$L^2$ Frölicher inequalities

We prove a Frölicher inequality between $L^2$ Betti and $L^2$ Hodge numbers on normal coverings of compact complex manifolds. This is achieved by building an injection using suitable spectral projectors associated to the self adjoint operators $(D_h{)}^2:=(\bar{\mathrm{\partial }}+{\bar{\mathrm{\partial }}}^{\ast }+h\mathrm{\partial }+h{\mathrm{\partial }}^{\ast }{)}^2$ for $h\in [0,1]$. As a by-product, we find a new proof of the classical Frölicher inequality on compact complex manifolds which does not rely at all on spectral sequences.