σ) is compact → by Rellich’s theorem. For an example of an a C1(Σ) with a / H5/2(Σ), take M to be the unit disc, so Σ= S1, and define a from the lacunary∈ Fourier series ∈ ∞ k k a(z) := 2−2k(z2 + z−2 ) C1(S1) H5/2(S1). (25) ∈ \ kX=0 For a as in Equation (25), the regular realisation (1 + ∆)Ba on the unit disc provides a concrete k km example where the Weyl law holds by Theorem 7.4 but the assumption dom(DB) H (M; E) for some k > n/m from [2] fails. ⊆ Corollary 7.7. Let D :C∞(M; E) C∞(M; F ) be an elliptic differential operator of order m> 0 → and B H(ˇ D) a regular and m-semiregular boundary condition. Then the sequence of singular ⊆ 2 2 values of the operator DB :L (M; E) 99K L (M; F ) satisfies that m m µ (D )= √c † k n + o(k n ), as k + , k B D D → ∞ where −2m/n 1 ∗ −n/2m † cD D = n TrE((σD(x, ξ) σD(x, ξ)) )dxdξ , n(2π) ˆ ∗ S M and σD denotes the interior principal symbol of D.

~~2 ∗ ∗ Proof. We have that µk(DB) = λk(DBDB) and by Corollary 6.8, DBDB is regular if B is regular ∗ and m-semiregular. The corollary follows from Theorem 7.4 since DBDB is non-negative.~~

4 s+3/2 s+1/2 In the sense that for ε ∈ (−1, 1) outside of a discrete set of points ΛDN + a + ε : H (Σ) → H (Σ) is invertible. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 59

Remark 7.8. A consequence of Corollary 7.7 is that if D is a formally self-adjoint, elliptic differential operator of order m> 0 and B H(ˇ D) is a Lagrangian, regular and m-semiregular boundary ⊆ condition, then DB has discrete spectrum consisting of real eigenvalues that asymptotically behaves as m m λ (D ) = c k n + o(k n ), as k + , | k B | |D| → ∞ where −m/n 1 −n/m c|D| = n TrE( σD(x, ξ) )dxdξ . n(2π) ˆ ∗ | | S M For more details on the spectral asymptotics of the sequences of negative and positive eigenvalues, see [42, Chapter 4.5]. Remark 7.9. The assumption that B is regular in Theorem 7.4 can in general not be dropped. To give an example thereof, we assume for simplicity that D is as in Theorem 7.4 and DDir is invertible. ′ m′ m Following the notations of [34, 36], for a small m < m we take a regular T0 Ψ (Σ; E C ) m′−m m ∈ ⊗ with [T0, PC ] Ψ (Σ; E C ) and define T := PCT0PC. The operator T is a densely defined ∈ ⊗ m′− 1 m Fredholm operator with dom(T )= P H 2 (Σ; E C ). Assume for simplicity that T CD →CD C ⊗ is invertible. Let DT be the realisation of D constructed as in [34, Chapter II, Proposition 1.2]. m′− 1 m The realisation DT is not regular, but γ(dom(DT )) H 2 (Σ; E C ), cf. [36, Corollary 6.10]. By [34, Chapter II, Theorem 1.4], we have that ⊆ ⊗ D−1 D = T −1, T − Dir −1 2 ′ where T is “extended by zero” to an operator on L (M; E)= D ran(Dmin). If m

~~m′ µ (D ) & k n−1 , as k . k T → ∞~~

~~8. Rigidity results for the index of elliptic differential operators~~

In this section we prove a rigidity result for the Fredholm index of Fredholm realisations DB. As noted in Remark 2.13, the index formula of Theorem 2.12 implies homotopy invariance for the choice of Fredholm boundary condition when D is ﬁxed. We now extend this observation to more general homotopies.

We topologise the space of pseudodifferential operators by declaring the symbol mappings and its local splittings defined from Kohn-Nirenberg quantisation, as well as the inclusion of the smoothing operators, to be continuous. The space of order m differential operators C∞(M; E) C∞(M; F ) will be topologised as linear operators between the Fréchet topologies of C∞(M; E) and→ C∞(M; F ). Equivalently, we can topologise the space of order m differential operators C∞(M; E) C∞(M; F ) in terms of the C∞-topology on the coefficients of the operator. →

~~Theorem 8.1. Assume that (Dt)t∈[0,1] is a family of elliptic diﬀerential operators of order m> 0 such that~~

~~(i) The family is continuous in the topology on the space of order m diﬀerential operators C∞(M; E) C∞(M; F ). → (ii) dom(Dt,max) is independent of t.~~

Let [0, 1] t P B(H(ˇ D)) be a norm continuous family of projectors such that B := (1 ∋ 7→ t ∈ t − P )H(ˇ D) satisﬁes that (B , ) is a Fredholm pair in H(ˇ D) for all t [0, 1]. Then it holds that t t CDt ∈

~~ind(D0,B0 ) = ind(D1,B1 ).~~

~~In order to prove this theorem, we ﬁrst prove the following two lemmas. 60 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN~~

Lemma 8.2. Assume that (Dt)t∈[0,1] is a family of elliptic diﬀerential operators of order m> 0 as 2 in Theorem 8.1. Then viewing Dt,max as a continuous linear mapping dom(D0,max) L (M; F ) it holds that t D deﬁnes a norm-continuous mapping [0, 1] B(dom(D ),→L2(M; F )). 7→ t,max → 0,max

Proof. The proof of this lemma is slightly technical and relies on the construction of approximate Calderón projectors following Theorem 3.8 and 3.18. The family (Dt)t∈[0,1] extends to a continuous family of elliptic differential operators on a closed manifold containing M as a domain, this is as in the proof of Theorem 3.18 above. We follow the notation in the proof of Theorem 3.18. Since the family (Dt)t∈[0,1] is continuous in the topology on the space of order m differential operators, we can construct a continuous family (Tˆ ) Ψ−m(M; F, E) such that Tˆ is a parametrix of t t∈[0,1] ⊆ cl t Dˆ for all t [0, 1]. Following Equation (6), we define an approximate Poisson operator t ∈ m−1 m−1−j ˜ (ξ )m−1 = ( i)l+1Tˆ [A ξ δ(l) ]. Kt j j=0 − t m−l−j−1 j ⊗ t=0 j=0 X Xl=0 Using the same arguments as in [65], it holds that ˜ : H(ˇ D) dom(D ) continuously for Kt → t,max all t [0, 1]. Indeed, letting Dt denote the Poisson operator of Dt (see [65] or Theorem 3.18) ∈ ˜ K ˇ ∞ ˜ above, we have that t Dt : H(D) C (M; E). We therefore also have that γ t PCt −∞ m K − K → ◦ K − ∈ Ψ (Σ; E C ) is a smoothing operator, and in particular γ ˜t is an approximate Calderón projection as⊗ in Theorem 3.8. ◦ K

Consider the family of bounded operators Rt := ˜t γ : dom(Dmax) dom(Dmax). We note that for f C∞(M; E), K ◦ → ∈ m−1 m−1−j D R f = ( i)l+1(D Tˆ 1)[A γ (f) δ(l) ], t t − t t − m−l−j−1 j ⊗ t=0 j=0 X Xl=0 because distributions supported on Σ vanish when restricted to M. However, since Tˆt is a con- −∞ tinuous family of parametrices, the mapping [0, 1] t (DˆtTˆt 1) Ψ (Mˆ ; Fˆ) is continuous. ∋ 7→ −ˆ ∈† We conclude that t DtRt factors over aγ : dom(Dmax) H(D ), the continuous inclusion 7→ → .( H(ˆ D†) ֒ ′(Σ; F Cm) and a continuous family of operators ′(Σ; F Cm) C∞(M; F → D ⊗ s D ⊗ → As such, [0, 1] t DtRt B(dom(Dmax), H (M; F )) is norm-continuous for all s R, and in particular [0, 1]∋ t7→ D R ∈ B(dom(D ), L2(M; F )) is norm-continuous. ∈ ∋ 7→ t t ∈ max ˜ ˇ ∞ m Since t Dt : H(D) C (M; E), we have that (1 Rt) : dom(Dmax) H (M; E). Indeed, K − K → m − → we can write dom(Dmax) = H (M; E) + ker(Dt,max) by Example 3.21 and 1 Rt is smoothing ˜ ˇ ∞ − on ker(Dt,max) because of t Dt : H(D) C (M; E). Using the semi-classical nature of K − K → m the arguments in [65], it follows that [0, 1] t 1 Rt B(dom(Dmax), H (M; E)) is norm- ∋ 7→ m − ∈ 2 continuous. It is clear that [0, 1] t Dt B(H (M; E), L (M, E)) is continuous. Therefore [0, 1] t D (1 R ) B(dom(D∋ 7→), L2(M,∈ E)). ∋ 7→ t − t ∈ max

Since the map t Dt = DtRt + Dt(1 Rt) is a sum of continuous maps [0, 1] 2 7→ − → B(dom(Dmax), L (M, E)), the map t Dt,max deﬁnes a norm-continuous mapping [0, 1] 2 7→ → B(dom(D0,max), L (M; F )).

~~Lemma 8.3. Assume that (Dt)t∈[0,1] is a family of elliptic diﬀerential operators of order m > 0 as in Theorem 8.1 and deﬁne the family of self-adjoint operators~~

0 D D˜ := t,min , t D† 0 t,max −1 for t [0, 1]. Then for each λ R 0 , the mapping t (λi + D˜ t) is a norm-continuous mapping∈ [0, 1] B(L2(M; F E∈)). \{ } 7→ → ⊕ REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 61

Proof. By the assumptions of Theorem 8.1, the domain of D˜ t is constant in t. We have for t ,t [0, 1] that 1 2 ∈ (λi + D˜ )−1 (λi + D˜ )−1 = (λi + D˜ )−1(D˜ D˜ )(λi + D˜ )−1 t1 − t2 t1 t2 − t1 t2 Since (λi+D˜ )−1 :L2(M; F E) dom(D˜ ) = dom(D ) dom(D† ) with the uniform norm t ⊕ → t t,min ⊕ t,max bound λ −1 as t varies, the lemma follows from Lemma 8.2 (applied to (D†) ) and the fact | | t t∈[0,1] that t D clearly deﬁnes a norm-continuous mapping [0, 1] B(Hm(M; E), L2(M; F )). 7→ t,min → 0

m Proof of Theorem 8.1. The ellipticity of each Dt ensures that dom(Dt,min) = H0 (M; E) is con- ˇ stant in t. We remark that item (ii) ensures that H(Dt) ∼= dom(Dt,max)/dom(Dt,min) = m ˇ dom(Dt,max)/H0 (M; E) is independent of t, justifying the notation H(D) for the Cauchy data space.

~~The only remaining property to verify in order to invoke Theorem B.32 below is that the mappings~~

† 0 Dt,min 0 D [0, 1] t and [0, 1] t t,min , ∋ 7→ D† 0 ∋ 7→ D 0 t,max t,max † are continuous in the gap topology. Lemma 8.3 applied to the families (Dt)t∈[0,1] and (Dt )t∈[0,1] furnishes us with this fact.

Corollary 8.4. Let D be an elliptic diﬀerential operator and B a regular boundary condition. As- sume that B is graphically decomposed with respect to the Calderon projector by (V+, W+, V−, W−,g) as in Deﬁnition 5.3. Then it holds that ind(D ) = dim(W ) dim(W ) + dim(ker(D ) dim(ker(D† ). B + − − min − min

~~Proof. We deﬁne the continuous path of regular boundary conditions (Bt)t∈[0,1] from the continuous path of data for a graphical decomposition (V+, W+, V−, W−,tg)t∈[0,1]. Note that~~

∗ −1 ∗ 1 m ∗ B0 = W+ V− and B = a (V H 2 (Σ; E C ) W ). (26) ⊕ 0 † + ∩ ⊗ op ⊕ − m−1/2 m We also note that V+ W+ = D H (Σ; E C ) by the deﬁnition of a graphical decomposition with respect to the Calderon⊕ C projection.∩ By Theorem⊗ 8.1, we have that

~~ind(DB) = ind(DB0 ) = ind(DB1 ).~~

The proof now follows from Theorem 2.12 upon proving that ind(B0, D) = dim(W+) dim(W−). ∗ −1 C ∗ − This follows from the fact that B0 D = W+ and B † = a (W ) by Equation (26). ∩C 0 ∩CD † −

Theorem 8.5. Assume that (Dt)t∈[0,1] is a family of elliptic diﬀerential operators of order m> 0 which is continuous in the topology on the space of order m diﬀerential operators C∞(M; E) ∞ m− 1 m → C (M; F ). Let (Pt)t∈[0,1] B(H 2 (Σ; E C )) be a norm continuous family of projectors such that ⊆ ⊗

− 1 m (1) For all t [0, 1], Pt extends to a bounded operator on H 2 (Σ; E C ). (2) For all t ∈ [0, 1], α 1/2,m 1/2 the operator ⊗ ∈ ∈ {− − } P (1 P ): Hα(Σ; E Cm) Hα(Σ; E Cm), (27) Ct − − t ⊗ → ⊗

~~is a Fredholm operator, where PCt denotes the Calderon projector for Dt.~~

~~m− 1 m Then for all t [0, 1], B := (1 P )H 2 (Σ; E C ) is a regular boundary condition for D and ∈ t − t ⊗ t the index ind(Dt,Bt ) is independent of t. In particular,~~

~~ind(D0,B0 ) = ind(D1,B1 ). 62 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN~~

The proof goes along similar lines as that of Theorem B.32 but adapted to regular realisations with domains contained in Sobolev spaces. We note that Theorem 8.5 in particular holds for a 0 m continuous family (Pt)t∈[0,1] Ψcl(Σ; E C ) of pseudodiﬀerential projectors such that for all t [0, 1] ⊆ ⊗ ∈ ∞ ∗ ∗ p (D ) (1 σ0(P )) C (S Σ; Aut(π E)). + t − − 0 t ∈

Proof. We note that each Bt is a regular boundary condition by Theorem 4.14. Since (Dt)t∈[0,1] is a continuous family of elliptic differential operators, it defines a bounded adjointable map of C[0, 1]-Hilbert C∗-modules D : Hm(M; E) C[0, 1] L2(M; F ) C[0, 1]. ⊗ → ⊗ 1 ˇ m− 2 m ˇ ∗ We write Ct := PtH (Σ; E C ) and let CC([0,1]) denote the C[0, 1]-Hilbert C -submodule m− 1 m ⊗ of H 2 (Σ; E C ) C([0, 1]) with fibre Cˇ , it is well-defined by norm continuity of the family ⊗ ⊗ t (P ) . We write Pγ : Hm(M; E) C([0, 1]) Cˇ for the bounded adjointable mapping t t∈[0,1] ⊗ → C([0,1]) obtained from composing the trace mapping with the family (Pt)t∈[0,1].

Following Proposition B.30 and Theorem B.32, we define the operator L2(M; F ) C[0, 1] ⊗ Du Lˇ : Hm(M; E) C[0, 1] , Luˇ = ⊗ → ⊕ Pγu Cˇ This operator is an adjointable C[0, 1]-linear map because D and Pγ are adjointable. We note that Lˇ if Fredholm in each fibre by Proposition B.30 (cf. [5, Proposition A.1]). By an argument along the same lines as in Theorem B.32, the adjointable C[0, 1]-linear map Lˇ : Hm(M; E) C[0, 1] ⊗ → L2(M; F ) C[0, 1] Cˇ is Fredholm with a well defined index ind(Lˇ) K0([0, 1]). Again arguing as in Theorem⊗ B.32⊕, homotopy invariance of K-theory implies that ∈ ˇ ∗ ˇ ind(Dt,Bt ) = ind(Lt)= rt ind(L), where r denotes the inclusion t [0, 1], is independent of t. t { }⊆

We end this section with an outlook towards index theory. It poses an interesting problem to compute ind(DBP ) where D is an elliptic differential operator D of order m > 0 and BP := m− 1 m (1 P )H 2 (Σ; E C ) is a regular pseudo-local boundary condition on D defined from P Ψ0−(Σ; E Cm). While⊗ Theorem 2.12 and Corollary 5.7 give formulas, a more worthwhile goal∈ cl ⊗ would be an explicit formula for the index of DB in terms of the interior principal symbol of D, the germ of D on Σ, the differential operator a defining the boundary pairing and the projection P . The Atiyah-Patodi-Singer index theorem and the index theorem in the extended Boutet de Monvel calculus of [63] indicates that such an explicit computation is possible. Under some mild assumptions on P , the special case of first order elliptic operators is susceptible to heat kernel techniques by the results of [43].

There is an interesting special case that the authors of this paper have not found an explicit solution to in the literature. A classical situation is that D/ is a Dirac operator and P is a spectral projection of the boundary operator of D/ constructed from a normal vector field to the boundary. Atiyah-Patodi-Singer [4] computed the index in the case of product structure at the boundary and [41] considered the general case. For more details, see also [15, 59]. Assume that we leave this classical realm and instead use a general vector field transversal to the boundary to define the adapted boundary operator A. Then A is not necessarily self-adjoint, but only bisectorial (cf. [6]). + Is there an explicit formula for the index ind(DBP ) for P = χ (A)? Preferably, such a formula only uses local data and spectral data of the bisectorial operator A. The results of [43] indicates that there might be a positive answer. The importance of this question is seen in that similar questions can be asked on manifolds with Lipschitz boundary, where a smooth vector field transversal to the boundary (almost everywhere) makes sense. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 63

~~9. Symbol computations with Calderon´ projectors in the first order case~~

In this section, we study the way in which the Calderón projection of a first order elliptic operator relates to the the positive spectral projection of an adapted boundary operator. We show that the difference of these projectors are an operator of order 1 and continuous, but in general not compact on the Cauchy data space. −

~~Near the boundary,~~

D = σ∂xn + A1, (28) where A = (A (x )) is a family of first order differential operators acting from E Σ to 1 1 n xn∈[0,1] |Σ → F Σ Σ. By ellipticity of D, σ is a bundle isomorphism and A1 is a family of elliptic differential operators.| → We note that an adapted boundary operator for D is precisely a differential operator A on E such that A σ−1A is of order zero. |Σ − 1|xn=0

Recall the construction of E+(D) and p+(D) from the subsection 3.1. Lemma 9.1. Let D be a first order elliptic differential operator and A an adapted boundary operator on E . Then χ+(A) is a classical order zero pseudo-differential operator whose principal |Σ symbol is p+(D).

Proof. Since χ+(A) is constructed as a sectorial projection, it is a classical order zero pseudo- diﬀerential operator, see [46]. Its principal symbol is the same as χ+(a) where a is the principal symbol of A. By construction, a = σ−1a where a is the principal symbol of A appearing 1 1 1|xn=0 in the collar decomposition D = σ∂xn + A1. We conclude that σ∂ (D)= σ∂xn + a1 = σ(∂xn + a). It is clear that for (x′, ξ′) S∗Σ and v E , Equation (5), that in this case takes the form ∈ 0 ∈ x ′ ′ σ(∂t + a(x , ξ ))v(t)=0, t> 0, (v(0) = v0, + ′ ′ has an exponentially decaying solution if and only if v0 χ (a(x , ξ ))Ex, namely v(t) = −a(x′,ξ′)t ∈ e v0. Similarly, there is an exponentially increasing solution if and only if v0 − ′ ′ + − ∈ χ (a(x , ξ ))Ex. So E+(D) = χ (a)E and E−(D) = χ (a)E and we can conclude that the + principal symbol of χ (A) equals p+(D).

Remark 9.2. By Proposition 3.4, we know that E+(D) = 0 and E−(D) = 0. In fact, each ﬁbre is non-zero if the dimension exceeds 1. For ﬁrst order operators,6 this can6 be proven more directly ′ ′ ′ ′ ′ ′ ′ ′ using the symmetry a(x , ξ ) = a(x , ξ ) which implies that p+(D)(x , ξ )=1 p+(D)(x , ξ ) so p (D) =0 and a similar argument− − implies p (D) =0. − − + 6 − 6 Lemma 9.3. Let D be a Dirac type operator and A an adapted boundary operator on E Σ such that |

~~D = σ(∂xn + A), near the boundary. Then χ+(A) P Ψ−∞(Σ; E) is a smoothing operator. − CD ∈~~

Proof. It suffices to prove that χ+(A) Q Ψ−∞(Σ; E) is a smoothing operator for Q being the approximate Calderón projection from− Theorem∈ 3.8. Since the construction in Theorem 3.8 is microlocal at the boundary we can assume that M = Σ [0, ). In this case, any f ker Dmax can be written in the form × ∞ ∈ f(t)=e−tAf(0). −tA + + As such, g(t) =e χ (A)g and the Calderón projection PCD = χ (A) is an approximate Calderón projection.K Lemma 9.4. Let D be a first order elliptic differential operator with adapted boundary operator A. The first term in the Calderón projection is given by + σ0(PCD )=p+(D)= χ (σ1(A)). 64 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

In coordinates near a point x Σ, we have that 0 ∈ ′ ′ ′ ′ σ−1(PCD )(x , ξ )= Resz=λt1(x , ξ , iλ), z:Re(Xz)>0 where ′ ′ − − ′ ′ −1 ′ ′ − ′ ′ −1− t1(x ,ξ , iλ)= (λ cA(x ,ξ )) (cA,0(x )+ cR0,0(x , 0))(λ cA(x ,ξ )) − − ′ ′ −1 ′ − ′ ′ −1 ′ ′ − ′ ′ −1 − (λ cA(x ,ξ )) cA x , (λ cA(x ,ξ )) dx′ cA(x ,ξ ))(λ cA(x ,ξ ))

~~′ ′ −2 ∂cR0 ′ ′ ′ ′ −1 − (λ − cA(x ,ξ )) (x , 0,ξ )(λ − cA(x ,ξ )) . ∂xn~~

Here, we have written D = σ(∂xn + A + R0) for an xn-dependent ﬁrst order diﬀerential operator R0 on Σ, cA denotes the principal symbol of A, cA,0 the sub-leading term (and similarly for A), ′ ′ and where we use the notation cA(x ,ν M) := cA(x ,ν)M for an endomorphism valued covector ν M. ⊗ ⊗

Proof. By the formula for the full symbol of the approximate Calder´on projection from Theorem 3.8, we have that Qf = γ T [σf δ ], 0 ◦ 0 ⊗ t=0 where T0 is a parametrix for D. In particular, we note that Qf = γ T [f δ ], 0 ◦ ⊗ t=0 where T is a parametrix for σ−1D. The operator T is a classical pseudodiﬀerential operator of order 1 in a neighbourhood of M in an extended manifold. In coordinates near a point x0 Σ, a full symbol− t of T has an asymptotic expansion t ∞ t and each t = t (x, ξ) is smooth∈ in x ∼ j=0 j j j and homogeneous of degree 1 j in ξ. Using the methodology of [50, Chapter XX], cf. Remark P 3.9, the full symbol of Q can− be− computed by

′ ′ ′ ′ qj (x , ξ )= Resξn=ztj (x , 0, ξ , ξn), z:Im(Xz)>0 where (x′, ξ′) denote the associated coordinates on Σ. As such, we shall need to compute the two ′ ′ terms in the asymptotic expansion of T . By an abuse of notation, we will write tj (x , ξ , ξn) for ′ ′ tj (x , 0, ξ , ξn).

To compute the asymptotic expansion of T , we write the full symbol of D as cD + cD,0 where cD is the principal symbol (i.e. a homogeneous degree 1 polynomial) and cD,0 is ξ-independent. We can write D = σ(Dt + A + R0) for an xn-dependent ﬁrst order diﬀerential operator R0 on Σ with R being a multiplication operator. We use similar notation for A and R and note that 0|xn=0 0 ′ ′ ′ ′ ′ ′ ′ cD(x , xn, ξ , ξn) = σ(x , xn)(iξn + cA(x , ξ )+ cR0 (x , xn, ξ )), and ′ ′ ′ ′ (cD,0(x , xn) = σ(x , xn)(cA,0(x )+ cR0,0(x , xn)). ′ ′ The assumption that A is adapted ensures that cR0 (x , 0, ξ ) = 0. To compute the parametrix of σ−1D, we use the symbols

′ ′ ′ −1 ′ ′ ′ ′ ′ ′ c˜D(x , xn, ξ , ξn) := σ(x , xn) cD(x , xn, ξ , ξn)= iξn + cA(x , ξ )+ cR0 (x , xn, ξ , ξn), ′ ′ −1 ′ ′ ′ (c˜D,0(x , xn) := σ(x , xn) cD,0(x , xn)= cA,0(x )+ cR0,0(x , xn). From the well known parametrix construction, the full symbol t ∞ t of T is computed ∼ j=0 j inductively from that t (x, ξ)= c (x, ξ)−1 and 0 D P 1 t = t c˜ t t ∂αc˜ Dαt . j+1 − 0 D,0 j − 0 α! ξ D x k k+|α|=Xj+1, k≤j In particular, t = t c˜ t t ∂αc˜ Dαt . 1 − 0 D,0 0 − 0 ξ D x 0 |αX|=1 REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 65

At xn = 0, we have that ′ ′ ′ ′ −1 t0(x , ξ , ξn) =(iξn + cA(x , ξ )) , and ′ ′ t1(x ,ξ ,ξn)= − ′ ′ −1 ′ ′ ′ ′ −1 = (iξn + cA(x ,ξ )) (cA,0(x )+ cR0,0(x , 0))(iξn + cA(x ,ξ )) + n−1 ′ ′ −1 ′ ′ ′ −1 ∂cA ′ ′ ′ ′ −1 +(iξn + cA(x ,ξ )) X cA(x , dxj )(iξn + cA(x ,ξ )) (x ,ξ ))(iξn + cA(x ,ξ )) + ∂xj j=1

′ ′ −1 ′ ′ ′ −1 ∂cR0 ′ ′ ′ ′ −1 +(iξn + cA(x ,ξ )) c˜D((x , 0), dxn)(iξn + cA(x ,ξ )) (x , 0,ξ )(iξn + cA(x ,ξ )) = ∂xn − ′ ′ −1 ′ ′ ′ ′ −1 = (iξn + cA(x ,ξ )) (cA,0(x )+ cR0,0(x , 0))(iξn + cA(x ,ξ )) + ′ ′ −1 ′ ′ ′ −1 ′ ′ ′ ′ −1 +(iξn + cA(x ,ξ )) cA x , (iξn + cA(x ,ξ )) dx′ cA(x ,ξ ))(iξn + cA(x ,ξ )) +

′ ′ −2 ∂cR0 ′ ′ ′ ′ −1 +(iξn + cA(x ,ξ )) (x , 0,ξ )(iξn + cA(x ,ξ )) . ∂xn Lemma 9.5. Let A be an elliptic first order differential operator on Σ. The first term in the positive spectral projection is given by + + σ0(χ (A)) =χ (σ1(A)). In coordinates near a point x Σ, we have that 0 ∈ + ′ ′ σ−1(χ (A)) = Resλ=zb1(x , ξ , λ), z:Re(Xz)>0 where in the notations of Lemma 9.4 ′ ′ ′ ′ −1 ′ ′ ′ −1 b1(x ,ξ ,λ)= − (λ − cA(x ,ξ )) cA,0(x )(λ − cA(x ,ξ )) + (29) − − ′ ′ −1 ′ − ′ ′ −1 ′ ′ − ′ ′ −1 (λ cA(x ,ξ )) cA x , (λ cA(x ,ξ )) dx′ cA(x ,ξ ))(λ cA(x ,ξ )) .

Proof. Consider B := (λ A)−1 as a pseudodiﬀerential with a parameter. Picking coordinates near a point x Σ, we write the− full symbol with parameter of B as b ∞ b , where b = b (x′, ξ′, λ) 0 ∈ ∼ j=0 j j j is smooth in x′ and homogeneous of degree 1 j in (ξ′, λ). By [46, Theorem 3.3], the full symbol + ∞− − P of χ (A) has the asymptotic expansion j=0 χj where ′ ′ ′ ′ χj (x , ξ )=P Resλ=zbj(x , ξ , λ). z:Re(Xz)>0 As such, we shall need to compute the ﬁrst two terms in the asymptotic expansion of B.

Using the notations of Lemma 9.4 and its proof, we can compute bj inductively from that b (x′, ξ′, λ) = (λ c (x′, ξ′))−1 and 0 − A 1 b = b c b b ∂αc Dαb . j+1 − 0 A,0 j − 0 α! ξ A x k k+|α|=Xj+1, k≤j In particular, b = b c b b ∂αc Dαb , 1 − 0 A,0 0 − 0 ξ A x 0 |αX|=1 which is readily computed to reproduce the formula in Equation (29). Definition 9.6. Let D be an elliptic first order differential operator with adapted boundary operator −1 A. We say that A is a well-adapted boundary operator if A = σ A1 xn=0, where A1 is as in Equation (28). |

We note that upon ﬁxing a choice of normal to the boundary, there is a uniquely determined well adapted boundary operator. In the notations of Lemma 9.4, A is well adapted if and only if ′ cR0,0(x , 0) = 0 which holds if and only if R0 xn=0 = 0. The following Proposition is immediate from Lemma 9.4 and 9.5. | 66 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Proposition 9.7. Let D be a ﬁrst order elliptic diﬀerential operator with adapted boundary oper- + ator A. Then χ (A) PCD is of order 1 and its principal symbol is in the notation of Lemma 9.4 given by − − σ (χ+(A) P )= Res p(x′, ξ′, λ), −1 − CD λ=z z:Re(Xz)>0 where p(x′, ξ′, λ) =(λ c (x′, ξ′))−1c (x′, 0)(λ c (x′, ξ′))−1+ − A R0,0 − A ′ ′ −2 ∂cR0 ′ ′ ′ ′ −1 + (λ cA(x , ξ )) (x , 0, ξ )(λ cA(x , ξ )) , − ∂xn −

If A is well adapted, then σ (χ+(A) P ) is constructed from residues of the simpler symbol −1 − CD ′ ′ ′ ′ −2 ∂cR0 ′ ′ ′ ′ −1 p(x , ξ , λ) =(λ cA(x , ξ )) (x , 0, ξ )(λ cA(x , ξ )) . − ∂xn − Corollary 9.8. Let D be a ﬁrst order elliptic diﬀerential operator with well adapted boundary operator A.

~~∂cR0 + −2 If Σ commutes with cA, then χ (A) PC Ψ (M; E) and acts compactly on • ∂xn | − D ∈ cl Hˇ A(D). ∂cR0 If Σ anti-commutes with cA, then • ∂xn |~~

~~+ 1 −2 − ∂cR0 σ−1(χ (A) PCD )= cA χ (cA) Σ. − 4 ∂xn |~~

′ ′ Proof. The ﬁrst item follows from Proposition 9.7 if we can prove that Resλ=zp(x , ξ , λ) = 0 for ∂cR0 all z. Since A is well adapted, and cA commutes with Σ, Proposition 9.7 ensures that ∂xn |

~~′ ′ ′ ′ −3 ∂cR0 ′ ′ p(x , ξ , λ) =(λ cA(x , ξ )) (x , 0, ξ ). − ∂xn~~

~~∂cR0 In particular, the residues of p vanish if Σ commutes with cA. ∂xn |~~

~~∂cR0 On the other hand, if Σ anti-commutes with cA ∂xn |~~

~~′ ′ ′ ′ −2 ′ ′ −1 ∂cR0 ′ ′ p(x , ξ , λ) =(λ cA(x , ξ )) (λ + cA(x , ξ )) (x , 0, ξ ). − ∂xn Using Proposition 9.7 we compute that in this case~~

+ ′ ′ 1 ′ ′ −2 − ′ ′ ∂cR0 ′ ′ σ−1(χ (A) PCD )(x , ξ )= cA(x , ξ ) χ (cA(x , ξ )) (x , 0, ξ ). − 4 ∂xn Proposition 9.9. Let D be an elliptic ﬁrst order diﬀerential operator with adapted boundary operator A. Then χ+(A) P is compact on Hˇ (D) if and only if − CD A σ (χ−(A))σ (χ+(A) P )σ (χ+(A))=0, 0 −1 − CD 0 as elements of C∞(S∗Σ; End(π∗E)).

Proof. We write χ+(A) P =χ+(A)(χ+(A) P )χ+(A)+ χ−(A)(χ+(A) P )χ−(A)+ − CD − CD − CD + χ+(A)(χ+(A) P )χ−(A)+ χ−(A)(χ+(A) P )χ+(A) − CD − CD =χ+(A)(χ+(A) P )χ+(A)+ χ−(A)(χ+(A) P )χ−(A)+ − CD − CD + χ+(A)(χ+(A) P )χ−(A)+ χ−(A)(χ+(A) P )χ+(A) − CD − CD Since χ+(A) P Ψ−1(Σ; E) the ﬁrst two terms are compact for any adapted boundary operator − CD ∈ cl A. The third term is a pseudodiﬀerential operator of order 1 and it acts compactly on Hˇ (D) − A REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 67

1 − 1 if and only if it acts compactly H 2 (Σ; E) H 2 (Σ; E) and as such the third term is compact on → Hˇ A(D) for any adapted boundary operator A. The fourth term is a pseudodifferential operator − 1 of order 1 and it acts compactly on Hˇ A(D) if and only if it acts compactly H 2 (Σ; E) 1 − → H 2 (Σ; E) and as such the fourth term is compact on Hˇ A(D) if and only if its symbol vanish, i.e. σ (χ−(A))σ (χ+(A) P )σ (χ+(A)) = 0. 0 −1 − CD 0 Proposition 9.10. Let D be a first order elliptic differential operator with well adapted boundary ∂cR0 + operator A and assume that Σ anti-commutes with cA. Then χ (A) PC is compact on ∂xn | − D Hˇ A(D) if and only if

~~− ∂cR0 χ (cA) Σ =0. ∂xn |~~

− + + Proof. By Proposition 9.9, it suﬃces to prove that χ (cA)σ−1(χ (A) PCD )χ (cA) = 0 if and − ∂cR0 ∂cR0 − only if χ (cA) Σ = 0 when Σ anti-commutes with cA. But by Corollary 9.8, ∂xn | ∂xn |

~~− + + 1 −2 − ∂cR0 + χ (cA)σ−1(χ (A) PCD )χ (cA)= cA χ (cA) Σχ (cA) − 4 ∂xn |~~

~~1 −2 − ∂cR0 = cA χ (cA) Σ, 4 ∂xn |~~

~~∂cR0 where we in the last line used that Σ anti-commutes with cA. The proposition follows from ∂xn | that cA is invertible.~~

~~Example 9.11. Let us give an easy example showcasing when χ+(A) P fails to be compact − CD on Hˇ (D). Take M to be the unit disc and E = F = M C2. A ×~~

We ﬁrst consider the rather straight forward example of the Dirac type operator D0 given in polar coordinates by 0 ∂ + i ∂ D = r r θ = σ(∂ + A + R ), 0 ∂ + i ∂ 0 r 00 − r r θ where 0 1 i∂θ 0 −1 σ = , A = − and R00 = (r 1)A. 1 0 0 i∂θ − − It is readily veriﬁed that A is a well adapted boundary operator, and in this case it is self-adjoint. A short computation with the Cauchy-Riemann equation in polar coordinates show that P = χ+(A) CD0 for the operator D0.

To jazz up this example a bit, we deﬁne R := R iασ∂ , 0 00 − θ where α = α(r) is a smooth function α C∞(0, 1] with α(1) = 0. We deﬁne ∈ c iα∂ ∂ + i ∂ D := σ(∂ + A + R )= θ r r θ . α r 0 ∂ + i ∂ iα∂ − r r θ θ The reader should note that Dα is formally self-adjoint if and only if α is real valued and A is well + adapted for any choice of α. Let us compute σ−1(χ (A) PCD ) in this case. For simplicity, we write ξ for the cotangent variable on Σ= S1. We have that−

−1 ξ 0 −1 (λ + ξ) 0 cA(ξ)= − , and (λ cA) (ξ)= . 0 ξ − 0 (λ ξ)−1 − We have c (r, ξ) = (r−1 1)c ασξ and compute that R0 − A − ∂c R0 (ξ)= c (ξ) α′(1)σξ. ∂r |Σ − A − 68 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

∂c Although Corollary 9.8 does not give an immediate answer because R0 does not commute nor ∂r |Σ anticommute with cA in this case, we have a sum of a commuting and anticommuting term and a short computation gives us that

∂c p(ξ, λ)= (λ c (ξ))−2 R0 (r, ξ) (λ c (ξ))−1 − − A ∂r |r=1 − A =(λ c (ξ))−2c (ξ)(λ c (ξ))−1 + (λ c (ξ))−2α′(1)σξ(λ c (ξ))−1 − A A − A − A − A =(λ c (ξ))−3c (ξ)+ α′(1)ξ(λ c (ξ))−2(λ + c (ξ))−1σ. − A A − A A The ﬁrst term has no residues, and the residues of the second term is computed as in Corollary 9.8 to produce

α′(1)ξ α′(1) σ (χ+(A) P )(ξ)= c−2χ−(c )σ = χ−(c )σ −1 − CD 4 A A 4ξ A ′ χ(0,∞) (ξ) α (1) 0 ξ = χ (ξ) , 4 (−∞,0) 0 ! − ξ using that χ (ξ) 0 χ−(c )= σ (χ−(A)) = (0,∞) . A 0 0 χ (ξ) (−∞,0) We see that σ (χ+(A) P ) =0 if and only if α′(1) =0. Moreover, since −1 − CD 6 6 χ−(c )σ (χ+(A) P )χ+(c ) A −1 − CD A ′ χ(0,∞) (ξ) α (1) 0 ξ χ(−∞,0)(ξ) 0 = χ (ξ) 4 (−∞,0) 0 ! 0 χ(0,∞)(ξ) − ξ ′ χ(0,∞) (ξ) α (1) 0 ξ = χ (ξ) , 4 (−∞,0) 0 ! − ξ + ˇ ′ we see that χ (A) PCD fails to be compact on HA(D) if and only if α (1) =0 by Proposition 9.9. Let us summarise− the salient features of this example in a proposition. 6

~~2 Proposition 9.12. Assume that M is the unit disc, E = F = M C and take Dα as in Example 9.11 for an α with α′(1) = 0. We let A denote its well adapted boundary× operator. Then it holds that 6~~

~~+ (i) σ−1(χ (A) PCD ) =0. + − 6 ˇ (ii) χ (A) PCD fails to be compact on HA(D). − + − 1 1 2 ∞ 1 2 (iii) The space χ (A)H 2 (S , C ) is a ﬁnite-dimensional subspace of C (S , C ). ∩CD~~

~~Proof. The ﬁrst two statements were already proven in Example 9.11. It remains to prove that last.~~

′ α (1) − 1 −1 1 2 We deﬁne the operator Λ := (1 + ∆) 2 Ψ (S , C ). Let us ﬁrst prove the claim that 4 ∈ cl L := χ+(A) P + σΛχ−(A) Ψ−1(S1, C2) is elliptic. We have that − CD ∈ cl χ (ξ) χ (ξ) α′(1) 0 (0,∞) (−∞,0) α′(1) 0 1 σ (L)= ξ − ξ = |ξ| . −1 χ(−∞,0)(ξ) χ(0,∞)(ξ) 1 4 0 ! 4 ξ 0 ! − ξ − ξ − This is invertible for all ξ = 0, so L is elliptic. 6 REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 69

+ − 1 1 2 Consider the space χ (A)H 2 (S , C ) . We have the following equivalence ∩CD + + − 1 1 2 χ (A)f = f f χ (A)H 2 (S , C ) D ∈ ∩C ⇐⇒ (PCD f = f χ+(A)f = f ⇐⇒ (χ+(A) P )f =0 ( − CD χ+(A)f = f ⇐⇒ (Lf =0. In the last equivalence we used that if χ+(A)f = f, then χ−(A)f = 0 and χ−(A)f = 0 implies that 1 1 + + − 2 1 2 + − 2 1 2 (χ (A) PCD )f = Lf. We can conclude that χ (A)H (S , C ) D = χ (A)H (S , C ) ker(L).− But since L is elliptic, ker(L) C∞(S1, C2) is a ﬁnite-dimensional∩C subspace and item (iii)∩ follows. ⊆

Although Proposition 9.12 shows that ker(D − 1 ) may fail to be infinite dimensional in gen- χ+(A)H 2 eral, we have instead the following. Corollary 9.13. We have that for any spectral cut r R (i.e., r R such that A := A r is ∈ ∈ r − invertible bisectorial) and any pseudo-differential Calderón projection PCD , + − 1 χ (Ar)H 2 (Σ; E) D + − 1 and + C P χ (A )H 2 (Σ; E) χ (A) D CD r C are finite dimensional.

c Proof. Using Corollary 3.28, we have that D = kerPCD deﬁnes a regular boundary condition. C ± c + − 1 Therefore, from Theorem 5.6 with the choice of = χ (A ) we have that ( ,χ (A )H 2 (Σ; E)) P± r CD r is a Fredholm pair in Hˇ A(D). Therefore,

~~Hˇ A(D) c + − 1 = D + − 1 + χ (A )H 2 (Σ; E) ∼ C P χ (A )H 2 (Σ; E) CD r CD r via Lemma A4 in [6].~~

For the remaining quotient, we invoke Theorem 5.6 with a choice of + = PCD , and since − 1 P the generalised APS condition χ (Ar)H 2 (Σ; E) is a regular boundary condition, we have that − 1 (χ (Ar)H 2 (Σ; E), D) is a Fredholm pair in Hˇ A(D). Therefore, using [6, Lemma A.4], we obtain that C + − 1 Hˇ A(D) χ (Ar)H 2 (Σ; E) − 1 = + . χ (A )H 2 (Σ; E)+ ∼ χ (Ar) D r CD C This ﬁnishes the proof.

If a boundary condition B defines a realisation with infinite dimensional kernel, then this is equivalent to saying that B D is infinite dimensional. Although Corollary 9.13 “measures” the amount 1 ∩C + − 2 of χ (A)H (Σ; E) in D (as seen through the projection PCD ), it does not give us information about such a B. The followingC gives a more precise description in the way of a necessary condition.

~~1 Proposition 9.14. Let B be a boundary condition. If dim ker(DB)= , then B H 2 (Σ; E) and dim χ+(A )B = . ∞ 6⊂ r ∞~~

1 Proof. If B H 2 , then by Proposition 2.6, we have that ker(DB) is ﬁnite. Therefore, the contra- ⊂ 1 positive yields that for ker(D ) inﬁnite, we have that B H 2 (Σ; E). B 6⊂ + − + Next, assume that dim χ (Ar)B < . For u B, we have that u = χ (Ar)u + χ (Ar)u, and we − 1 ∞ ∈ always have that χ (A )u H 2 (Σ; E). Setting r ∈ ′ − 1 + B := χ (A )H 2 (Σ; E) χ (A )B, r ⊕ r 70 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

′ ′ + we therefore have an inclusion B B . The space B is a boundary condition since χ (Ar)B was ⊂ − 1 assumed to be ﬁnite dimensional. Now, letting B0 = χ (Ar)H 2 (Σ; E), the map + ,ker(D ′ )/ ker(D ) ֒ dom(D ′ )/dom(D ) = χ (A )B B B0 → B B0 ∼ r ′ + is an injection given that dom(DB ) = dom(DB0 )+dom(Dχ (Ar)B) This allows us to conclude that + dim ker(D ′ )/ ker(D ) dim χ (A )B, B B0 ≤ r and since ker(D ′ ) = ker(D ′ )/ ker(D ) ker(D ), B ∼ B B0 ⊕ B0 we have that + dim ker(D ) dim ker(D ′ ) dim χ (A )B + dim ker(D ) < . B ≤ B ≤ r B0 ∞ 1 Taking the contrapositive and combining with our earlier deduction that B H 2 (Σ; E) concludes the proof. 6⊂

~~Appendix A. Lemmas on dimensions and subspaces~~

Lemma A.1. Let and ∗ be Banach spaces and suppose that , : ∗ C is a perfect B B h· ·i B×B → pairing. Suppose that = X Y and let PX,Y denote the bounded projection X and similarly B ∗⊕ ∗ ∗ B∗ 7→ ∗ ∗ for PY,X =1 PX,Y . Let X = PX,Y , the dual map to PX,Y and similarly Y = PY,X . Then, the following− hold: B B

∗ ∗ ∗ ∗ ∗ (i) P ∗ ∗ = P and P ∗ ∗ = P . In particular = X Y . X ,Y X,Y Y ,X Y,X B ⊕ (ii) The pairing , restricts to perfect pairings X X∗ C, as well as Y Y ∗ C. (iii) The annihilatorsh· ·i × → × → Y ⊥ := b∗ ∗ : b,y =0 y Y = X∗, and X⊥ = Y ∗. { ∈B h i ∀ ∈ }

~~Proof. The item (i) is immediate from the deﬁnition of the dual projectors.~~

We prove (ii). Let x X and x∗ X∗. Then, there exists C < such that ∈ ∈ ∞ ∗ ∗ ∗ ∗ x, x C x x ∗ = P x P ∗ ∗ x ∗ = x x ∗ . | h i|≤ k kBk kB k X,Y kBk X ,Y kB k kX k kX Again, from the fact that , is a perfect paring between ∗, we have a C < such that h· ·i B×B 1 ∞ ∗ x, b ∗ ∗ x C1 sup | h ∗ i | :0 = b . (30) k k≤ b ∗ 6 ∈B k kB Now, note that ∗ ∗ ∗ x, b = P x, b = x, P ∗ ∗ b . (31) h i h X,Y i h X ,Y i Moreover, there exists C2 such that ∗ ∗ ∗ P ∗ ∗ b + P ∗ ∗ b C b , k X ,Y k k Y ,X k≤ 2k k and therefore, 1 1 ∗ C2 ∗ . (32) b ≤ P ∗ ∗ b k k k X ,Y k Combining (31) and (32) and using (30), we obtain ∗ ∗ x, b ∗ ∗ ∗ ∗ ∗ x, x C1C2 sup | h i∗ | :0 = PX ,Y b = C1C2 sup x, x . | h i|≤ PX∗,Y ∗ b B∗ 6 ∈B x∗∈X∗ | h i | ∗ k k 1=kx kX∗ A symmetric argument yields that x∗ . sup x, x∗ . k k x∈X | h i | 1=kxkX REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 71

Lastly, we prove (iii). First, suppose that that b∗ Y ⊥. That is, 0 = b∗,y for all y Y . ∈ ∗ h i ∈ Since every y = PY,X b for some b ,0 = PY ∗,X∗ b ,b for all b . This implies that ∗ ∗ ∈⊥ B ∗ h i ∈ B b ker P ∗ ∗ = X and therefore, Y X . ∈ Y ,X ⊂ Conversely, let x∗ X∗. Then, for y Y , we have that ∈ ∈ x∗,y = P ∗ x∗,y = x∗, P y =0. h i X,Y h X,Y i That is, X∗ Y ⊥. The statement that X⊥ = Y ∗ follows similarly. ⊂

~~Next we prove two results concerning the dimension of certain subspaces.~~

0 Lemma A.2. Let P Ψcl(M;E) be a classical zeroth order pseudo-differential projection in the Douglis-Nirenberg calculus∈ acting on sections of an R-graded Hermitian vector bundle E on a closed s manifold Σ with dim(Σ) > 0. If σ0(P ) =0, 1, then for any s R, the space P H (Σ;E) has infinite dimension and infinite codimension. 6 ∈

~~In particular, in the setting of Lemma 2.28 with dim(M) > 1, then for any admissible cut r R, ∈ the spaces χ±(A )L2(Σ; E) are inﬁnite dimensional. Moreover, the space Hˇ (D) = 0 . r A 6 { }~~

s Proof. We have that P acts compactly on H (Σ;E) if and only if σ0 (P ) = 0. We conclude that s 6 P H (Σ;E) has infinite dimension if and only if σ0 (P ) = 0. The infinite codimensionality of s 6 P H (Σ;E) is by a similar argument equivalent to σ0(P ) = 1. 0 6 Lemma A.3. Let A be any elliptic first order differential operator acting on sections of a vector bundle E on a closed manifold Σ such that its resolvent set is non-empty and in some point the principal symbol of A has an eigenvalue with positive real part. Then there exists two sequences of + − eigenvalues (λj )j∈N and (λj )j∈N of A such that Re(λ±) , as j . j → ±∞ → ∞

In particular, in the setting of Lemma 2.28 with dim(M) > 1, then for any admissible cut r R and ∈ an s R, the spaces χ±(A )Hs(Σ; E) are inﬁnite dimensional. Moreover, the space Hˇ (D) = 0 . ∈ r A 6 { }

∗ − 1 Proof. Deﬁne the zeroth order classical pseudodiﬀerential operator F := A(1 + A A) 2 . The spectrum of F is contained in the unit disc. From the fact that the symbol mapping is a - homomorphism, it follows that the spectrum of F contains the spectrum of its symbol, which we∗ will denote by σF . In fact, the spectrum of σF coincides with the essential spectrum of F .

From the fact that A is ﬁrst order, it follows that λ spec(σ (x, ξ)) if and only if λ ∈ A − ∈ spec(σA(x, ξ)). This implies the spectrum of σF is symmetric about the imaginary axis. Since the principal− symbol of A has an eigenvalue with positive real part in some point, it follows that F has essential spectrum on both sides of the imaginary axis.

Now, A is assumed to have non-empty resolvent set, hence its must be discrete by [67]. The essential spectrum of F consists of all limits points of the image of the spectrum of A under the 2 − 1 homeomorphism z z(1 + z ) 2 of the complex plane onto the open unit disc. Since F has essential spectrum on→ both sides| | of the imaginary axis, the spectrum of A has limit points on both sides of the imaginary axis. It follows that the spectrum of A must go to inﬁnity on both sides of the imaginary axis. By [67], each element of the spectrum is an eigenvalue and the statement follows.

Now, we have that each eigenspace Eig(λ ) C∞(Σ; E) and therefore, χ±(A )Hs(Σ; E) are inﬁnite- i ⊂ r dimensional. Therefore Hˇ (D) = 0. A 6 72 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~Appendix B. Bar-Ballmann’s¨ approach from an abstract perspective~~

Although the main results of this paper concern elliptic diﬀerential operators on manifolds with boundary, a large proportion of the main tools can be formulated in a general abstract formalism. We believe several of these results to be known to specialists, but nevertheless have decided to formulate the B¨ar-Ballmann approach in complete abstraction. Similar ideas to those appearing in this appendix can be found in [13]. In particular, several of our main results have abstract counterparts that hold in potentially non-elliptic settings.

B.1. Preliminaries. Throughout this section, we fix a closed densely defined linear operator between separable Hilbert spaces H1 H2. For reasons that will later become apparent we shall → † denote this operator by T . We also fix an adjoint T : H H which is closed and densely min min 2 → 1 defined. In the main part of this paper, we were exclusively concerned with Tmin being the mini- † mal closure of an elliptic differential operator on a manifold with boundary and Tmin the minimal closure of its formal adjoint. We introduce the notation † ∗ † ∗ Tmax := (Tmin) and Tmax := (Tmin) . Note that T T and T † T † . min ⊆ max min ⊆ max † Definition B.1. A pair = (Tmin,Tmin) as above is said to be a formally adjointed pair (FAP). • T A realisation T of a FAP is an extension T of Tmin contained in Tmax. If T is closed as • an operator, we say that T is a closed realisation of . The Cauchy data space of a FAP is defined as theT Hilbert space • T dom(Tmax) HˇT := . dom(Tmin) The quotient map dom(T ) Hˇ will be denoted by γ. max → T

The reader can note that also in this abstract setting, whenever H(ˇ ) is a Banach space equipped ˇ T ˇ Hˇ with a surjective map γ : dom(Tmax) H( ) with kernel dom(Tmin) then H( ) ∼= T via an isomorphism induced from the boundary→ mappings.T We can therefore speak ofT concrete Cauchy data spaces also in this abstract setting.

† † † If = (Tmin,Tmin) is a FAP, then := (Tmin,Tmin) is also a FAP. A symmetric closed operator S T T † will, unless otherwise indicated, be identified with the FAP (S,S). In particular, Smin = S = Smin ∗ and S = S∗ = S† . We use the notation Hˇ := Hˇ dom(S ) . max max S (S,S) ≡ dom(S) † ˜ Proposition B.2. Let = (Tmin,Tmin) be a FAP. Define the densely defined operator T on H H by T ⊕ 0 T T˜ := min . T † 0 min Then, the following hold.

(i) The operator T˜ is closed and symmetric, and its adjoint is given by 0 T T˜∗ := max . T † 0 max Both T˜ and T˜∗ are Z/2-graded in the grading of H H making the ﬁrst summand even and the second odd. ⊕ (ii) The equality dom(T˜∗) = dom(T † ) dom(T ) induces an equality max ⊕ max Hˇ˜ = Hˇ † Hˇ . T T ⊕ T (iii) The von Neumann decomposition of dom(T˜∗) induces an isomorphism of Hilbert spaces ∗ ∗ Hˇ † Hˇ = ker(T˜ + i) ker(T˜ i). T ⊕ T ∼ ⊕ − REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 73

~~We omit the proof as it follows from standard techniques for unbounded operators. The reader can ﬁnd relevant details in [70]. Lemma B.3. Let S be a closed symmetric operator.~~

(i) The sesquilinear form ∗ ∗ ∗ ∗ ω : dom(S ) dom(S ) C, ω (v , v ) := S v , v H v ,S v H , S,0 × → S,0 1 2 h 1 2i − h 1 2i descends to a well-deﬁned sesquilinear non-degenerate pairing ω : Hˇ Hˇ C, S S × S → satisfying the symmetry condition ω (ξ , ξ )= ω (ξ , ξ ). S 1 2 − S 2 1 Hˇ ∗ ∗ (ii) Under the isomorphism S ∼= ker(S + i) ker(S i), the pairing ωS corresponds to the sesquilinear form ⊕ − ω :(ker(S∗ + i) ker(S∗ i)) (ker(S∗ + i) ker(S∗ i)) C, S,vN ⊕ − × ⊕ − → v v v 0 1 v ω 1,+ , 2,+ := 2i 1,+ , 2,+ . S,vN v v v 1 0 v 1,− 2,− 1,− − 2,−H ⊕H

Proof. The analogous symmetry condition ωS,0(v1, v2)= ωS,0(v2, v1) for ωS,0 is readily verified. To prove item (i), we take v′ dom(S∗) and v dom(S)− and compute that ∈ ∈ ω (v, v′) := S∗v, v′ v,S∗v′ = S∗v, v′ Sv,v′ =0, S,0 h i − h i h i − h i since v dom(S) and S∗ S. This computation, the symmetry condition on ω and its ∈ ⊇ S,0 sesquilinearity implies that ωS is a well-defined sesquilinear form satisfying the stated symmetry condition. To show that ω is non-degenerate, if v dom(S∗) satisfies ω ([v], ξ) = 0 for all ξ Hˇ S ∈ S ∈ S then S∗v, v′ H = v,S∗v′ H for all v′ dom(S∗) so v dom(S∗∗) = dom(S) so [v] = 0 in Hˇ . h i h i ∈ ∈ S Let us prove item (ii). We use the convention that the inner product is linear in the second leg. For v , v ker(S∗ i), 1,± 2,± ∈ ± ωS(v1,+ + v1,−,v2,+ + v2,−) = S∗(v + v ), v + v v + v ,S∗(v + v ) h 1,+ 1,− 2,+ 2,−i − h 1,+ 1,− 2,+ 2,− i = iv + iv , v + v v + v , iv + iv h− 1,+ 1,− 2,+ 2,−i − h 1,+ 1,− − 2,+ 2,−i =2i v , v 2i v , v h 1,+ 2,−i− h 1,− 2,+i v v =ω 1,+ , 2,+ . S,vN v v 1,− 2,− Proposition B.4. Let = (T ,T † ) be a FAP. The sesquilinear form T min min ω : dom(T † ) dom(T ) C, ω (v , v ) := T † v , v v ,T v , T ,0 max × max → T ,0 1 2 h max 1 2i − h 1 max 2i descends to a well-defined sesquilinear non-degenerate pairing

~~ω : Hˇ † Hˇ C, T T × T → satisfying the symmetry condition~~

~~ω (ξ , ξ )= ω † (ξ , ξ ). T 1 2 − T 2 1~~

Proof. That ωT is well-deﬁned and satisﬁes the symmetry condition follows from Proposition B.2 and Lemma B.3. The proof that ωT is non-degenerate goes along the same lines as in the proof of Lemma B.3 and is omitted. 74 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

B.2. The Cauchy data space and boundary conditions. For a subspace B Hˇ , we deﬁne ⊆ T the realisation TB by dom(T ) := v dom(T ): γ(v) B . B { ∈ max ∈ } † Proposition B.5. Let (Tmin,Tmin) be a FAP. The construction B TB sets up a one-to-one † 7→ Hˇ correspondence of (closed) realisations of (Tmin,Tmin) and (closed) subspaces of T .

Proof. A short argument involving the ﬁve lemma shows that given a (closed) subspace B Hˇ , ⊆ T TB is the unique (closed) realisation T with γdom(T )= B. A similar argument shows that given a (closed) realisation T , the subspace B := γ(dom(T )) Hˇ is the unique (closed) subspace B ⊆ T 0 for which T = TB0 .

Deﬁnition B.6. A subspace B HˇT is called a generalised boundary condition and a closed subspace B Hˇ a boundary condition⊆ . ⊆ T If B is a generalised boundary condition, we deﬁne its adjoint boundary condition as ∗ B := w H † : ω (w, v)=0 v B . { ∈ T T ∀ ∈ }

In this terminology, Proposition B.5 can be phrased as there being a one-to-one correspondence between closed realisations of a formally adjointed closed operator and its boundary conditions. The terminology adjoint boundary condition is motivated by the next proposition. † Proposition B.7. Let (Tmin,Tmin) be a FAP and B a generalised boundary condition. Then it holds that ∗ † (TB) = TB∗ .

∗ † † Proof. We ﬁrst show that (T ) T ∗ . If v domT ∗ then for any w domT we have that B ⊇ B ∈ B ∈ B † † T ∗ v, w v,T w = T v, w v,T w = ω (γ(w),γ(v)) =0. h B i − h B i h max i − h max i T ∗ ∗ † Therefore, v dom(T ) and (T ) v = T ∗ v. ∈ B B B

∗ † ∗ † † † We prove (TB) TB∗ . As TB Tmin, (TB) Tmax so consider a v dom(Tmax) dom(TB∗ ), i.e. † ⊆ ⊇∗ ⊆ ∈ \ a v dom(Tmax) with γ(v) / B . Nondegeneracy of ωT implies that there exists a w dom(TB) with∈ω (γ(v),γ(w)) = 0, or∈ equivalently that there exists a w dom(T ) with ∈ T 6 ∈ B T † v, w v,T w = (T )∗v, w v,T w =0. h max i − h max i h B i − h B i 6 ∗ ∗ † This shows that v dom(T ) so we conclude that (T ) T ∗ . ∈ B B ⊆ B Proposition B.8. Let S be a closed symmetric operator.

∗ (i) For any generalised boundary condition B, SB = SB∗ . (ii) A boundary condition B for S deﬁnes a self-adjoint realisation SB if and only if B is Lagrangian (i.e. B∗ = B). (iii) A boundary condition B for S deﬁnes a symmetric realisation SB if and only if B is isotropic (i.e. B B∗). ⊆

† Proof. By deﬁnition, Smin = Smin = S for the FAP constructed from a closed symmetric operator and therefore the proposition follows from Proposition B.7. Proposition B.9. Let = (T ,T † ) be a FAP and recall the notation from Proposition B.2. T min min (i) There is a one-to-one correspondence between closed realisations of and self-adjoint Z/2- T graded realisations of T˜ given by

~~0 TB TB † . ←→ T ∗ 0 B REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 75~~

~~(ii) There is a one-to-one correspondence between closed subspaces B HˇT and Z/2-graded ∗ ⊆ Lagrangian subspaces B˜ Hˇ˜ = dom(T˜ )/dom(T˜) given by ⊆ T B B∗ B. ←→ ⊕~~

Proof. By Proposition B.7 and B.8, item (i) follows from item (ii). Let us prove item (ii). We note that ′ v1 v1 ′ ′ ω , = ω (v , v )+ ω † (v , v ), T˜ v v′ T 1 2 T 2 1 2 2 ′ ′ ∗ for v , v Hˇ † and v , v Hˇ . Therefore, given a B Hˇ , B˜ := B B is a Z/2-graded 1 1 ∈ T 2 2 ∈ T ⊆ T ⊕ isotropic subspace (i.e. B˜ B˜∗). Given (v , v ) B˜∗, the condition ⊆ 1 2 ∈ v v′ ω 1 , 1 =0, T˜ v 0 2 for all v′ B∗ implies that v B and the condition 1 ∈ 2 ∈ v 0 ω 1 , =0, T˜ v v′ 2 2 for all v′ B implies that v B∗ which proves that B˜∗ B˜. We conclude that B˜ is a Z/2-graded 2 ∈ 1 ∈ ⊆ Lagrangian subspace. Conversely, given a Z/2-graded Lagrangian subspaces B˜ Hˇ˜ the fact that ⊆ T B˜ is graded implies that B˜ = B′ B for closed subspaces B′ Hˇ and B Hˇ and the fact that ⊕ ⊆ T † ⊆ T B˜ is Lagrangian implies that B′ = B∗ by an argument as above. We conclude that B B∗ B produces a one-to-one correspondence as stated. ↔ ⊕

† Proposition B.10. Let = (Tmin,Tmin) be a FAP and B1 B2 be boundary conditions. Then T T and there is aT short exact sequence of Hilbert spaces⊆ B1 ⊆ B2 0 dom(T ) dom(T ) B2 0. → B1 → B2 → B1 → In particular, the following special cases hold for a boundary condition B:

~~(i) dom(TB) = B, dom(Tmin) ∼ Hˇ (ii) dom(Tmax) = T . dom(TB) ∼ B~~

~~This follows from standard considerations with the open mapping theorem and the ﬁve lemma.~~

~~Deﬁnition B.11. Let = (T ,T † ) be a FAP. We deﬁne the Hardy space as T min min := γ ker T Hˇ . CT max ⊆ T~~

~~It is readily veriﬁed that the Hardy space ﬁts into a short exact sequence of vector spaces γ 0 ker(T ) ker(T ) 0. → min → max −→CT → This also follows from the next proposition.~~

~~† Proposition B.12. Let = (Tmin,Tmin) be a FAP and B be a boundary condition. Then there is a short exact sequence ofT vector spaces 0 ker(T ) ker(T ) B 0. → min → B →CT ∩ →~~

~~Proof. This follows from the fact that ker(Tmin) = ker(Tmax) dom(Tmin) and ker(TB) = ker(T ) dom(T ). ∩ max ∩ B 76 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN~~

Let us brieﬂy comment on the relation to boundary triplets. We compare our result to the setup of [34]. For more details, history and context for boundary triplets, see also [10, 23]. We remark † that if a FAP = (Tmin,Tmin) admits a realisation T which is invertible, which is equivalent to T ∗ conj ∗ † 0 res(T ) = res(T ) , then T is also invertible and therefore, ker Tmin = ker Tmin = 0. Thus, ∈ † γ : ker T and γ : ker T † are vector space isomorphisms. | max →CT | max →CT † Proposition B.13. Let = (Tmin,Tmin) be a FAP admitting a realisation Tβ which is invert- −1 T ∗ −1 † † ible. Deﬁne prβ := Tβ Tmax : dom(Tmax) dom(Tβ) and prβ† := (Tβ ) Tmax : dom(Tmax) ∗ → → dom(Tβ ). Then, the following hold.

~~(i) It holds that~~

dom(Tmax) = dom(Tβ)+ker Tmax, and † ∗ † dom(Tmax) = dom(Tβ )+ker Tmax, and the mappings pr ⊕γ(1−pr ) dom(T ) β β dom(T ) and max −−−−−−−−−→ β ⊕CT pr ⊕γ(1−pr ) † β† β† ∗ dom(T ) dom(T ) † , max −−−−−−−−−−−→ β ⊕CT are isomorphisms. (ii) The isomorphisms from (i) induce isomorphisms

Hˇ = Hˇ and Hˇ † = Hˇ † † , T ∼ T ,β ⊕CT T ∼ T ,β ⊕CT Hˇ Hˇ ∗ where T ,β := γdom(Tβ) and T †,β := γdomTβ are closed subspaces. (iii) There is a one-to-one correspondence between boundary conditions B of and triples T T T (W, V, ), where V T and W T † are closed subspaces and : V 99K W is a closed densely deﬁned operator.⊆ C ⊆ C

B.3. Characterising Fredholm boundary conditions. In this section, we show how to characterise Fredholm boundary conditions for a FAP satisfying the extra condition that Tmin and † † Tmin have closed range and ker(Tmin) and ker(Tmin) being finite-dimensional. For the sake of terminology, we define this notion as follows. † Definition B.14. Let = (Tmin,Tmin) be a FAP. We say that has closed range if Tmin and † T T Tmin have closed range. We say that is a Fredholm FAP if it has closed range and ker(Tmin) and † T ker(Tmin) are finite-dimensional.

The reader should note that by the closed range theorem, = (T ,T † ) is a FAP with closed T min min range if and only if Tmin and Tmax have closed range. For the purpose of studying elliptic diﬀerential operators, the following suﬃcient condition for closed range and Fredholmness will be useful. ֒ ( Proposition B.15. Let = (T ,T † ) be a FAP. Assume that the inclusion ι : dom(T T min min min → H1 is a compact inclusion. Then, the following holds.

~~(i) ker(Tmin) is ﬁnite-dimensional. † (ii) Tmin and Tmax have closed range. (iii) H = ker(T ) ⊥ ran(T † ) and H = ker(T † ) ⊥ ran(T ). 1 min ⊕ max 2 max ⊕ min~~

~~In particular, if both dom(T ) ֒ H and dom(T † ) ֒ H are compact inclusions, then is a min → 1 min → 2 T Fredholm FAP and we additionally have that H = ker(T † ) ⊥ ran(T ). 2 min ⊕ max~~

~~Proof. For u dom(T ), we have the estimate ∈ min u ιu H + T u H . k kTmin ≃k k 1 k min k 2 REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 77~~

Then, by Proposition A3 in [5] (ii), we conclude that Tmin has finite dimensional kernel and closed ∗ † image. The fact that ran(Tmin) = ran(Tmax) is closed follows from the closed range theorem. † H Remark B.16. Note that, even without the assumption dom(Tmin) ֒ 2 is a compact embedding H † → it still holds that 2 = ker(Tmin) ran(Tmax). The compactness of the embedding ensures that ⊕ † ran(Tmax)= ran(Tmax) as well as the finite dimensionality of ker(Tmin). Lemma B.17. Let K Z be a closed subspace of a Banach space Z and let X Z a subspace. If ⊂ ⊂ X Z K X is closed, the map Φ: (K X) K defined by Φ[u]X = [u]Z is well-defined ∩ ∩ → (K∩X) K and injective.

~~Proof. To show the map is well-deﬁned, suppose [x]X = [y]X . Then, x y (K X) (K∩X) (K∩X) − ∈ ∩ ⊂ K and therefore, [x]Z = [y]Z . K K~~

For injectivity, suppose that ϕ[u]X = 0, which is the same as [u]Z = 0. Then, u = (K∩X) K u 0 K and since u X, we obtain that u K X. That is, u (K X) and therefore, − ∈ ∈ ∈ ∩ ∈ ∩ [u]X = 0. (K∩X)

† Proposition B.18. Let = (Tmin,Tmin) be a FAP and B a generalised boundary condition. The map T dom(TB) ΦB : ran(TB), ΦB[u] := Tmaxu, ker(TB) → is a bounded bijection. Moreover, the map dom(TB) dom(Tmax) ker(TB) → ker(Tmax) given by

~~[u]dom(TB) [u]dom(Tmax) ker(TB) 7→ ker(Tmax) is well-deﬁned and is a bounded injection.~~

~~Proof. To show that this map is bounded, we note that~~

~~ΦB[u] = Tmaxu inf u x + Tmaxu k k k k≤ x∈ker(TB) k − k k k~~

~~= inf u x + TBu = [u] dom(TB) . x∈ker(TB) k − k k k k k ker(TB)~~

Surjectivity is clear. For injectivity, note that if ΦB[u] = 0, then Tmaxu = 0, so u ker(Tmax). But since u dom(T ), we have that u ker(T ) = dom(T ) ker(T ). ∈ ∈ B ∈ B B ∩ max The map dom(TB) dom(Tmax) ker(TB) → ker(Tmax) is a well-deﬁned injection since ker(T ) = dom(T ) ker(T ) by invoking Lemma B.17. B B ∩ max Lemma B.19. Let = (T ,T † ) be a FAP and Y dom(T ) a subspace. Consider the T min min ⊂ max generalised boundary condition B := γ(Y ) Hˇ . Then, dom(T ) = dom(T )+ Y . ⊆ T B min

Proof. Since Y, dom(T ) dom(T ), we have that dom(T ) dom(T )+ Y . min ⊂ B B ⊃ min To prove the opposite containment, let u dom(T ), which is if and only if γ(u) B = γ(Y ). ∈ B ∈ Therefore, there exists y Y such that γ(u) = γ(y). So, u y dom(Tmin) and y Y so u = (u y)+ y dom(T ∈)+ Y . − ∈ ∈ − ∈ min Lemma B.20. Let = (T ,T † ) be a FAP. For a generalised boundary condition B, we have T min min that ran(TB) = ran(TB+CT ) and that dom(TB+CT ) = dom(TB) + ker(Tmax). 78 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

Proof. It is clear that ran(T ) ran(T ). Set Y = dom(T ) + ker(T ) and apply Lemma B ⊂ B+CT B max B.19 to obtain that dom(TB+CT ) = dom(TB) + ker(Tmax), since dom(Tmin) dom(TB). Then, for u ran(T ), we can ﬁnd v dom(T ) such that u = T v. By⊂ what we have just ∈ B+CT ∈ B+CT max proved, we have v = xB + y dom(TB) + ker(Tmax) and therefore, u = Tmaxv = TmaxxB. I.e., u ran(T ). ∈ ∈ B † ′ Lemma B.21. Let = (Tmin,Tmin) be a FAP. For a generalised boundary condition B such that B′, we have thatT the inclusion map CT ⊂ ( ֒ dom(Tmaxdom(TB′ ) ker(Tmax) → ker(Tmax) is an isometry. If has closed range, the normed space dom(TB′ ) is closed if and only T ker(Tmax) ′ if B is a boundary condition, i.e., closed in HˇT .

Proof. Note that the induced norm on dom(TB′ ) is ker(Tmax) 2 2 2 2 2 [u] = inf u x + TB′ u = inf u x + Tmaxu , k k x∈ker(Tmax) k − k k k x∈ker(Tmax) k − k k k where the norm after the ultimate equality is the norm on dom(Tmax) . If B′ is closed, ker(Tmax) dom(TB′ ) then dom(TB′ ) is closed and therefore, is a Banach space, i.e. it is closed in ker(Tmax) dom(Tmax) dom(TB′ ) . On the other hand, if is closed, then dom(TB′ ) is closed ker(Tmax) ker(Tmax) dom(TB′ ) since dom(TB′ ) = ker(Tmax). ∼ ker(Tmax) ⊕ Lemma B.22. Let = (T ,T † ) be a FAP with closed range. For a generalised boundary condi- T min min dom(Tmax) tion B, ran(TB) is closed if and only if the image of the map Ξ : dom(TB) → ker(Tmax) given by dom(TB) dom(Tmax) ֒ (Ξ : dom(TB → ker(TB) → ker(Tmax) has closed range.

Proof. We ﬁrst start by noting that, by Proposition B.18, the open mapping theorem yields that the we have that the map Φmax as deﬁned in Proposition B.18 is homeomorphism between dom(Tmax) and ran(Tmax). ker(Tmax)

~~Moreover, by the deﬁnition of ΦB, we have the following commuting diagram.~~

~~ι dom(TB) 1 dom(Tmax) ker(TB) ker(Tmax)~~

~~ΦB Φmax~~

~~ι2 ran(TB) ran(Tmax)~~

~~Note that ran(TB) ran(Tmax), so ι2 is simply the inclusion map. Since Φmax is a Banach space isomorphism, we have⊂ that~~

−1 dom(TB) Φmax : ran(TB) ι1 | ran(TB) → ker(TB) dom(TB) is bounded. Therefore, ran(TB) is closed if and only if ι1 is closed. The latter ker(TB) is, indeed, the image of Ξ so this proves the lemma. † Theorem B.23. Let = (Tmin,Tmin) be a FAP with closed range. For a generalised boundary condition B, ran(T ) isT closed if and only if B + is a boundary condition. B CT REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 79

Proof. Lemma B.20 yields that ran(TB) = ran(TB+CT ). Then, Lemma B.22 says that this is closed if and only if the range of the map Ξ is closed. Since B + , it is easy to see that the range CT ⊂ CT dom(TB+C ) of Ξ is precisely T . Therefore, by Lemma B.21, we obtain that ran(TB) is ker(Tmax) closed if and only if B + is closed, i.e., that B + is a boundary condition. CT CT Corollary B.24. Let = (T ,T † ) be a FAP with closed range and B a generalised boundary T min min condition. Then the Hardy space is closed in Hˇ . CT T

Proof. The corollary follows directly from Theorem B.23 with B = 0. Lemma B.25. Let = (T ,T † ) be a FAP with closed range. When ran(T ) is closed, T min min B H H Hˇ Hˇ 2 2 T † T ran(T ) ∼= ran(T ) (B + ) ∼= ker(Tmin) (B + ). B max ⊕ CT ⊕ CT

~~Proof. By assumption ran(Tmax) is closed and therefore, H H2 = 2 ran(Tmax) ∼ ran(Tmax) ⊕ H ran(Tmax) = 2 ran(TB). ∼ ran(Tmax) ⊕ ran(TB) ⊕~~

~~From Lemma B.20, we have that ran(TB) = ran(TB+CT ) and TB+CT is a closed operator from~~

~~Lemma B.21. Thus, note that ker(TB+CT ) = ker(Tmax) and therefore,~~

~~dom(TB+CT ) dom(TB+CT ) = = ran(TB+CT ) = ran(TB). ker(Tmax) ker(TB+CT ) ∼ Thus, Hˇ ran(Tmax) = dom(Tmax) = T ran(TB) ∼ dom(TB+C ) ∼ (B + T ) T C from Proposition B.10.~~

~~† Proposition B.26. Let = (Tmin,Tmin) be a Fredholm FAP. For a general boundary condition B, the following are equivalent:T~~

(i) ran(TB) has finite algebraic codimension in H2; ⊥ (ii) TB has closed range and ran(TB) is finite dimensional; ∗ (iii) TB has closed range and ker(TB) is finite dimensional; (iv) B + has finite algebraic codimension in Hˇ ; CT T (v) B + is closed and has finite algebraic codimension in in Hˇ . CT T If any of these hold, then H Hˇ ∗ 2 † T ker(TB) ∼= ran(T ) ∼= ker(Tmin) (B + ). B ⊕ CT

Proof. The equivalence of (i)-(iii), as well as (iv)-(v) are standard. The statements (v) and (ii) are equivalent because of Lemma B.20 coupled with Lemma B.25. The formula follows from the fact ⊥ ∗ that ran(TB) ∼= ker(TB) along with Lemma B.25.

Before proceeding with studying Fredholm properties of realisations, we make a remark about the maximal domain in terms of the Hardy space and a suitable realisation. Corollary B.27. Let = (T ,T † ) be a FAP with closed range and B be a boundary condition. T min min Then B is a complement to in Hˇ if and only if T has closed range and satisﬁes that CT T B ∗ † ker(Tmin) = ker(TB) and ker(Tmin) = ker(TB∗ ). In this case, dom(T ) = dom(T ) ker(T ). max B ⊕ker Tmin max 80 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~The reader should note that closed complements of the Hardy space in HˇT always exist for a FAP with closed range since HˇT is a Hilbert space and the Hardy space is closed by Corollary B.24.~~

Proof. Assume that B is a complement to T in HˇT . The operator TB has closed range by Theorem B.23. Let P : Hˇ denote the projectionC along B. We then arrive at a short exact sequence T →CT P ◦γ 0 dom(T ) dom(T ) 0. (33) → B → max −−−→CT → In particular, we have the short exact sequence γ 0 ker(T ) ker(T ) 0, → B → max −→CT → showing that ker(Tmin) = ker(TB). Moreover, the short exact sequence (33) implies that the map ΦB from Proposition B.18 is a bijection, and so ran(TB) = ran(Tmax). Since B is a complement to ∗ † , B + is closed, so ran(T ) = ran(T ) implies that ker(T ) = ker(T ∗ ). CT CT B max min B

~~Assume now that TB has closed range and satisﬁes that~~

~~∗ † ker(Tmin) = ker(TB) and ker(Tmin) = ker(TB∗ ).~~

The ﬁrst equality ensures that B T = 0. The second equality and Proposition B.26 implies that B + = Hˇ . Therefore B is a∩C complement to . CT T CT The equality dom(T ) = dom(T ) ker(T ) follows from Equation (33). max B ⊕ker Tmin max

~~Now, we characterise Fredholm boundary conditions via the language of Fredholm pairs.~~

~~† Theorem B.28. Let = (Tmin,Tmin) be a Fredholm FAP and B be a boundary condition. The following are equivalent:T~~

~~(i) (B, ) is a Fredholm pair in Hˇ ; CT T (ii) TB is a Fredholm operator.~~

~~If either of these equivalent conditions hold, we have that~~

~~∗ HˇT B † = . T ∼ (B + T ) ∩C C Moreover, ind(T ) = ind(B, ) + dim ker(T ) dim ker(T † ). B CT min − min~~

Proof. Since B is closed, TB is a closed operator and ker(TB) is a closed subspace of H1. If (i) holds, then B + T is closed and by Theorem B.23, we have that ran(TB) is closed. Similarly, C ⊥ if (i) holds then B + T has finite codimension and ran(TB) has finite codimension in H2 by Proposition B.26. Moreover,C B is finite dimensional, so Proposition B.12 yields that ker(T ) ∩CT B is finite dimensional and therefore, TB is Fredholm.

On the other hand if (ii) holds, i.e., TB is Fredholm, then ran(TB) = ran(TB+C) is closed, so Theorem B.23 implies that B + is closed. Moreover, by Proposition B.12 we have that B = CT ∩CT ∼ ker(TB) ⊥ is finite dimensional. Moreover ran(TB) has finite codimension and so by ker(Tmin) Proposition B.26 (v), B + has finite codimension in Hˇ . That is, (B, ) is a Fredholm pair CT T CT in HˇT .

~~From a similar calculation, we obtain that~~

~~∗ † † ∗ ker(T ) = ker(T ∗ ) = ker(T ) (B † ). B B ∼ min ⊕ ∩CT REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 81~~

By the formula in Proposition B.26, we obtain the desired formula. Plugging this into the index, we obtain ind(T ) = dim ker(T ) dim ker(T ∗) B B − B † ∗ = dim ker(T ) + dim(B ) dim ker(T ) dim(B † ) min ∩CT − min − ∩CT = ind(B, ) + dim ker(T ) dim ker(T † ). CT min − min

We say that a boundary condition of a FAP is a Fredholm boundary condition if the associated realisation is a Fredholm operator. Corollary B.29. Let = (T ,T † ) be a Fredholm FAP. Let B B be generalised bound- T min min 1 ⊆ 2 ary conditions with B1 being a Fredholm boundary condition. Then, B2 is a Fredholm boundary condition if and only if B2/B1 is ﬁnite dimensional. In this case, the following index formula holds: ind(T ) = ind(T ) + dim B2 . B2 B1 B1

Proof. Assume that B1 has ﬁnite codimension in B2, so clearly B2 will be closed. It is easy to see that B is ﬁnite dimensional and since B B , we have that 2 ∩CT 1 ⊂ 2 HˇT HˇT dim (B + ) dim (B + ). 2 CT ≤ 1 CT That is, (B2, T ) is a Fredholm pair and therefore, we have that B2 is a Fredholm boundary condition. C

~~Assume now that B2 is a Fredholm boundary condition, i.e. that B2 is closed and (B2, T ) is a Fredholm pair. Note that C~~

HˇT HˇT = (B2 + T ) ∼ (B2 + T ) C ⊕ C Hˇ (B2 + T ) T = (B1 + T ) ∼ C (B1 + T ) (B2 + T ) C ⊕ C ⊕ C and therefore, Hˇ Hˇ Hˇ T = (B2 + T ) T = B2 T . (B1 + T ) ∼ C (B1 + T ) (B2 + T ) ∼ B1 (B2 + T ) C C ⊕ C ⊕ C HˇT HˇT Since (B2, T ) and (B1, T ) are Fredholm pairs, (B + ) and (B + ) are ﬁnite- C C 2 CT 1 CT dimensional and we conclude that B2 is ﬁnite dimensional. B1

~~For the index formula, we note that it suﬃces to prove that ind(B , ) = ind(B , ) + dim B2 . 2 CT 1 CT B1 This identity follows from the computation~~

HˇT ind(B2, T ) = dim(B2 T ) dim (B + ) C ∩C − 2 CT (B2 T ) = dim(B1 T ) + dim ∩C (B ) ∩C 1 ∩CT Hˇ (B )⊥,B2 dim T + dim 2 T ⊥,B (B1 + T ) ∩C (B ) 1 − C 1 ∩CT ⊥,B2 (B2 T ) (B2 T ) = ind(B1, T ) + dim + dim ⊥,B1 . ∩C (B1 T ) ∩C (B ) C ∩C 1 ∩CT This concludes the proof.

In addition to Theorem B.28, a useful way to obtain Fredholm boundary conditions is given by the following Proposition which follows from [5, Proposition A.1]. It is adapted from [5, Corollary 8.8], although [5] only considered the context of regular boundary conditions. The reader can, in the context of elliptic boundary value problems, compare the construction of the next proposition 82 LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN

~~to standard constructions in the Boutet de Monvel calculus [16, 42, 60]. Again, the provenance of the next result dates back further than [5] and can be found in for instance [42, Lemma 4.3.1].~~

Proposition B.30. Let = (T ,T † ) be a Fredholm FAP with a boundary condition B. Choose T min min a closed complementary subspace Cˇ of B in HˇT with projection Pˇ : HˇT Cˇ deﬁned along B (for instance the orthogonal complement). The map →

~~H2 Tmaxu Lˇ ˇ : dom(T ) , Lˇ ˇ u = , P max → ⊕ P Pγuˇ Cˇ satisﬁes:~~

ˇ (i) TB has closed range if and only if LPˇ has closed range. ˇ (ii) The range of TB has finite codimension if and only if the range of LPˇ has finite codimension, in which case ˇ dim coker(TB) = dim coker(LPˇ ). ˇ (iii) ker TB has finite dimension if and only if ker LPˇ has finite dimension, in which case ˇ dim ker(TB) = dim ker(LPˇ ). ˇ (iv) LPˇ is Fredholm if and only if TB is Fredholm.

~~We also note the following: Fredholm boundary conditions must necessarily be inﬁnite dimensional.~~

~~Proposition B.31. Let = (T ,T † ) be a Fredholm FAP such that has inﬁnite codimension T min min CT in Hˇ . If B is a Fredholm boundary condition then dim B = dim B∗ = . T ∞~~

Proof. We prove via the contrapositive. Let B HˇT be a finite dimensional subspace. Indeed, this is a boundary condition and B + is closed⊂ since is closed. Now, CT CT Hˇ = (B ) (B )⊥ T ∩CT ⊕ ∩CT = (B ) (B )⊥,CT CT ∩CT ⊕ ∩CT B + = B (B )⊥,CT = (B ) (B )⊥,B (B )⊥,CT . CT ⊕ ∩CT ∩CT ⊕ ∩CT ⊕ ∩CT From this, we obtain ⊥ HˇT (B T ) = ⊥,CT T ∼ ∩C (B ) C ∩CT and ⊥ HˇT (B T ) = ⊥,B ⊥,CT . B + T ∼ ∩C (B ) (B ) C ∩CT ⊕ ∩CT Therefore, ⊥ HˇT ⊥,B (B T ) HˇT (B ) = ⊥,CT = . B + T T ∼ ∩C (B ) ∼ T C ⊕ ∩C ∩CT C ⊥,B Since (B T ) is finite dimensional and T has infinite codimension in HˇT , we have that Hˇ ∩C C T is infinite dimensional. That is, B is not Fredholm. By the contrapositive, if B is B + T Fredholm,C then dim B = . Since B∗ is Fredholm if and only if B is Fredholm, a symmetric argument yields that dim B∞∗ = . ∞

B.4. Rigidity results for the index. Let us prove a homotopy invariance result for the index of Fredholm realisations of Fredholm FAPs. Theorem B.32. Assume that = (T ,T † ) is a family of Fredholm FAPs H H Tt t,min t,min t∈[0,1] 1 → 2 such that REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 83

(i) The mappings † 0 Tt,min 0 T [0, 1] t and [0, 1] t t,min ∋ 7→ T † 0 ∋ 7→ T 0 t,max t,max are continuous in the gap topology on the space of densely deﬁned self-adjoint operators (i.e. the resolvents are norm continuous with respect to t). ii) dom(Tt,min) and dom(Tt,max) are constant in t, so HˇT := dom(Tt,max)/dom(Tt,min) is independent of t up to a canonical Banach space isomorphism.

Let [0, 1] t P B(Hˇ ) be a norm continuous family of projectors such that B := (1 P )Hˇ ∋ 7→ t ∈ T t − t T satisﬁes that (B , ) is a Fredholm pair in Hˇ for all t [0, 1]. Then it holds that t CTt T ∈

~~ind(T0,B0 ) = ind(T1,B1 ).~~

~~The proof uses standard techniques from KK-theory and the index theory of Fredholm operators on Hilbert C∗-modules. For an overview, see [54] and [31].~~

H Proof. We write T for the Hilbert space dom(Tt0,max) for some fixed t0 and note that Tt,max : H H is continuous for each t. By assumption i) above, the family (T ) defines an T → 2 t,max t∈[0,1] adjointable C[0, 1]-linear map that we denote by Tmax : HT ,C([0,1]) H2,C([0,1]) where HT ,C([0,1]) ∗ → and H2,C([0,1]) denote the C[0, 1]-Hilbert C -module defined from the indicated Hilbert spaces. We ˇ ˇ ˇ ∗ ˇ also set Ct := PtHT and let CC([0,1]) denote the C[0, 1]-Hilbert C -submodule of C([0, 1]) HT ˇ ⊗ with fibre Ct, it is well-defined by norm continuity of the family (Pt)t∈[0,1].

~~Following Proposition B.30, we deﬁne the operator~~

H2 Tt,maxu Lˇt : HT , Lˇtu = , → ⊕ Ptγu Cˇt for t [0, 1]. This operator defines an adjointable C[0, 1]-linear map Lˇ : H H ∈ T ,C([0,1]) → 2,C([0,1]) ⊕ Cˇ because T : H H and ξ (t P ξ) is an adjointable map C([0,1]) max T ,C([0,1]) → 2,C([0,1]) 7→ 7→ t H(ˇ D) Cˇ . C([0,1]) → C([0,1]) We claim that the adjointable C[0, 1]-linear map Lˇ : H H Cˇ is Fredholm. For T ,C([0,1]) → 2,C([0,1]) ⊕ each t [0, 1], Lˇ is Fredholm, so we can find finite-dimensional vector bundles E , E [0, 1] ∈ t t,1 t,2 → with E being a subbundle of H such that E = ker Lˇ and E being a subbundle of t,1 T ,C([0,1]) t,1|t t t,2 H such that the induced map E cokerLˇ is surjective. In particular, there is a finite- 2,C([0,1]) t,2|t → t rank perturbation of Lˇ as a mapping H E H Cˇ E which is invertible T ,C([0,1]) ⊕ t,2 → 2,C([0,1]) ⊕ ⊕ t,1 when restricted to the fibre over t. By an openness argument, this finite-rank perturbation of Lˇ is invertible on a small open neighbourhood Ut of t. In particular, Lˇ is Fredholm when restricted to Ut. By compactness of [0, 1], Lˇ is Fredholm.

Since, Lˇ is a Fredholm operator between C[0, 1]-Hilbert C∗-modules, it has a well-deﬁned index 0 0 0 Z in K ([0, 1]). Under the isomorphisms K ([0, 1]) ∼= K (pt) ∼= , homotopy invariance of K-theory implies that ˇ ∗ ˇ ∗ ˇ ˇ ind(L0)= r0 indC([0,1])(L)= r1 indC([0,1])(L) = ind(L1), where rt denotes the inclusion t [0, 1]. We can now conclude the theorem from Proposition B.30 which implies that { } ⊆ ˇ ˇ ind(T0,B0 ) = ind(L0) = ind(L1) = ind(T1,B1 ).

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~~Lashi Bandara, Institut fur¨ Mathematik, Universitat¨ Potsdam, D-14476, Potsdam OT Golm, Germany URL: http://www.math.uni-potsdam.de/~bandara Email address: [email protected]~~

Magnus Goffeng, Centre for Mathematical Sciences, University of Lund, Box 118, 221 00 LUND, Sweden URL: https://www.lunduniversity.lu.se/lucat/user/e604494124f99d9bb6048e890306f7a4 Email address: [email protected]

~~Hemanth Saratchandran The University of Adelaide Adelaide, South Australia, 5005, Australia URL: https://researchers.adelaide.edu.au/profile/hemanth.saratchandran Email address: [email protected]~~