UDC 517.43+513.83

MATHEMATICS

A. S. DYNIN

ON THE INDEX OF FAMILIES OF PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH BOUNDARY

(Presented by Academician I. G. Petrovskii on 18 XI 1968)

In the present paper a homotopic method is indicated for computing the index of a locally trivial family of Fredholm structures of pseudodifferential operators on manifolds with boundary, studied in \((^4)\), and in particular of families of elliptic differential boundary-value problems. This clarifies the homotopic structure of the space of Fredholm structures of pseudodifferential operators (cf. the program of I. M. Gel'fand \((^7)\)).

Let us note that the index of a family of pseudodifferential operators on closed manifolds was computed in the work of B. Shih \((^3)\).

1. Let \(M \to X\) be a locally trivial fibration over a Hausdorff compactum \(X\), whose typical fiber is a smooth compact Riemannian manifold with boundary (smoothness everywhere means belonging to the class \(\mathscr C^\infty\)). Associated with this fibration are: the fibration \(M' \to X\) of the boundaries \(M_x'\) of the fibers \(M_x\) of the fibration \(M\) (here and below \(x \in X\)); the fibration \(T \to X\) of the tangent fibrations \(T_x\) of the manifolds with boundary \(M_x\); the fibration \(N \to X\) of the normal fibrations \(N_x\) of the boundaries \(M_x'\); the fibration \(T' \to X\) of the tangent fibrations \(T_x'\) of the boundaries \(M_x'\).

Let \(E \to X\) denote the fibration of Hermitian fibrations \(E_x\) over \(M_x\), associated with the fibration \(M \to X\). This fibration generates over \(X\) the Banach fibrations \(\mathscr H_0^{s,p}(M;E)\), \(\mathscr H^{s,p}(M;E)\), \(\mathscr B^{s,p}(M';E|M')\), \(-\infty < s < \infty\), \(1 < p < \infty\), whose fibers are the Banach spaces \(\mathscr H_0^{s,p}(M_x;E_x)\), \(\mathscr H^{s,p}(M_x;E_x)\), \(\mathscr B^{s,p}(M_x';E|M_x')\) \((^4)\). Here and below a vertical bar denotes restriction.

For \(ps > 1\) the restriction epimorphism is defined:
\[ \delta:\mathscr H^{s,p}(M;E)\to \mathscr B^{s-1/p,p}(M';E|M'). \]
For \(ps < p-1\) the dual cosection homomorphism is defined
\[ \delta':\mathscr B^{s+1-1/p,p}(M';E|M')\to \mathscr H_0^{s,p}(M;E). \]

Let \(M_1=M\setminus M'\). A homomorphism of fibrations
\[ P:\mathscr C_0^\infty(M_1;E_1)\to \mathscr C^\infty(M_1;E_2) \]
will be called pseudodifferential of class \(\mathscr P^r(M;E_1,E_2)\), if \(P_x\) are pseudodifferential operators of classes \(\mathscr P^r(M_x;E_{1x},E_{2x})\). These homomorphisms extend (see \((^4)\)) to homomorphisms
\[ P_{(0)}^{(s)}:\mathscr H_0^{s,p}(M;E_1)\to \mathscr H^{s-r,p}(M;E_2). \]

The family of symbols \(\sigma(P_x)\) of the operators \(P_x\) determines a homomorphism of fibrations \(\bar\sigma(P):E_1^\pi\to E_2^\pi\), where \(E_1^\pi,E_2^\pi\) are the lifts of the fibrations \(E_1,E_2\) to the space \(T_0\) of nonzero elements of the fibration \(T\). Similarly, the family of indicators \(\iota(P_x)\) of the pseudodifferential operators \(P_x\) \((^4)\) determines a homomorphism \(\iota(P)\) of the corresponding Banach fibrations over \(T_0'\), the space of nonzero elements from \(T'\).

For what follows fix a number \(p\) strictly between \(1\) and \(\infty\). Let \(P\in\mathscr P^r(M;E_1,E_2)\), \(P_1\in\mathscr P^{r_1}(M;F_1,E_2)\), \(P_2\in\mathscr P^{r_2}(M;E_1,F_2)\), where \(F_1,F_2\) are fibrations over \(X\) of Hermitian fibrations \(F_{1x},F_{2x}\) over \(M_x\). For any \(s\) such that \(s_1=s-r+r_1<1-1/p\), \(s_2=s-r_2>1/p\),

define an augmentation of the pseudodifferential homomorphism

\[ \mathfrak{P}^{s}_{(0)}:\mathcal{H}^{s,p}_{0}(M;E_{1})\times \mathcal{B}^{s_{1}+1-1/p,p}(M';F_{1}|M')\to \]

\[ \to \mathcal{H}^{s,p}(M;E_{2})\times \mathcal{B}^{s_{2}-1/p,p}(M';F_{2}|M') \]

of the form

\[ \mathfrak{P}^{s}_{(0)}(u,v)=\bigl(P^{s}_{(0)}u+P^{s_{1}}_{1(0)}\delta^{*}v,\ \delta P^{s_{2}}_{2(0)}u\bigr). \]

The augmentation \(P^{s}_{(0)}\) is called Fredholm if all the corresponding augmentations \(\mathfrak{P}^{s}_{(0)x}\) are Fredholm (4). For Fredholm augmentations the index is defined in the sense of (5).

1. A pseudodifferential homomorphism \(P\in\mathcal{P}^{r}(M;E_{1},E_{2})\) will be called simple if the restriction of the homomorphism \(\bar{\sigma}(P)\) to \(T|M'\) has the form
   \(\bar{\sigma}(P)(\xi)=(\nu_{\xi}+i|\tau_{\xi}|)^{r}\beta^{\pi}\), where \(\xi\in T|M'\), \((\nu_{\xi},\tau_{\xi})\) are the (scalar) normal and tangential components of the vector \(\xi\), and \(\beta^{\pi}\) is the lifting of some isomorphism \(\beta\in\operatorname{Iso}(E_{1}|M',E_{2}|M')\).

Let \(U\to X\) be a fibration of tubular neighborhoods \(U_{x}\) of the boundaries \(M'_{x}\) in \(M_{x}\). Let \(\varphi\) be a continuous function on \(U\) such that \(\varphi|U_{x}\in\mathcal{E}^{\infty}(U_{x})\), \(0\leq\varphi\leq1\), \(\varphi|M'=1\). Introduce pseudodifferential homomorphisms
\(P^{r}_{t}(E)\in\mathcal{P}^{r}(M;E,E)\) with real \(t\), for which

\[ \bar{\sigma}(P^{r}_{t}(E))(\xi)= \left[(\nu_{\xi}-i|\tau_{\xi}|)^{t}(\nu_{\xi}+i|\tau_{\xi}|)^{r-t}\varphi(m_{\xi}) +i|\xi|^{r}(1-\varphi(m_{\xi}))\right]1_{E}, \]

where \(\xi\in T_{0}\), \(m_{\xi}\) is the projection of \(\xi\) onto \(M\), \((\nu_{\xi},\tau_{\xi})\) are the (scalar) normal and tangential components of \(\xi\) for \(m_{\xi}\in U\), and \(1_{E}\) is the identity endomorphism in \(E\).

Define: the augmentation \([\mathfrak{P}^{r}_{r-s}(E)]^{s}_{(0)}\) by means of \(P_{1}=0,\ P_{2}=0\); the augmentation
\([\mathfrak{P}^{r}_{r-s-1}(E)]^{s}_{(0)}\) by means of
\(P_{1}=0,\ P_{2}=P^{r}_{r-s}(E)\); the augmentation
\([\mathfrak{P}^{r}_{r-s+1}(E)]^{s}_{(0)}\) by means of
\(P_{1}=P^{r}_{r-s}(E),\ P_{2}=0\).

Lemma 1. The augmentations
\([\mathfrak{P}^{r}_{r-s}(E)]^{s}_{(0)}\),
\([\mathfrak{P}^{r}_{r-s-1}(E)]^{s}_{(0)}\),
\(\mathfrak{P}^{r}_{r-s+1}(E)]^{s}_{(0)}\)
are Fredholm. Their indices are equal to \(0\in K(X)\).

Theorem 1. Let \(\mathfrak{P}^{s}_{(0)}\) be a Fredholm augmentation of a homomorphism
\(P\in\mathcal{P}^{r}(M;E_{1},E_{2})\). Then there exist such fibrations
\(\widetilde{E}=E_{1}\oplus E,\ \widetilde{F}_{1}=F_{1}\oplus F,\ \widetilde{F}_{2}=F_{2}\oplus F\), that the Fredholm augmentation

\[ \mathfrak{P}^{s}_{(0)}\oplus [\mathfrak{P}^{r}_{r-s}(E)]^{s}_{(0)} \oplus [\mathfrak{P}^{r}_{r-s+1}(\widetilde{F}_{2})] \oplus [\mathfrak{P}^{r}_{r-s-1}(\widetilde{F}_{1})]^{s}_{(0)} \]

is homotopic in the space of Fredholm augmentations to the Fredholm augmentation
\(\widetilde{\mathfrak{P}}^{s}_{(0)}\) of some simple pseudodifferential operator \(\widetilde{P}\).

The indicated homotopy is constructed effectively, on the basis of linearization (cf. (1)).

Remark. From the theorem there follows, in particular, the homotopic triviality of the restriction of the homomorphism \(\sigma(P)\) to every tangent sphere of the manifolds with boundary \(M_{x}\). In the case where \(P\) is a differential operator admitting an elliptic boundary-value problem, this was proved by M. Atiyah and R. Bott in (1). Theorem 1 may be regarded not only as a generalization, but also as a refinement of the result of M. Atiyah and R. Bott in view of the special character of the homotopy.

Theorem 2. The index of the augmentation \(\mathfrak{P}^{s}_{(0)}\) is equal to the index of the augmentation \(\widetilde{\mathfrak{P}}^{s}_{(0)}\).

1. M. Atiyah and I. Singer computed the index of a Fredholm pseudodifferential homomorphism for empty \(M'\) and one-point \(X\). One of their methods (6) directly generalizes to the computation of the index in the sense of (5) of a Fredholm simple pseudodifferential homomorphism. For this purpose consider an embedding \(i\) of the fibration \(M\) into a trivial finite-dimensional vector fibration \(V\) over \(X\). Following (6), introduce an almost complex structure in the fiberwise tangent fibration \(TW\) of a normal tubular neighborhood \(W\) of the space \(M\) in \(V\). Then the Thom isomorphism \(\varphi_{i}:K(T)\to K(TW)\) is defined (6). The embedding \(k:W\to V\) generates the imbedding

embedding \(k_{!}: K(TW)\to K(TV)\). On the other hand, if the zero section \(j:X\to V\) of the bundle \(V\) is regarded as an embedding in \(V\) of the identity family of zero-dimensional manifolds \(X\to X\), then there is a Thom isomorphism \(\varphi_j:K(X)\to K(TV)\).

If \(P\) is an elliptic simple homomorphism, then the homomorphism \(\bar\sigma(P)\) determines a certain element \([\bar\sigma(P)]\in K(T)\) \((^6)\).

Theorem 3. Let \(P\in P^r(M;E_1,E_2)\) be a Fredholm simple pseudodifferential homomorphism. Then the index of the homomorphism \(P_0^s\) is equal to
\[ \varphi_j^{-1}k_{!}\varphi_i[\bar\sigma(P)]. \]

Let \(F_1^{\pi'}, F_2^{\pi'}\) be the lifts of the bundles \(F_1|M'\), \(F_2|M'\) to \(T_0'\). If \(P_0^s\) is a Fredholm framing of a simple morphism \(P\) by means of \(P_1,P_2\), then the product of homomorphisms
\[ \delta_{l(0)}^s(P_2)\,t_{(0)}^s(P)^{-1}\,t_{(0)}^{s_1}(P_1)\delta^* \]
defines an isomorphism \(F_1^{\pi'}\to F_2^{\pi'}\), which may be regarded as the symbol of a certain Fredholm homomorphism
\[ P'\in P^{r_1+r_2-r}(M';F_1|M',F_2|M'). \]

Theorem 4. If \(\mathfrak P_{(0)}^s\) is a Fredholm framing of a simple homomorphism \(P\), then the index \(\operatorname{ind}\mathfrak P_{(0)}^s\) is equal to the sum of the indices
\[ \operatorname{ind} P_{(0)}^s+\operatorname{ind} P'{}_{(0)}^s . \]

Theorems 1–4 give a homotopy method for computing the index of any Fredholm framing.

1. We shall say that a homomorphism \(P\in \mathscr P^r(M;E_1,E_2)\) satisfies condition \((\mathscr D)\) if all operators \(P_x\) satisfy condition \((\mathscr D)\) from \((^4)\). In this case a \((\mathscr D)\)-homomorphism is defined
   \[ P^s:\mathscr H^{s,p}(M;E_1)\to \mathscr H^{s,p}(M;E_2). \]

Let \(P_1\in P^{r_1}(M;F_1,E_1)\), \(P_2\in \mathscr P^{r_2}(M;E_1,F_2)\). If \(P\) and \(P_2\) satisfy condition \((\mathscr D)\), then one can define a \((\mathscr D)\)-framing
\[ \begin{aligned} \mathfrak P^s:\mathscr H^{s,p}(M;E_1)\times \mathscr B^{s+1-1/p,p}(M';F_1|M')\to{}\\ \to \mathscr H^{s-r,p}(M;E_2)\times \mathscr B^{s_2-1/p,p}(M';F_2|M') \end{aligned} \]
of the form
\[ \mathfrak P^s(u,v)=\bigl(P^su+P_{1,0}^{s_1}\delta^*v,\ \delta P_2^{s_2}u\bigr), \]
under the conditions \(s_1=s-r+r_1<1-1/p,\ s_2=s-r_2>1/p\).

Lemma 2. If \(P\in \mathscr P^r(M;E_1,E_2)\) satisfies condition \((\mathscr D)\), then the product of homomorphisms \(PP_r^0(E_1)\) belongs to \(\mathscr P^r(M;E_1,E_2)\). (Here \(P_r^0(E_1)\) is the homomorphism from item 2.)

Therefore to a \((\mathscr D)\)-framing \(\mathfrak P^s\) one can assign the framing \((\mathfrak P_0)_{(0)}^s\) of the homomorphism
\[ P_0=PP_s^0(E_1) \]
by means of
\[ P_{10}=P_1,\qquad P_{20}=P_2P_s^0(E_1). \]
Since the homomorphism \([P_s^0(E_1)]_0^s\) is Fredholm and its index is equal to \(0\) (by Lemma 1), it follows that

Theorem 5. A \((\mathscr D)\)-framing \(\mathfrak P^s\) is Fredholm if and only if the framing \((\mathfrak P_0)_{(0)}^s\) is Fredholm. Moreover, the indices of the two framings coincide.

Remark. Theorem 5 indicates, in particular, a method for computing the index of an elliptic differential boundary-value problem, which gives a solution of I. M. Gel'fand’s problem \((^7)\) (cf. \((^1)\)).

1. The results set forth generalize to Fredholm framings of systems of A. Douglis—L. Nirenberg pseudodifferential homomorphisms.

Central Economics and Mathematics Institute
Academy of Sciences of the USSR

Received
1 IX 1968

References

\(^1\) M. Atiyah, R. Bott, in Differential Analysis (Proc. Bombay Symposium), Oxford, 1964.
\(^2\) M. I. Vishik, G. I. Eskin, Mat. Sb., 74, No. 3, 327 (1967).
\(^3\) W. Shi, Bull. Am. Math. Soc., 72, No. 6, 981 (1966).
\(^4\) A. S. Dynin, DAN, 186, No. 2 (1969).
\(^5\) L. Illusie, C. R., 260, 6499 (1965).
\(^6\) N. F. Atiyah, I. Singer, UMN, 23, No. 5, 99 (1968).
\(^7\) I. M. Gel'fand, UMN, 15, No. 3, 121 (1960).