M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces

The paper studies properties of acoustic operators in bounded Lipschitz domains $\mathrm{\Omega }$ with m-dissipative generalized impedance boundary conditions. We prove that such acoustic operators have a compact resolvent if and only if the impedance operator from the trace space $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ to the other trace space $H^{-1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations.