  We study many-body-localized (MBL) systems that are weakly coupled to
thermalizing environments, focusing on the spectral functions of local
operators. We argue that these spectral functions carry signatures of
localization even away from the limit of perfectly isolated systems. We find
that, in the limit of vanishing coupling to a bath, MBL systems come in two
varieties, with either discrete or continuous local spectra. Both varieties of
MBL systems exhibit a "soft gap" at zero frequency in the spatially-averaged
spectral functions of local operators, which serves as a diagnostic for
localization. We estimate the degree to which coupling to a bath broadens these
spectral features, and find that characteristics of incipient localization
survive as long as the system-bath coupling is much weaker than the
characteristic energy scales of the system. Since perfect isolation is
impossible, we expect the ideas discussed in this paper to be relevant for all
experiments on many-body localization.
