This paper begins by studying some spectral properties of the transfer operators of sampled-data systems described by applying the lifting technique. Through a "nonasymptotic" characterization of the transfer operator, its spectrum is determined in terms of finite-dimensional eigenvalue problems. Then, it is shown that a close connection with such eigenvalue problems and the exponential stability condition can be exploited to study the robust internal ( exponential) stability problem of sampled-data systems. Since the transfer operator is relevant to input-output characteristics, the relationship between input-output stability and internal stability is also discussed in the context of sampled-data systems.