Etched multilayers obtained by forming a laminar grating pattern within interferential multilayer mirrors are used in the soft x-ray range to improve the spectral resolution of wavelength dispersive spectrometers equipped with periodic multilayers. We describe the fabrication process of such an etched multilayer dispersive element, its characterization through reflectivity measurement and simulations, and its implementation in a high-resolution Johann-type spectrometer. The specially designed patterning of a Mo/B4C multilayer is found fruitful in the range of the C K emission as the diffraction pattern narrows by a factor 4 with respect to the non-etched structure. This dispersive element with an improved spectral resolution was successfully implemented for electronic structure study with an improved spectral resolution by x-ray emission spectroscopy. As first results we present the distinction between the chemical states of carbon atoms in various compounds, such as graphite, SiC and B4C, by the different shape of their C K emission band.