Interplay between relativistic and thermal plasma in relaxed galaxy clusters

Galaxy clusters are complex ecosystems, where galaxies with powerful active galactic nuclei, thermal plasma, magnetic fields, and cosmic rays can interact and flourish in spectacular astrophysical phenomena. Decades of studies have investigated these systems, and as the boundaries of our knowledge were pushed forward, new questions emerged awaiting to be answered. The main focus of this Thesis is the interplay between the relativistic and thermal plasma of the intra-cluster medium. The study of the intra-cluster medium (ICM) allows to understand the way energy is injected from large scale dynamics and dissipated in different channels. These channels include viscous heating and the generation of non-thermal components that, eventually, generate diffuse radio emission. Furthermore, studying the ICM provides complementary insights into the physics of cluster galaxy evolution. We mainly focus on the case of relaxed galaxy clusters, where the connection between the central, massive, radio-loud galaxies and the diffuse radio emission is still unclear. In this context, we investigate also the interactions between the ICM and the cluster galaxies. The cornerstone of our works is a multi-wavelength analysis based on radio and X-ray observations, which features brand-new Low-Frequency Array (LOFAR) observations and new approaches to the study of the spatial correlation between these emissions.