Toeplitz operators (also called localization operators) are a generalization
of the well-known anti-Wick pseudodifferential operators studied by Berezin and
Shubin. When a Toeplitz operator is positive semi-definite and has trace one we
call it a density Toeplitz operator. Such operators represent physical states
in quantum mechanics. In the present paper we study several aspects of Toeplitz
operators when their symbols belong to some well-known functional spaces (e.g.
the Feichtinger algebra) and discuss (tentatively) their separability
properties with an emphasis on the Gaussian case.