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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.
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J28 Eta-invariants, Chern-Simons invariants