Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

Abstract

The paper deals with first order self-adjoint elliptic differential operators on a compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. We compute the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required.

Appendix A. Criteria of Graph Continuity

In this appendix we give some general conditions describing if a family of closed operators (in particular, differential operators on a manifold with boundary) is graph continuous. We use the results of Sect. A.5 in the main part of the paper for two purposes. First, Proposition 3.3 arises as a particular case of Proposition A.11. Second, Proposition A.10 and Lemma A.12 provide the continuity of the family of global boundary value problems used in the proof of Lemma 11.5.

After the main part of the appendix (namely, the case of Hilbert spaces) was written, the author discovered that some of the results of Sects A.3 and A.4, though in a different form and with different proofs, are contained in the appendix to the recent paper of Booss-Bavnbek and Zhu [3]. In particular, our Proposition A.5 is a corollary of [3, Proposition A.6.2] and our Proposition A.7 is a special case of [3, Corollary A.6.4]. Nevertheless, we leave these results and their proofs in the paper for the sake of completeness, and also because their statements better meet our needs. For Hilbert spaces, our proofs have the advantage of not using elaborated estimates and inequalities. We also add the more general case of Banach spaces to the appendix with the purpose of better matching the results of [3], though we use only Hilbert spaces in the main part of the paper.

It is worth noticing that our Proposition A.4 gives an equivalent definition of the gap topology on the space ${{\,\mathrm{Gr}\,}}(H)$ of all complemented closed linear subspaces of a Banach space H. Namely, the gap topology on ${{\,\mathrm{Gr}\,}}(H)$ coincides with the quotient topology induced by the map ${{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$, $P\mapsto {{\,\mathrm{Im}\,}}P$, where ${{\,\mathrm{Proj}\,}}(H)$ is the space of all idempotents in ${\mathcal {B}}(H)$ with the norm topology. The author does not know if this fact was noted before.

1.1 A.1 Complementary Pairs of Subspaces

Subspaces of a Banach Space Let H be a Banach space. Denote by ${\mathcal {B}}(H)$ the space of all bounded linear operators on H with the norm topology. Denote by ${{\,\mathrm{Proj}\,}}(H)$ the subspace of ${\mathcal {B}}(H)$ consisting of all idempotents.

A closed subspace $L\subset H$ is called complemented if there is another closed subspace $M\subset H$ such that $L\cap M=0$, $L+M=H$; such pair (L, M) is called a complementary pair. Equivalently, $L\subset H$ is complemented if it is the image of some $P\in {{\,\mathrm{Proj}\,}}(H)$; (L, M) is a complementary pair if it is equal to $({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Ker}\,}}P)$ for some $P\in {{\,\mathrm{Proj}\,}}(H)$.

We denote by ${{\,\mathrm{Gr}\,}}(H)$ the set of all complemented closed linear subspaces of H, and by ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ the set of all complementary pairs of subspaces of H. We will also write ${{\,\mathrm{Gr}\,}}^2(H)$ instead of ${{\,\mathrm{Gr}\,}}(H)^2$ for convenience.

For $(L,M)\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$ we denote by $P_{L,M}$ the projection of H onto L along M. For $M\in {{\,\mathrm{Gr}\,}}(H)$ denote by ${{\,\mathrm{Gr}\,}}^M(H) = \left\{ L\in {{\,\mathrm{Gr}\,}}(H) :(L,M)\in {{\,\mathrm{Gr^{(2)}}\,}}(H) \right\}$ the set of all complement subspaces for M.

Proposition A.1

Let H be a Banach space and $P,Q\in {{\,\mathrm{Proj}\,}}(H)$. Then the following two conditions are equivalent:

• 1. Both $({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Im}\,}}Q)$ and $({{\,\mathrm{Ker}\,}}P, {{\,\mathrm{Ker}\,}}Q)$ are complementary pairs of subspaces.

• 2. $P-Q$ is invertible.

If this is the case, then for the projection S on ${{\,\mathrm{Im}\,}}P$ along ${{\,\mathrm{Im}\,}}Q$ and the projection T on ${{\,\mathrm{Ker}\,}}P$ along ${{\,\mathrm{Ker}\,}}Q$ we have:

$$\begin{aligned} S=P(P-Q)^{-1}, \quad T=(P-1)(P-Q)^{-1}, \quad (P-Q)^{-1}= S-T, \end{aligned}$$

(A.1)

and $P+Q=(2S-1)(P-Q)$ is also invertible.

Proof

$(1\Rightarrow 2)$ Let $({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Im}\,}}Q), ({{\,\mathrm{Ker}\,}}P, {{\,\mathrm{Ker}\,}}Q) \in {{\,\mathrm{Gr^{(2)}}\,}}(H)$. Denote by S, T the elements of ${{\,\mathrm{Proj}\,}}(H)$ corresponding these two pairs of complementary subspaces. Using the identities $SP=P$, $TQ=T$, $SQ=0$, and $(1-T)(1-P)=0$, we obtain

$$\begin{aligned} (S-T)(P-Q) = T+P-TP = 1-(1-T)(1-P) = 1. \end{aligned}$$

Similarly, we have

$$\begin{aligned} (P-Q)(S-T) = Q+S-QS = 1-(1-Q)(1-S) = 1. \end{aligned}$$

Therefore, $P-Q$ is invertible with $S-T$ the inverse operator.

$(2\Rightarrow 1)$ Let $P-Q$ be invertible. It vanishes on the intersections ${{\,\mathrm{Im}\,}}P\cap {{\,\mathrm{Im}\,}}Q$ and ${{\,\mathrm{Ker}\,}}P\cap {{\,\mathrm{Ker}\,}}Q$, so these intersections are trivial. Consider the operators $S=P(P-Q)^{-1}$ and $S'=-Q(P-Q)^{-1}$. We have ${{\,\mathrm{Im}\,}}S = {{\,\mathrm{Im}\,}}P$, ${{\,\mathrm{Im}\,}}S' = {{\,\mathrm{Im}\,}}Q$, and $S+S'=1$, so ${{\,\mathrm{Im}\,}}P+{{\,\mathrm{Im}\,}}Q = H$. Similarly, consider the operators $T=(P-1)(P-Q)^{-1}$ and $T'=(1-Q)(P-Q)^{-1}$. We have ${{\,\mathrm{Im}\,}}T = {{\,\mathrm{Ker}\,}}P$, ${{\,\mathrm{Im}\,}}T' = {{\,\mathrm{Ker}\,}}Q$, and $T+T'=1$, so ${{\,\mathrm{Ker}\,}}P+{{\,\mathrm{Ker}\,}}Q = H$. All four subspaces ${{\,\mathrm{Im}\,}}P$, ${{\,\mathrm{Im}\,}}Q$, ${{\,\mathrm{Ker}\,}}P$, ${{\,\mathrm{Ker}\,}}Q$ are closed. Therefore, both $({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Im}\,}}Q)$ and $({{\,\mathrm{Ker}\,}}P, {{\,\mathrm{Ker}\,}}Q)$ lie in ${{\,\mathrm{Gr^{(2)}}\,}}(H)$.

The first equality of (A.1) implies $(2S-1)(P-Q) = 2P-(P-Q) = P+Q$. Note that invertibility of $P+Q$ implies ${{\,\mathrm{Im}\,}}P + {{\,\mathrm{Im}\,}}Q = H$, but does not imply ${{\,\mathrm{Im}\,}}P \cap {{\,\mathrm{Im}\,}}Q = 0$. $\square$

Subspaces of a Hilbert Space If H is a Hilbert space, then each closed subspace of H is complemented, so ${{\,\mathrm{Gr}\,}}(H)$ is the set of all closed subspaces of H. The map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ has a natural section taking a closed subspace $L\subset H$ to the orthogonal projection $P_L$ of H onto L. Applying Proposition A.1, we obtain the following result.

Proposition A.2

Let H be a Hilbert space. Then the following statements hold:

• 1. The pair (L, M) of closed subspaces of H is complementary if and only if $P_L - P_M$ is invertible. If this is the case, then

  $$\begin{aligned} P_{L,M}=P_L(P_L - P_M)^{-1}. \end{aligned}$$

  (A.2)

• 2. Let $P\in {{\,\mathrm{Proj}\,}}(H)$. Then the operator $P+P^*-1$ is invertible, and the orthogonal projection on the image of P is given by the formula

  $$\begin{aligned} P^{\mathrm {ort}}=P(P+P^*-1)^{-1}. \end{aligned}$$

  (A.3)

Proof

1. If $(L,M)\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$, then also $(L^{\bot },M^{\bot })\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$. Applying Proposition A.1 to the pair of orthogonal projections $P_L$ and $P_M$, we obtain the first claim of the Corollary.

2. $1-P^*$ is the projection on $({{\,\mathrm{Im}\,}}P)^{\bot }$ along $({{\,\mathrm{Ker}\,}}P)^{\bot }$. Applying Proposition A.1 to the pair of projections P and $1-P^*$, we see that $P+P^*-1 = P-(1-P^*)$ is invertible and $P(P+P^*-1)^{-1}$ is the projection on ${{\,\mathrm{Im}\,}}P$ along $({{\,\mathrm{Im}\,}}P)^{\bot }$. $\square$

1.2 A.2 The Gap Topology on ${{\,\mathrm{Gr}\,}}(H)$

For a Hilbert space H the map $L\mapsto P_L$ given by the orthogonal projection allows us to identify ${{\,\mathrm{Gr}\,}}(H)$ with the subspace ${{\,\mathrm{Proj}\,}}^{\mathrm {ort}}(H)\subset {{\,\mathrm{Proj}\,}}(H)$ of orthogonal projections in H. The gap topology on ${{\,\mathrm{Gr}\,}}(H)$ is induced by the norm topology on ${{\,\mathrm{Proj}\,}}(H)\subset {\mathcal {B}}(H)$.

For a Banach space H there is no natural section ${{\,\mathrm{Gr}\,}}(H)\rightarrow {{\,\mathrm{Proj}\,}}(H)$, so the definition of the gap topology on ${{\,\mathrm{Gr}\,}}(H)$ is slightly more complicated in this case. Usually the gap topology on ${{\,\mathrm{Gr}\,}}(H)$ is defined as the topology induced by the gap metric

$$\begin{aligned} {\hat{\delta }}(L_1, L_2)&= \max _{i\ne j} \left\{ \sup \left\{ {{\,\mathrm{dist}\,}}(u, L_j):u\in L_i, \left\| u\right\| =1 \right\} \right\} ,\nonumber \\ {\hat{\delta }}(0, 0)&= 0, \quad {\hat{\delta }}(0, L) = 1 \text { for } L\ne 0. \end{aligned}$$

(A.4)

For a Hilbert space H these two definitions of the gap topology coincide.

Proposition A.4 below gives an equivalent definition of the gap topology on the Grassmanian of a Banach space in terms of projections, resembling the definition of the gap topology for Hilbert spaces.

The gap topology on ${{\,\mathrm{Gr}\,}}(H)$ induces the topology on ${{\,\mathrm{Gr}\,}}^2(H)$ and on its subspace ${{\,\mathrm{Gr^{(2)}}\,}}(H)$.

Proposition A.3

Let H be a Banach space. Then the following statements hold:

• 1. The map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ is continuous.

• 2. The map $\varphi :{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr^{(2)}}\,}}(H)$ taking $P\in {{\,\mathrm{Proj}\,}}(H)$ to $({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Ker}\,}}P)\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$ is a homeomorphism.

• 3. ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ is open in ${{\,\mathrm{Gr}\,}}^2(H)$.

We first give the proof in the case of a Hilbert space H, because it is simpler and because we need only this case in the main part of the paper as well as in the proofs of all the results below in the context of Hilbert spaces. After proving the “Hilbert case” we give the proof of the general “Banach case”.

Proof

1. Suppose first that H is a Hilbert space. The map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ is continuous. Indeed, it is the composition of the two maps ${{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Proj}\,}}^{\mathrm {ort}}(H)$ and ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}^{\mathrm {ort}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$, where the first map is given by formula (A.3) and ${{\,\mathrm{Proj}\,}}^{\mathrm {ort}}(H)$ is the subspace of ${{\,\mathrm{Proj}\,}}(H)$ consisting of orthogonal projections. The first map is continuous and the second map is an isometry, so their composition is also continuous.

The composition with the involution $P\mapsto 1-P$ takes the map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ to the map ${{\,\mathrm{Ker}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$, so the second map is also continuous. Therefore, $\varphi$ is continuous. Obviously, $\varphi$ is bijective.

The inverse map ${{\,\mathrm{Gr^{(2)}}\,}}(H)\rightarrow {{\,\mathrm{Proj}\,}}(H)$ is given by formula (A.2) and therefore is continuous. Thus the map ${{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr^{(2)}}\,}}(H)$ is a homeomorphism.

To prove that ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ is open in ${{\,\mathrm{Gr}\,}}^2(H)$, take arbitrary $(L,M)\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$. The operator $P_L-P_M$ is invertible by Corollary A.2. Choose $\varepsilon >0$ such that $2\varepsilon$-neighbourhood of $P_L-P_M$ in ${\mathcal {B}}(H)$ consists of invertible operators. Then for any $L',M'\in {{\,\mathrm{Gr}\,}}(H)$ such that $\left\| P_L-P_{L'}\right\| <\varepsilon$, $\left\| P_M-P_{M'}\right\| <\varepsilon$ we have

$$\begin{aligned} \left\| (P_L-P_M)-(P_{L'}-P_{M'})\right\| \leqslant \left\| P_L-P_{L'}\right\| + \left\| P_M-P_{M'}\right\| < 2\varepsilon , \end{aligned}$$

so $P_{L'}-P_{M'}$ is invertible. Applying again Corollary A.2, we obtain $(L',M')\in {{\,\mathrm{Gr^{(2)}}\,}}(H)$. This completes the proof of the proposition for Hilbert spaces.

2. Let now H be an arbitrary Banach space. The continuity of the map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ follows from the inequality ${\hat{\delta }}({{\,\mathrm{Im}\,}}P, {{\,\mathrm{Im}\,}}Q) \leqslant \left\| P-Q\right\|$. As above, this implies that $\varphi$ is a continuous bijection. The continuity of the map ${{\,\mathrm{Gr^{(2)}}\,}}(H)\rightarrow {{\,\mathrm{Proj}\,}}(H)$, $(L,M)\mapsto P_{L,M}$ follows from [14, Lemma 0.2]. By [5, Lemma 1 and Theorem 2], ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ is open in ${{\,\mathrm{Gr}\,}}^2(H)$. This completes the proof of the Proposition for Banach spaces. $\square$

Proposition A.4

Let H be a Banach space. Then the gap topology on ${{\,\mathrm{Gr}\,}}(H)$ coincides with the quotient topology induced by the map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$, $P\mapsto {{\,\mathrm{Im}\,}}P$.

Proof

The projection $p_1:{{\,\mathrm{Gr}\,}}^2(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$ onto the first factor is an open continuous map. By Proposition A.3, ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ is open in ${{\,\mathrm{Gr}\,}}^2(H)$, so the restriction of $p_1$ to ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ is also an open map. This restriction maps ${{\,\mathrm{Gr^{(2)}}\,}}(H)$ continuously and surjectively onto ${{\,\mathrm{Gr}\,}}(H)$. Therefore, the gap topology on ${{\,\mathrm{Gr}\,}}(H)$ coincides with the quotient topology induced by the map $p_1:{{\,\mathrm{Gr^{(2)}}\,}}(H)\rightarrow {{\,\mathrm{Gr}\,}}(H)$. To complete the proof, it is sufficient to apply the homeomorphism $\varphi :{{\,\mathrm{Proj}\,}}(H)\rightarrow {{\,\mathrm{Gr^{(2)}}\,}}(H)$ from Proposition A.3. $\square$

1.3 A.3 Injective Maps of Banach Spaces

Proposition A.5

Let $j\in {\mathcal {B}}(H,H')$ be an injective map of Banach spaces. Denote by ${{\,\mathrm{Gr}\,}}_j(H)$ the subspace of ${{\,\mathrm{Gr}\,}}(H)$ consisting of L with $j(L)\in {{\,\mathrm{Gr}\,}}(H')$. Then ${{\,\mathrm{Gr}\,}}_j(H)$ is open in ${{\,\mathrm{Gr}\,}}(H)$ and the natural inclusion $j_*:{{\,\mathrm{Gr}\,}}_j(H) \hookrightarrow {{\,\mathrm{Gr}\,}}(H')$, $L\mapsto j(L)$ is continuous.

Proof

By Proposition A.3, ${{\,\mathrm{Gr}\,}}^M(H)$ is open in ${{\,\mathrm{Gr}\,}}(H)$. Thus the statement of the proposition results from the following lemma. $\square$

Lemma A.6

Let $L\in {{\,\mathrm{Gr}\,}}_j(H)$, let $M'\in {{\,\mathrm{Gr}\,}}(H')$ be a complement subspace for $L'=j(L)$, and $M=j^{-1}(M')$. Then $L\in {{\,\mathrm{Gr}\,}}^M(H)\subset {{\,\mathrm{Gr}\,}}_j(H)$, and the restriction of $j_*$ to ${{\,\mathrm{Gr}\,}}^M(H)$ is continuous.

Proof of the Lemma

Denote by $Q'$ the projection of $H'$ onto $L'$ along $M'$. By the Closed Graph Theorem, the bounded linear operator $\left. j \right| _{L}:L\rightarrow L'$ is an isomorphism. Thus the composition $Q = (\left. j \right| _{L})^{-1}Q'j$ is a bounded operator on H. Obviously, Q is an idempotent, ${{\,\mathrm{Im}\,}}Q=L$, and $\ker Q=M$. This implies that L and M are complement subspaces of H.

Let $N\in {{\,\mathrm{Gr}\,}}^M(H)$, $N'=j(N)$. Then $Q_N = jP_{N,M}(\left. j \right| _{L})^{-1}Q'$ is a bounded operator acting on $H'$. The kernel of $Q_N$ is $M'$ and the restriction of $Q_N^2-Q_N$ to $L'$ vanishes, so $Q_N^2=Q_N$ and $Q_N\in {{\,\mathrm{Proj}\,}}(H')$. The image of $Q_N$ contains in $N'$ and $N'\cap M' = j(N\cap M)=0$. Therefore, $Q_N=P_{N',M'}$, $N' = {{\,\mathrm{Im}\,}}Q_N \in {{\,\mathrm{Gr}\,}}(H')$, and $N\in {{\,\mathrm{Gr}\,}}_j(H)$.

By Proposition A.3, the map $N\mapsto P_{N,M}$ is continuous. Thus the map ${{\,\mathrm{Gr}\,}}^M(H)\rightarrow {{\,\mathrm{Proj}\,}}(H')$, $N\mapsto Q_N$ is also continuous. Composing it with the continuous map ${{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}(H')\rightarrow {{\,\mathrm{Gr}\,}}(H')$, we obtain the continuity of the map $j_*:{{\,\mathrm{Gr}\,}}^M(H)\rightarrow {{\,\mathrm{Gr}\,}}(H')$, $N\mapsto j(N)={{\,\mathrm{Im}\,}}Q_N$. This completes the proof of the lemma and of Proposition A.5. $\square$

1.4 A.4 Closed Operators

Let H and $H'$ be Hilbert spaces. The space ${\mathcal {C}}(H,H')$ of closed linear operators from H to $H'$ is the subspace of ${{\,\mathrm{Gr}\,}}(H\oplus H')$ consisting of closed subspaces of $H\oplus H'$ which injectively projects on H. An element of ${\mathcal {C}}(H,H')$ can be identified with a linear (not necessarily bounded) operator A acting to $H'$ from (not necessarily closed or dense) subspace ${{\,\mathrm{dom}\,}}(A)$ of H such that the graph of A is a closed subspace of $H\oplus H'$.

All results of this subsection are valid for Banach spaces as well. However, in this case the space ${\mathcal {C}}(H,H')$ as we define it (namely, as a the subspace of ${{\,\mathrm{Gr}\,}}(H\oplus H')$) does not contain all closed linear operators from H to $H'$, but only those whose graphs are complemented subspaces of $H\oplus H'$. Nevertheless, families of such operators often arise in applications, so these results can be used for them as well. For example, for Banach spaces H, $H'$ and a linear operator A acting from ${\mathcal {D}}\subset H$ to $H'$, if ${{\,\mathrm{Ker}\,}}A\subset H$ and ${{\,\mathrm{Im}\,}}A\subset H'$ are closed complemented subspaces, then the graph of A is a closed complemented subspace of $H\oplus H'$. In particular, every (not necessarily bounded) Fredholm operator has this property.

Proposition A.7

Let H, $H'$ be Banach spaces. Then the map ${\mathcal {B}}(H,H')\times {{\,\mathrm{Gr}\,}}(H) \rightarrow {\mathcal {C}}(H,H')$ taking $(A,{\mathcal {D}})$ to $\left. A \right| _{{\mathcal {D}}}$ is continuous.

Proof

For each $A\in {\mathcal {B}}(H,H')$ we define the automorphism $J_A$ of $H\oplus H'$ by the formula $J_A(u\oplus u') = u\oplus (u'-Au)$. Both $A\mapsto J_A$ and $A\mapsto J_A^{-1}$ are continuous maps from ${\mathcal {B}}(H,H')$ to ${\mathcal {B}}(H\oplus H')$. The formula $f(A,Q)= J_A^{-1}QP_{H,H'}J_A$ defines the continuous map $f:{\mathcal {B}}(H,H')\times {{\,\mathrm{Proj}\,}}(H) \rightarrow {{\,\mathrm{Proj}\,}}(H\oplus H')$ (here $P_{H,H'}$ denote the projection of $H\oplus H'$ on H along $H'$). Since $J_A$ takes the graph of $\left. A \right| _{{\mathcal {D}}}$ to ${\mathcal {D}}\oplus 0$ for each ${\mathcal {D}}\in {{\,\mathrm{Gr}\,}}(H)$, f(A, Q) is the projection of $H\oplus H'$ onto the graph of $\left. A \right| _{{{\,\mathrm{Im}\,}}Q}$. In other words, we have the commutative diagram

where g is the map taking the pair $(A,{\mathcal {D}})$ to the graph of $\left. A \right| _{{\mathcal {D}}}$. The top and the right arrows of the diagram are continuous maps, while the left arrow is a quotient map by Proposition A.4. Therefore, g is also continuous. This completes the proof of the proposition. $\square$

Proposition A.8

Let W, H, $H'$ be Banach spaces, and let $j\in {\mathcal {B}}(W,H)$ be injective. Denote by ${\mathcal {C}}_j(W,H')$ the subspace of ${\mathcal {C}}(W,H')$ consisting of operators $A:{{\,\mathrm{dom}\,}}(A)\rightarrow H'$ such that the operator $j_*A:j({{\,\mathrm{dom}\,}}(A))\rightarrow H'$, $j_*A = A\cdot j^{-1}$ lies in ${\mathcal {C}}(H,H')$. Then the natural inclusion $j_*:{\mathcal {C}}_j(W,H') \hookrightarrow {\mathcal {C}}(H,H')$ is continuous.

Proof

Consider the following commutative diagram:

The spaces above are just subspaces of the spaces below, and ${\mathcal {C}}_j(W,H') = {\mathcal {C}}(W,H') \cap {{\,\mathrm{Gr}\,}}_j(W \oplus H')$. By Proposition A.5, the map $j_*:{{\,\mathrm{Gr}\,}}_j(W \oplus H') \rightarrow {{\,\mathrm{Gr}\,}}(H \oplus H')$ is continuous. So the restriction of $j_*$ to ${\mathcal {C}}_j(W,H') \subset {{\,\mathrm{Gr}\,}}_j(W \oplus H')$ is also continuous. This completes the proof of the proposition. $\square$

1.5 A.5 Differential and Pseudo-differential Operators

The results of the previous subsection can be used for differential and pseudo-differential operators acting between sections of vector bundles over M. To achieve continuity of the corresponding families of closed operators, the relevant topology on the space of differential operators will be the $C^0_b$-topology on their coefficients.

General Framework Let M be a smooth Riemannian manifold and E, $E'$ be smooth Hermitian vector bundles over M. For an integer $d\geqslant 1$, we denote by ${{\,\mathrm{Op^d}\,}}(E,E')$ the set of all pairs $(A,{\mathcal {D}})$ such that

• A is a bounded operator from $H^d(E)$ to $L^2(E')$,

• ${\mathcal {D}}$ is a closed subspace of $H^d(E)$, and

• the restriction $\left. A \right| _{{\mathcal {D}}}$ of A to the domain ${\mathcal {D}}$ is closed as an operator from $L^2(E)$ to $L^2(E')$.

We equip ${{\,\mathrm{Op^d}\,}}(E,E')$ with the topology induced by the inclusion

$$\begin{aligned} {{\,\mathrm{Op^d}\,}}(E,E') \hookrightarrow {\mathcal {B}}(H^d(E), L^2(E')) \times {{\,\mathrm{Gr}\,}}(H^d(E)). \end{aligned}$$

Here $L^2(E)$ is the Hilbert space of $L^2$-sections of E and $H^d(E)$ is the d-th order Sobolev space of sections of E.

Proposition A.9

The map ${{\,\mathrm{Op^d}\,}}(E,E') \rightarrow {\mathcal {C}}(L^2(E),L^2(E'))$ taking $(A,{\mathcal {D}})$ to $\left. A \right| _{{\mathcal {D}}}$ is continuous.

Proof

Take $W=H^d(E)$, $H=L^2(E)$, $H'=L^2(E')$, and let j be the natural embedding $W \hookrightarrow H$. By Proposition A.7, the map ${{\,\mathrm{Op^d}\,}}(E,E') \subset {\mathcal {B}}(W,H')\times {{\,\mathrm{Gr}\,}}(W) \rightarrow {\mathcal {C}}(W,H')$ is continuous. By definition of ${{\,\mathrm{Op^d}\,}}(E,E')$, the image of this map is contained in ${\mathcal {C}}_j(W,H')$. By Proposition A.8, the map $j_*:{\mathcal {C}}_j(W,H') \rightarrow {\mathcal {C}}(H,H')$ is continuous. Combining all these we obtain the continuity of the map ${{\,\mathrm{Op^d}\,}}(E,E') \rightarrow {\mathcal {C}}(H,H')$. $\square$

This general result can be applied to differential or pseudo-differential operators A of order d with domains ${\mathcal {D}}$ given by boundary conditions. We show below how Proposition A.9 can be applied to boundary value problems for first order differential operators, in particular local boundary value problems. We omit discussion of higher order operators because boundary conditions are slightly more complicated in that case; however, Proposition A.9 works for higher order operators as well.

Boundary Value Problems for First Order Operators Suppose now that M is a compact manifold. Denote by $E_{\partial }$ the restriction of E to the boundary $\partial M$.

Let $A\in {\mathcal {B}}(H^1(E), L^2(E'))$. In particular, A can be a first order differential operator with continuous coefficients. For a closed subspace ${\mathcal {L}}$ of $H^{1/2}(E_{\partial })$ we denote by $A_{{\mathcal {L}}}$ the operator A with the domain

$$\begin{aligned} {{\,\mathrm{dom}\,}}\left( A_{{\mathcal {L}}}\right) = \left\{ u\in H^1(E):\tau (u) \in {\mathcal {L}}\right\} , \end{aligned}$$

where $\tau :H^1(E) \rightarrow H^{1/2}(E_{\partial })$ is the trace map extending by continuity the restriction map $C^{\infty }(E) \rightarrow C^{\infty }(E_{\partial })$, $u\mapsto \left. u \right| _{\partial M}$.

Let $\widetilde{Op}(E,E')$ denotes the subspace of ${\mathcal {B}}(H^1(E), L^2(E')) \times {{\,\mathrm{Gr}\,}}(H^{1/2}(E_{\partial }))$ consisting of pairs $(A,{\mathcal {L}})$ such that the operator $A_{{\mathcal {L}}}$ is closed.

Proposition A.10

The map

$$\begin{aligned} \widetilde{Op}(E,E')\rightarrow {\mathcal {C}}(L^2(E),L^2(E')), \quad (A,{\mathcal {L}}) \mapsto A_{{\mathcal {L}}} \end{aligned}$$

is continuous.

Proof

The inverse image $\tau ^{-1}({\mathcal {L}})$ is a closed subspace of $H^1(E)$. Since $\tau$ is bounded and surjective, the map

$$\begin{aligned} \tau ^*:{{\,\mathrm{Gr}\,}}(H^{1/2}(E_{\partial })) \rightarrow {{\,\mathrm{Gr}\,}}(H^1(E)), \quad {\mathcal {L}}\mapsto \tau ^{-1}({\mathcal {L}}), \end{aligned}$$

is continuous. Hence the map $\widetilde{Op}(E,E')\rightarrow {{\,\mathrm{Op^1}\,}}(E,E')$ taking $(A,{\mathcal {L}})$ to $(A,\tau ^{-1}({\mathcal {L}}))$ is also continuous. It remains to apply Proposition A.9. $\square$

Local Boundary Value Problems for First Order Operators Denote by $\widetilde{{{\,\mathrm{Ell}\,}}}(E,E')$ the set of all pairs (A, L) such that

• A is a first order elliptic differential operator with smooth coefficients acting from sections of E to sections of $E'$, and

• L is a smooth subbundle of $E_{\partial }$ satisfying Shapiro–Lopatinskii condition (3.3).

Equip $\widetilde{{{\,\mathrm{Ell}\,}}}(E,E')$ with the $C^0$-topology on coefficients of operators and the $C^1$-topology on boundary conditions, that is, the topology induced by the inclusion

$$\begin{aligned} \widetilde{{{\,\mathrm{Ell}\,}}}(E,E') \hookrightarrow {\mathcal {B}}(H^1(E),L^2(E')) \times C^1({{\,\mathrm{Gr}\,}}(E_{\partial })). \end{aligned}$$

Here ${{\,\mathrm{Gr}\,}}(E_{\partial })$ denotes the smooth vector bundle over $\partial M$ whose fiber over $x\in \partial M$ is the Grassmanian ${{\,\mathrm{Gr}\,}}(E_x)$, and sections of ${{\,\mathrm{Gr}\,}}(E_{\partial })$ are identified with subbundles of $E_{\partial }$.

The Sobolev space $H^{1/2}(L)$ can be naturally identified with the closed subspace of $H^{1/2}(E_{\partial })$ via the map $H^{1/2}(L)\ni u \mapsto u\oplus 0 \in H^{1/2}(L)\oplus H^{1/2}(L^{\bot }) = H^{1/2}(E_{\partial })$. This allows us to associate with a pair $(A,L)\in \widetilde{{{\,\mathrm{Ell}\,}}}(E,E')$ the unbounded operator $A_L$ acting as A on the domain

$$\begin{aligned} {{\,\mathrm{dom}\,}}\left( A_L\right) = \left\{ u\in H^1(E):\tau (u) \in H^{1/2}(L)\right\} . \end{aligned}$$

By the classical theory of elliptic operators, $A_L$ is closed for every $(A,L)\in \widetilde{{{\,\mathrm{Ell}\,}}}(E,E')$. See, for example, Proposition 3.2, where it is proven for self-adjoint operators. Closedness of a non-self-adjoint $A_L$ can be proven along the same lines, or can be obtained directly from Proposition 3.2 by replacing a pair (A, L) with the pair $(A',L')\in \widetilde{{{\,\mathrm{Ell}\,}}}(E\oplus E')$, where $A'=\left( {\begin{matrix} 0 &{} A^t \\ A &{} 0\end{matrix}} \right)$ and $L' = L\oplus (\sigma _A(n)L)^{\bot } \subset E_{\partial }\oplus E_{\partial }'$.

Proposition A.11

The natural inclusion $\widetilde{{{\,\mathrm{Ell}\,}}}(E,E') \hookrightarrow {\mathcal {C}}\left( L^2(E),L^2(E')\right)$, $(A,L) \mapsto A_L$ is continuous.

Proof

It is an immediate corollary of the following lemma applied to $N=\partial M$ and $F=E_{\partial }$ and of Proposition A.10. $\square$

Lemma A.12

Let F be a smooth Hermitian vector bundle over a smooth closed Riemannian manifold N. Then the map

$$\begin{aligned} C^{\infty ,1}({{\,\mathrm{Gr}\,}}(F)) \rightarrow {{\,\mathrm{Gr}\,}}(H^{1/2}(F)), \end{aligned}$$

(A.5)

taking a smooth subbundle L of F to $H^{1/2}(L) \subset H^{1/2}(F)$, is continuous. Here $C^{\infty ,1}({{\,\mathrm{Gr}\,}}(F))$ denotes the space of smooth sections of ${{\,\mathrm{Gr}\,}}(F)$ with the $C^{1}$-topology, that is, the topology induced by the embedding $C^{\infty }({{\,\mathrm{Gr}\,}}(F)) \hookrightarrow C^{1}({{\,\mathrm{Gr}\,}}(F))$.

Proof

Operator of multiplication by a $C^1$-function $N\rightarrow {\mathbb {C}}$ is a bounded operator on $H^s(N)$ for every $s\in [0,1]$. In particular, it is bounded as an operator acting on $H^{1/2}(N)$, and the correspondent inclusion $C^1(N) \hookrightarrow {\mathcal {B}}\left( H^{1/2}(N)\right)$ is continuous. Passing to bundles, we obtain the natural continuous inclusion

$$\begin{aligned} C^1({{\,\mathrm{End}\,}}(F)) \hookrightarrow {\mathcal {B}}\left( H^{1/2}(F)\right) . \end{aligned}$$

(A.6)

The smooth map $P:{{\,\mathrm{Gr}\,}}({\mathbb {C}}^{n})\rightarrow {{\,\mathrm{End}\,}}({\mathbb {C}}^{n})$, $V\mapsto P_V$, induces the continuous map

$$\begin{aligned} P_* :C^{1}({{\,\mathrm{Gr}\,}}(F)) \hookrightarrow C^{1}({{\,\mathrm{End}\,}}(F)), \end{aligned}$$

which carries a subbundle L of F to the orthogonal projection $P_*L$ of F onto L. Composing it with the continuous inclusion (A.6), we obtain the continuous map

$$\begin{aligned} Q:C^{1}({{\,\mathrm{Gr}\,}}(F)) \hookrightarrow {\mathcal {B}}\left( H^{1/2}(F)\right) . \end{aligned}$$

For each smooth subbundle L of F the bounded operator Q(L) is an idempotent with the image $H^{1/2}(L)$. By Proposition A.3(1), the map

$$\begin{aligned} {{\,\mathrm{Im}\,}}:{{\,\mathrm{Proj}\,}}\left( H^{1/2}(F)\right) \rightarrow {{\,\mathrm{Gr}\,}}\left( H^{1/2}(F)\right) , \end{aligned}$$

is continuous. Composing it with Q, we obtain continuity of (A.5). This completes the proof of the lemma. $\square$