  We present a Green's function formalism for an interacting Bose-Einstein
condensate (BEC) satisfying the two required conditions: (i) the
infrared-divergent longitudinal susceptibility with respect to the BEC order
parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the
vanishing off-diagonal self-energy in the low-energy and low-momentum limit.
These conditions cannot be described by the ordinary mean-field Bogoliubov
theory, the many-body $T$-matrix theory, as well as the random-phase
approximation with the vertex correction. In this paper, we show that these
required conditions can be satisfied, when we divide many-body corrections into
singular and non-singular parts, and separately treat them as different
self-energy corrections. The resulting Green's function may be viewed as an
extension of the Popov's hydrodynamic theory to the region at finite
temperatures. Our results would be useful in constructing a consistent theory
of BECs satisfying various required conditions, beyond the mean-field level.
