On two ways of defining eta-invariants of elliptic operators

The eta-invariants of elliptic operators were originally defined by Atiyah, Patodi and Singer as spectral invariants. They have applications in index theory in domains with conic points and cylindrical ends, as well as geometry, topology, analysis, number theory, and mathematical physics. Later, Melrose defined eta-invariants for families of operators elliptic with parameter in the sense of Agranovich-Vishik. The talk will describe the relation between these two definitions of eta-invariants. Namely, for a differential operator $A$ of odd order $m$ on a smooth compact manifold without edge, the equality of its Atiyi—Patodi–Zinger eta-invariant and the Melrose eta-invariant of the family of operators with parameter $p^{m}-iA$ will be established. A similar equality is also true for some boundary value problems on manifolds with edge.
The results were obtained in joint work with K.N. Zhuikov and published in: K. N. Zhuikov, A. Yu. Savin, “On two methods of determining \eta-invariants of elliptic boundary-value problems,” Sovrem. Mat. Fundam. Napravl., 2024, vol. 70, No. 3, 403–416.