Millimeter wave (mmWave) band mobile communications can be a solution to the continuously increasing traffic demand in modern wireless systems. Even though mmWave bands are scarcely occupied, the design of a prospect transceiver should guarantee the efficient coexistence with the incumbent services in these bands. To that end, in this paper, multi-user underlay cognitive transceiver designs are proposed that enable the mmWave spectrum access while controlling the interference to the incumbent users. MmWave systems usually require large-scale antenna arrays to achieve satisfactory performance and thus, it is difficult to support fully digital transceiver designs due to high demands in hardware complexity and power consumption. Thus, in order to develop efficient solutions, the proposed approaches are based on a hybrid analog-digital (A/D) architecture. Transceiver designs are developed for both the uplink and the downlink regime of a multi-user cellular system. Efficient algorithmic solutions are proposed for the design of the analog and the digital counterparts of the precoding and the decoding matrices of the latter systems based on the Alternating Direction Method of Multipliers (ADMM). Simulations show that the performance of the proposed hybrid A/D approaches is very close to the one of the corresponding fully digital transceivers for typical experimental setups.