Elliptic operators in odd subspaces.

*(English. Russian original)*Zbl 0981.58018
Sb. Math. 191, No. 8, 1191-1213 (2000); translation from Mat. Sb. 191, No. 8, 89-112 (2000).

This paper is a continuation of the paper [Mat. Sb. 190, No. 8, 125-160 (1999; Zbl 0963.58008)], by the same authors, where the index problem is studied for operators acting on even subspaces of smooth sections. Now they consider the case of odd subspaces, which are defined as follows. Let \(E\) be a smooth vector bundle over a closed manifold \(M\), \(\pi:S^*M\to M\) the cosphere bundle, and \(\alpha:S^*M\to S^*M\) the involution given by \(\alpha(x,\xi)=(x,-\xi)\). Then a subbundle \(L\subset\pi^*E\) is called odd if \(\pi^*E=L\oplus\alpha^*L\); a projection of \(\pi^*E\) onto an odd subbundle \(L\) along \(\alpha^*L\) is called an odd projection; a pseudodifferential projection on \(C^\infty(M,E)\) of order zero is called odd if its principal symbol \(\sigma(P)\) is an odd projection of \(\pi^*E\); and a subspace \(\widehat{L}\subset C^\infty(M,E)\) is said to be odd if it is the image of some odd pseudodifferential projection \(P\) of order zero. In this case, the subbundle \(L={\text{Im}} \sigma(P)\) is called the principal symbol of \(\widehat{L}\). For instance, if \(A\) is an elliptic self-adjoint operator on \(M\) whose principal symbol satisfies \(\alpha^*\sigma(A)=-\sigma(A)\), then the subspace \(\widehat{L}_+(A)\) spanned by the eigenvectors of \(A\) corresponding to non-negative eigenvalues is odd.

The authors give the following stable classification of odd subspaces when the dimension of \(M\) is even. Two odd subspaces are said to be equivalent when they correspond by an invertible pseudodifferential operator whose principal symbol is even. The equivalence classes form an abelian semigroup \(\widehat({\text{Odd}}/\sim\), and the corresponding Grothendieck group is denoted by \(K{\widehat{\text{Odd}}/\sim)}\). On the other hand, two vector bundles \(E\) and \(F\) over \(M\) are said to be equivalent if there is an even elliptic symbol \(\sigma:\pi^*E\to\pi^*F\) with trivial (topological) index. This defines a quotient of the usual semigroup \(\text{Vect}(M)\) of vector bundles over \(M\), and the corresponding Grothendieck group is denoted by \(K(\text{Vect}(M)/\sim)\); indeed, this is a quotient of the usual \(K\)-theory group \(K(M)\). The ‘forgetful’ map \(j:K(\widehat{\text{Odd}}/\sim)\to K(\text{Vect}(M)/\sim)\) relates these groups. Another canonical map \(i:{\mathbb Z}\to K(\widehat{\text{Odd}}/\sim)\) is defined by \(i(k)=[\widehat{L}+k]-[\widehat{L}]\), where \(\widehat{L}\) is any odd subspace and \(\widehat{L}+k\) is the sum of \(\widehat{L}\) and any subspace of dimension \(k\) in its complement. Then a strong theorem of the paper states that the sequence \[ 0\to {\mathbb Z}\to K(\widehat{\text{Odd}}/\sim) \to K(\text{Vect}(M)/\sim)\to 0 \] defined by \(i\) and \(j\) becomes exact and split when tensoring with the abelian group \({\mathbb Z}[1/2]\) of dyadic rationals, and certain canonical splitting induces a dimension functional \(d:\widehat{\text{Odd}}(M)\to \mathbb Z[1/2]\).

To state the index formula of the paper, consider \(C^\infty\) vector bundles \(E\), \(F\) over a closed manifold \(M\) of even dimension, let \(\widehat{L}_1\subset C^\infty(M,E)\) and \(\widehat{L}_2\subset C^\infty(M,F)\) be odd subspaces with symbols \(L_1,L_2\), and let \(D:C^\infty(M,E)\to C^\infty(M,F)\) be a pseudodifferential operator satisfying \(D(\widehat{L}_1)\subset\widehat{L}_2\). Some typical definitions and results were generalized to the restriction \(D:\widehat{L}_1\to\widehat{L}_2\) by B. Yu. Sternin, V. E. Shatalov and B.-W. Schulze [Mat. Sb. 189, No. 10, 145-160 (1998; Zbl 0926.35033)]. For instance, its principal symbol is a homomorphism \(\sigma(D):L_1\to L_2\), which is an isomorphism just when \(D:\widehat{L}_1\to\widehat{L}_2\) is a Fredholm operator. In this case, the authors show the following index formula: \[ {\text{ind}}(D:\widehat{L}_1\to\widehat{L}_2)= {\text{ind}}(\widetilde{D})/2 +d(\widehat{L}_1)-d(\widehat{L}_2) , \] where \(\widetilde{D}:C^\infty(M,E)\to C^\infty(M,F)\) is any elliptic operator with principal symbol \(\sigma(\widetilde{D}):\pi^*E\to\pi^*F\) given by the formula \[ \sigma(D)\oplus\alpha^*\sigma(D): L_1\oplus\alpha^*L_1\to L_2\oplus\alpha^*L_2 . \] The celebrated index theorem of Atiyah-Singer describes \({\text{ind}}(\widetilde{D})\) in terms of \(\sigma(\widetilde{D})\), and thus in terms of \(\sigma(D):L_1\to L_2\). The dimension functional \(d\) is shown to be given by the Atiyah-Patodi-Singer \(\eta\)-invariant in the following case. If \(A\) is an elliptic selfadjoint operator on \(M\) of odd order whose complete symbol \[ a(x,\xi)\sim a_m(x,\xi)+ a_{m-1}(x,\xi)+\cdots \] satisfies \(a_{\alpha}(x,-\xi)=(-1)^{\alpha}a(x,\xi)\) for \(\xi\not=0\) and \(\alpha=m,m-1,m-2,\dots\), then \(d(\widehat{L}_+(A))=\eta(A)\). In particular, this proves a conjecture of P. B. Gilkey stating that the \(\eta\)-invariant is always a dyadic rational [Adv. Math. 58, 243-284 (1985; Zbl 0602.58041)] The authors also prove an index theorem for a general boundary-value problem in odd subspaces, in the sense of Sternin-Shatalov-Schulze [loc. cit.], where the dimension functional \(d\) is also used.

The authors give the following stable classification of odd subspaces when the dimension of \(M\) is even. Two odd subspaces are said to be equivalent when they correspond by an invertible pseudodifferential operator whose principal symbol is even. The equivalence classes form an abelian semigroup \(\widehat({\text{Odd}}/\sim\), and the corresponding Grothendieck group is denoted by \(K{\widehat{\text{Odd}}/\sim)}\). On the other hand, two vector bundles \(E\) and \(F\) over \(M\) are said to be equivalent if there is an even elliptic symbol \(\sigma:\pi^*E\to\pi^*F\) with trivial (topological) index. This defines a quotient of the usual semigroup \(\text{Vect}(M)\) of vector bundles over \(M\), and the corresponding Grothendieck group is denoted by \(K(\text{Vect}(M)/\sim)\); indeed, this is a quotient of the usual \(K\)-theory group \(K(M)\). The ‘forgetful’ map \(j:K(\widehat{\text{Odd}}/\sim)\to K(\text{Vect}(M)/\sim)\) relates these groups. Another canonical map \(i:{\mathbb Z}\to K(\widehat{\text{Odd}}/\sim)\) is defined by \(i(k)=[\widehat{L}+k]-[\widehat{L}]\), where \(\widehat{L}\) is any odd subspace and \(\widehat{L}+k\) is the sum of \(\widehat{L}\) and any subspace of dimension \(k\) in its complement. Then a strong theorem of the paper states that the sequence \[ 0\to {\mathbb Z}\to K(\widehat{\text{Odd}}/\sim) \to K(\text{Vect}(M)/\sim)\to 0 \] defined by \(i\) and \(j\) becomes exact and split when tensoring with the abelian group \({\mathbb Z}[1/2]\) of dyadic rationals, and certain canonical splitting induces a dimension functional \(d:\widehat{\text{Odd}}(M)\to \mathbb Z[1/2]\).

To state the index formula of the paper, consider \(C^\infty\) vector bundles \(E\), \(F\) over a closed manifold \(M\) of even dimension, let \(\widehat{L}_1\subset C^\infty(M,E)\) and \(\widehat{L}_2\subset C^\infty(M,F)\) be odd subspaces with symbols \(L_1,L_2\), and let \(D:C^\infty(M,E)\to C^\infty(M,F)\) be a pseudodifferential operator satisfying \(D(\widehat{L}_1)\subset\widehat{L}_2\). Some typical definitions and results were generalized to the restriction \(D:\widehat{L}_1\to\widehat{L}_2\) by B. Yu. Sternin, V. E. Shatalov and B.-W. Schulze [Mat. Sb. 189, No. 10, 145-160 (1998; Zbl 0926.35033)]. For instance, its principal symbol is a homomorphism \(\sigma(D):L_1\to L_2\), which is an isomorphism just when \(D:\widehat{L}_1\to\widehat{L}_2\) is a Fredholm operator. In this case, the authors show the following index formula: \[ {\text{ind}}(D:\widehat{L}_1\to\widehat{L}_2)= {\text{ind}}(\widetilde{D})/2 +d(\widehat{L}_1)-d(\widehat{L}_2) , \] where \(\widetilde{D}:C^\infty(M,E)\to C^\infty(M,F)\) is any elliptic operator with principal symbol \(\sigma(\widetilde{D}):\pi^*E\to\pi^*F\) given by the formula \[ \sigma(D)\oplus\alpha^*\sigma(D): L_1\oplus\alpha^*L_1\to L_2\oplus\alpha^*L_2 . \] The celebrated index theorem of Atiyah-Singer describes \({\text{ind}}(\widetilde{D})\) in terms of \(\sigma(\widetilde{D})\), and thus in terms of \(\sigma(D):L_1\to L_2\). The dimension functional \(d\) is shown to be given by the Atiyah-Patodi-Singer \(\eta\)-invariant in the following case. If \(A\) is an elliptic selfadjoint operator on \(M\) of odd order whose complete symbol \[ a(x,\xi)\sim a_m(x,\xi)+ a_{m-1}(x,\xi)+\cdots \] satisfies \(a_{\alpha}(x,-\xi)=(-1)^{\alpha}a(x,\xi)\) for \(\xi\not=0\) and \(\alpha=m,m-1,m-2,\dots\), then \(d(\widehat{L}_+(A))=\eta(A)\). In particular, this proves a conjecture of P. B. Gilkey stating that the \(\eta\)-invariant is always a dyadic rational [Adv. Math. 58, 243-284 (1985; Zbl 0602.58041)] The authors also prove an index theorem for a general boundary-value problem in odd subspaces, in the sense of Sternin-Shatalov-Schulze [loc. cit.], where the dimension functional \(d\) is also used.

##### MSC:

58J05 | Elliptic equations on manifolds, general theory |

58J20 | Index theory and related fixed-point theorems on manifolds |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |