# zbMATH — the first resource for mathematics

Subspaces defined by pseudodifferential projections, and some of their applications. (English. Russian original) Zbl 1055.58012
Dokl. Math. 61, No. 2, 235-238 (2000); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 371, No. 4, 448-451 (2000).
This nicely written short paper reviews the results of the authors from [Sb. Math. 191, No. 8, 1191–1213 (2000; Zbl 0981.58018)] and [Sb. Math. 190, No. 8, 1195–1228 (1999; Zbl 0963.58008)]. The subspaces from the title are images in $$C^\infty(M,E)$$ of pseudodifferential projections of order $$0$$ over a closed manifold $$M$$ which are “admissible”, in the sense that the parity of their symbol with respect to the antipodal map is opposite to the parity of $$\dim M$$. There exists a map $$d$$ from the semigroup of homotopy classes of such projections into $$\mathbb Z[1/2]$$ with the following property: If $$A$$ is an elliptic pseudodifferential operator with “parity” opposite to $$\dim M$$, and $$L_+(A)$$ is the image of the spectral projection coming from the nonnegative eigenvalues of $$A$$, then $$d(L_+(A))$$ coincides with the eta invariant of $$A$$. This implies that $$\eta(A)$$ belongs to $$\mathbb Z[1/2]$$, which answers positively a conjecture of P. Gilkey [Adv. Math. 58, 243–284 (1985; Zbl 0602.58041)]. The functional $$d$$ appears also in an index formula for elliptic operators acting on subspaces as above, as well as for the index of certain boundary-value problems.
##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 58J28 Eta-invariants, Chern-Simons invariants