  We present a theory for the linear dynamics of a weakly interacting Bose gas
confined inside a harmonic trap at finite temperature. The theory treats the
motions of the condensate and of the non-condensate on an equal footing within
a generalized random-phase approximation, which ({\it i}) extends the
second-order Beliaev-Popov approach by allowing for the dynamical coupling
between fluctuations in the thermal cloud, and ({\it ii}) reduces to an earlier
random-phase scheme when the anomalous density fluctuations are omitted.
Numerical calculations of the low-lying spectra in the case of isotropic
confinement show that the present theory obeys with high accuracy the
generalized Kohn theorem for the dipolar excitations and demonstrate that
combined normal and anomalous density fluctuations play an important role in
the monopolar excitations of the condensate. Mean-field theory is instead found
to yield accurate results for the quadrupolar modes of the condensate. Although
the restriction to spherical confinement prevents quantitative comparisons with
measured spectra, it appears that the non-mean field effects that we examine
may be relevant to explain the features exhibited by the breathing mode as a
function of temperature in the experiments carried out at JILA on a gas of
$^{87}$Rb atoms.
