  We prove some general Sobolev-type and related inequalities for positive
operators A of given ultracontractive spectral decay, without assuming e^{-tA}
is submarkovian. These inequalities hold on functions, or pure states, as
usual, but also on mixed states, or density operators in the quantum mechanical
sense. This provides universal bounds of Faber-Krahn type on domains, that
apply to their whole Dirichlet spectrum distribution, not only the first
eigenvalue. Another application is given to relate the Novikov-Shubin numbers
of coverings of finite simplicial complexes to the vanishing of the torsion of
some l^{p,2}-cohomology.
