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

Spectral density and Sobolev inequalities for pure and mixed states