On two methods of determining $\eta$-invariants of elliptic boundary-value problems

For a class of boundary-value problems with a parameter that are elliptic in the sense of Agranovich–Vishik, we establish the equality of the $\eta$-invariant defined in terms of the Melrose regularization and the spectral $\eta$-invariant of the Atiyah–Patodi–Singer type defined using the analytic continuation of the spectral $\eta$-function of the operator.